STOCHASTIC POPULATION DYNAMICS · the dynamics when R= 1:1 for di erent habitat sizes. These...

STOCHASTIC POPULATIONDYNAMICS

Sebastian Schreiber

Contents

Chapter 1. Introduction 41. What and Who? 42. Tales from a simple model 5Exercises 12

Part I. Environmental Stochasticity 13

Chapter 2. Random matrix models 141. Random matrix products 142. Sensitivity of r 213. General random products 274. Random affine models 27Exercises 33

Chapter 3. Nonlinear single species models 351. Unstructured models 352. Structured models 423. Ergodicity and stationary distributions 454. Exercises 48

Chapter 4. Multispecies models 491. The lottery model 492. Two species competition 523. Many species 58

Chapter 5. Approximation methods 651. The scalar case 652. The Multivariate Case 68

Index 70

Bibliography 71

3

CHAPTER 1

Introduction

1. What and Who?

1.1. What are these notes about? Since the pioneering work of Lotka [1925] and Volterra [1926]on competitive and predator–prey interactions, Thompson [1924], Nicholson and Bailey [1935] on host–parasite interactions, and Kermack and McKendrick [1927] on disease outbreaks, nonlinear differenceand differential equations have been used to understand population dynamics. However (with apologiesto John Gay), lest biologists believe ones theory untrue, models need to keep probability in view.All populations experience demographic and environmental stochasticity. Demographic stochasticitystems from populations and communities consisting of a finite number of interacting individuals whosefates aren’t perfectly correlated. Some individuals die while others survive as if coin flips determinedtheir fates. Environmental stochasticity stems from fluctuations in environmental conditions such astemperature and precipitation, or large stochastic shocks such as floods, fires, droughts, and hurricanes.As these environmental conditions influence survival, growth, and reproduction, these environmentalfluctuations result in demographic fluctuations that influence population dynamics.

These stochastic forces can impact population dynamics in a diversity of ways. Most fundamentally,for populations without immigration, demographic stochasticity in a finite world implies the populationwill go extinct in finite time, contrary to predictions from most deterministic models. This simpleobservation leads to a host of questions: What is the likelihood of extinction next year versus 50 yearsfrom now? Under what conditions is extinction going to be more rapid? For interacting populations,which species or genotypes are likely to be lost first? Despite extinction being inevitable, populationsmay persist for exceptionally long periods, in which case the nature of these transient dynamics are ofinterest. One may ask: To what extent are the transient dynamics driven by underlying deterministicfeedbacks? If so, how frequently, do stochastic forces causes the population state to drift betweendifferent attractors of the underlying deterministic dynamics?

When populations are sufficiently large to diminish the importance of demographic stochasticity,the interplay between deterministic feedbacks and environmental fluctuations can play a crucial role indetermining species richness and genetic diversity [Gillespie, 1973, Chesson and Warner, 1981, Turelli,1981, Chesson, 1994, Ellner and Sasaki, 1996, Abrams et al., 1998, Bjornstad and Grenfell, 2001,Kuang and Chesson, 2008, 2009]. For example, competition for limited resources [Gause, 1934] orsharing common predators [Holt, 1977] may result in species or genotypes being displaced. However,random forcing of these systems can reverse these trends and, thereby, enhance diversity [Gillespie andGuess, 1978, Chesson and Warner, 1981, Abrams et al., 1998]. Conversely, differential predation canmediate coexistence between competitors [Paine, 1966, Holt et al., 1994, Chesson and Kuang, 2008],yet environmental fluctuations can disrupt this coexistence mechanism. If species do coexist, one mightask: How do species interactions and environmental stochasticity determine the mean abundance of aspecies? Which species exhibit the greatest demographic fluctuations? Which species densities covarymost closely?

The goal of these lectures is to describe a theory of stochastic population dynamics in a mathemat-ically rigorous, but practical manner. The theory consists of a mixture of analytical characterizationsof short-term and long-term behavior of the stochastic population dynamics (e.g. existence of a sta-tionary distribution supporting all popuations), analytical and numerical methods for verifying these

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2. TALES FROM A SIMPLE MODEL 5

conditions (e.g. small noise approximations), numerical methods for exploring finer details of the sto-chastic dynamics (e.g. extinction likelihoods), and anaytical approximations for these finer details (e.g.diffusion type approximations).

1.2. Who is this written for and how much do you need to know? There are two targetaudiences for these lecture notes: theoretical population biologists and applied mathematicans. Toreach both audiences, the main presentation provides the minimal technical details to get the mainpunchlines. Proofs are given in the main text to the extent they provide useful insights into the mainideas. Additional technical details and background will be provided in the Appendices.

All ideas are motivated and illustrated by applications to real models from the biological literature.The impact of the theory on the models is developed in detail to highlight the biological implicationsand the many challenges that these models still contain.

The lectures assumes all readers are familiar with some basic concepts from dynamical systemsand probability theory. Specifically, from dynamical systems theory, readers should be familiar withdifference and differential equations, linearization and stability of equilibria, and attractors. From prob-ability, familiarity with expectations, probability distributions, the law of large numbers, the centrallimit theorem, and basic finite state Markov chain theory will be useful. For the more advanced readersthat want to get into the mathematical details, familarity with real analysis, measure theory, Markovchains on general state spaces, large deviation theory, and martingales will be useful.

2. Tales from a simple model

To introduce some of the themes for these lectures, we consider two stochastic variants of theBeverton-Holt model for a single species. In this model nt denotes the population density at time t.At low densities, individuals produce on average R offspring to replace themselves. Density-dependentfeedbacks result in a reduction in the fitness of the individuals. For the Beverton-Holt, this reductionis of the form R/(1 +ant) where a > 0 is a measure of the strength of intra-specific competition. Thus,the population dynamics are given by

(1.1) nt+1 = ntR

1 + ant.

The dynamics of this classical are well-known. If R > 1 (i.e. at low densities individuals more thanreplace themselves) and n0 > 0, then

limt→∞

nt =R− 1

awhere (R − 1)/a is the unique positive equilibrium of the model. Alternatively, if R ≤ 1 and n0 ≥ 0,then limt→∞ nt = 0. Notice that extinction in this latter case only occurs in the limit as time marchesto ∞. For all finite time, the population density is positive provide it initially was positive.

2.1. Demographic stochasticity. A fundamentally strange aspect of the classical Beverton-Holtmodel is nt can be any non-negative real number. We skated around this issue by calling nt a ”density”but even densities can not take on any value in a finite world. Namely, for given habitat size S, thepossible densities are 0, 1/S, 2/S, . . . . So what in the world does the model represent? Here we brieflyconsider this question.

Natural populations consist of a finite number, discrete individuals. To account for this featurein the model, lets consider an individual based model where a simple set of individual-based rulesdetermine the population dynamics. To this end, let Nt be the number (not density!!) of individualsat time t. These individuals live in some part of the world with a finite amount of space. Let S be thetotal amount (e.g. area or volume) of this space. Then Nt/S is the density of individuals. Consistentwith the deterministic model, lets assume that each individual produces on average R/(1 + aNt/S)number of offspring. But how is this average achieved? While there a countless number of answers to


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Figure 1. Multiple realizations of a branching process with R = 0.9 (left) and R = 1.1 (right).

this questions, lets keep the answer simple by assuming that each individual, independent of the othersproduces a Poisson number of offspring with mean R/(1 + aNt/S). During the time step between tand t+ 1, let Xi be the number of offspring that individual 1 ≤ i ≤ Nt produces to replace themselves.Then, the population size in the next time step is

Nt+1 = X1 + · · ·+XNt .

Equivalently, Nt+1 is a Poisson distributed random variable with mean NtR/(1 + aNt/S). What canone say about such a model?

Density-independence. Consider the very special case where there is no density-dependent feedbackin the population dynamics i.e. a = 0. Then our stochastic model is an example of a classical branchingprocess which we discuss later in greater detail (see XX.XXX). A basic result from this classical theoryimplies that if the mean number of offspring R is less than or equal to one, then Nt goes to 0 in finitetime with probability one (see left hand panel of Figure 1). Unlike the deterministic model, extinctionoccurs in finite time.

On the other hand, if the mean number of offspring R is strictly greater than one, then there isa positive probability that Nt goes to 0 in finite time and a positive complementary probability thatlimt∞Nt/R

t exists and is positive (see right hand panel of Figure 1). In particular, the population sizegrows to infinity at a geometric rate on this later event. Unlike the deterministic model, extinctionstill occurs despite individuals replacing themselves on average. However, the probability of extinctionoccurring decreases exponentially with population size.

Of course, exponential growth can not continue unabated. In the words of economist KennethBoulding,

Anyone who believes that exponential growth can go on forever in a finite world iseither a madman or an economist.

Density-dependence. To avoid being called a madman (or perhaps worse an economist), suppose thatthere is a density-dependent feedback in the model i.e a > 0. Figure 2 illustrates several simulations ofthe dynamics when R = 1.1 for different habitat sizes. These simulations illustrate that as the habitatsize S gets larger the stochastic dynamics of the densities Nt/S follow the deterministic dynamics moreand more closely. In fact, what one can show (see Section XX.XX) is that for any ε > 0 and T > 0and N0/S = n0, the stochastic dynamics Nt and the deterministic dynamics nt satisfy

P[|Nt/S − nt| ≤ ε] ≥ 1− ε for 0 ≤ t ≤ T


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Figure 2. Multiple realizations of the density-dependent branching process with S = 20,100, 1, 000, and 10, 000.Parameters: N0 = S/20, a = 0.1, R = 1.1.

provided that S is sufficiently large. In words, for any interval of time T , the stochastic dynamics arehighly likely to be close to the deterministic dynamics provided the habitat size is sufficiently large.This close correspondence, however, ultimately breaks up as the stochastic model goes to extinction infinite time with probability one i.e. P[Nt = 0 for some t] = 1.

While extinction is inevitable for Nt, the time to extinction tends to increase exponenentially withthe habitat size S and, consequently, can be an extremely long time for even moderate habitat sizes s(seeupper left hand panel of Figure 3). Therefore, it is natural to ask what can we say about this “meta-stable” behavior? It turns out there is a fair amount we can say. For example, the metastable behavioris characterized by so-called quasi-stationary distributions which describe the long-term behavior ofthe model conditioned on non-extinction. One way of thinking about the quasi-stationary distribution

is that there is a random variable, call N , taking values on the natural numbers such that

limt→∞

P[Nt = i|Nt 6= 0] = P[N = i]

Figure 2 plots quasi-stationary distributions for several habitat sizes. In these lectures, we will discusshow to compute these quasi-stationary distributions, to relate them to the dynamics of deterministicmodels when S is large, how to compute “intrinsic” extinction statistics associated with these quasi-stationary distributions, and how to approximate them when S is large but not too large. In particular,we will answer the following question raised by Peter Jaegers


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Figure 3. Intrinsic mean times to extinction and quasi-stationary distributions for thedensity-dependent branching process with S = 10, 100, and 500. Parameters: a = 0.1,R = 1.1.

“Any population allowing individual variation in reproduction, ultimately dies out–unless it grows beyond all limits, an impossibility in a bounded world. Deterministicpopulation mathematics on the contrary allows stable asymptotics. Are these arti-facts or do they tell us something interesting about quasi-stationary stages of real orstochastic populations?”

2.2. Environmental stochasticity. While demographic stochasticity is an “intrinsic” form ofstochasticity (i.e. generated within the system itself), environmental stochasticity is an “extrinsic”form of stochasticty (i.e. an “external forcing” of the system). If the habitat size S is sufficiently large(really infinite), then we can ignore demographic stochasticity and focus on the external forcing of thesystem. For example, in our Beverton-Holt model, the population may be experiencing environmentalfluctuations which influence the maximal reproductive output R of all individuals e.g. fluctuationsin precipitation for annual plants. We can model this environmental forcing with the following non-autonomous difference equation:

(1.2) nt+1 = ntRt+1

1 + ant

whereRt+1 represents how the environmental conditions from time t to time t+1 determine the fecundityof individuals at low population densities. Environmental fluctuations could lead to fluctuations in the


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Figure 4. Realizations of nt+1 = Rt+1nt with R1 = 2 and 1/5 with equal probability.The dashed line on the right is the expected values of the log density.

intraspecific competition coefficient a, but for simplicity, lets ignore this possibility for now. If the Rt

are sequence of random variables, then (1.2) is an example of a stochastic difference equation. For now,lets assume that R1, R2, . . . are independent and identically distributed.

2.2.1. Density-independent model. In the absence of density dependent feedbacks i.e. a = 0, ourstochastic difference equation (1.2) becomes

nt+1 = Rt+1nt

The solution of this difference equation is (check for yourself!)

nt = RtRt−1 . . . R1n0

What can we say about the long-term behavior of this equation? If we are interested in the expectedpopulation density E[nt] and n0 is given, then

E[nt] = E[RtRt−1 . . . R1n0]

= E[RtRt−1 . . . R1]n0 as n0 is a constant

= E[Rt]E[Rt−1] . . .E[R1]n0 by independence of the Rt

= E[R1]tn0 as Rt are identically distributed

Hence, the expected population density grows by a factor E[R1] every time step. This suggest that ifE[R1] > 1, then the population grows and persists, and otherwise it would decrease toward extinction.This turns out to be misleading. For example, suppose that R1 = 2 and 1/5 with equal probability.Then E[R1] = 1.1 and we expect the popualtion to grow. However, the left hand side of Figure 4 shows20 simulations of nt and in all of them, the population size is tending toward zero. Why?

To understand the discrepancy between the simulations and the expectation that we computed inthe previous example, we can make use of the Law of Large Numbers. Let xt = log nt. Then

xt = log(RtRt−1 . . . R1n0)

=t∑

s=1

logRs + log n0

If logR1 has finite mean r and finite variance, then the Law of Large Numbers implies

limt→∞

xtt

= E[logR1] = r with probability one.


Hence, if r < 0, then log nt → −∞ with probability one, and if r > 0, then log nt → ∞. How doesthis help with our specific example where Rt = 2 and 1/5 with equal probability? For this example,E[logR1] = 1

2log 2

5< 0. Hence, nt → 0 with probability one. Indeed the right hand side of Figure 4

illustrates this trend by plotting many realizations on a log scale. This difference stems from thefact that the geometric mean of R1, e

r, is smaller than the arithmetic mean E[R1]. Thus, we haverediscovered the main conclusion of a PNAS paper by Lewontin and Cohen [1969] who wrote

Even though the expectation of population size may grow infinitely large with time,the probability of extinction may approach unity, owing to the difference between thegeometric and arithmetic mean growth rates.

2.2.2. Density-dependent model. Again to avoid being called a madman or economist, lets addsome density-dependence back into the model i.e. allow a > 0. Some simulations for this model withlog-normally distributed values of Rt are shown in Figure 5. What might we be able to say aboutthis model? For populations to persist, it seems that it would necessary that they have a tendency toincrease at low densities. But at low densities the model is given approximately the density-independentmodel nt+1 = Rt+1nt which exhibits a tendency of population increase only if r = E[logR1] > 0. Forthe stochastic Beverton-Holt model (see XX.XX), the condition r > 0 implies that there is a positivestationary distribution. Namely, a positive random variable n such that if n0 is distributed like n, thennt is distributed like n for all time i.e. P[a ≤ nt ≤ b|n0 = n] = P[a ≤ n ≤ b] for all a < b.

Similar to the basic limit theorems for finite-state Markov chains, this stationary distribution ncharacterizes the long-term behavior of the model in two ways. First, given n0 > 0, nt converges to nin probability: for all intervals [a, b],

limt→∞

P[a ≤ nt ≤ b] = P[a ≤ n ≤ b].

As n is a positive random variable, this implies that nt is unlikely to be close to 0 for large t. Fur-thermore, this result also implies that the long-term statistical behavior of nt, from the ensembleperspective, is independent of n0 (provide it is positive.)

The second way n characterizes the long-term behavior is from the “typical trajectory” perspective.Namely, given any interval [a, b] and n0 > 0, the asymptotic fraction of time spent in the interval bya “typical” realization of the stochastic process equals the probability that n is the interval. Moreprecisely,

limt→∞

#{1 ≤ s ≤ t such that a ≤ ns ≤ b}t

= P[a ≤ n ≤ b]

Note the important implication for simulations. To estimate the stationary distribution it suffices (withprobability one) to run the model one time for a long time. Both of these forms of convergence areillustrated in Figure 5.

When the environmental fluctuations are sufficiently small, we can approximate the long-termbehavior by linearizing the system around the stable equilibrium of the deterministic system (seeSection XX.XX). These approximations yield another type of linear stochastic difference equation forwhich we can say a lot about their behavior. The approximation for the stationary distribution isshown as red curves in Figure 5.

Finally, what if r < 0? Then, limt→∞ nt = 0 with probability one. However, unlike the case ofdemographic stochasiticty, a density of zero is only approached in the infinite time horizon.

2.3. Where to hence? This introduction provided a quick overview about the types of resultsthat are the focus of these lectures. The remainder of these lectures will flush out the details in asufficiently general context to be applicable to models accounting for population structure (e.g. spatial,age, or stage structure) and interactions between multipe populations (e.g. predator-prey interactionsin ecological communities or competition amoung multiple genotypes). Furthermore, we will examinemodels which simultaneously account for demographic and environmental stochasticity.


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Figure 5. Runs of BH with environmental stochasticity. Histograms for first two rowscorrespond to a single run of the model. Final histogram corresponds to the distributionat time 5, 000 for 1, 000 runs of the model. Red curves show an analytically basedapproximation of the stationary distribution.Parameters: N0 = S/20, a = 0.1, R = 1.1.

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EXERCISES 12

Exercises

(1) For the demographic stochasticity version of the Beverton-Holt model, prove that for all ε > 0,n0 ≥ 0, and T ≥ 1, there exists S∗ such that

P[|nt −Nt/S| ≤ ε|N0 = n0S] ≥ 1− ε for 0 ≤ t ≤ T

whenever S > S∗. Hint: Use Chebshev’s inequality.(2) Recently, human activities have isolated the acorn woodpecker population in Water Canyon

from other acorn woodpecker populations. This isolation coupled with their recent declineto ten breeding pairs has caused an public outcry. People fear, extinction is just aroundthe corner for this beloved bird. The Nature Conservancy has hired you as a consultant toevaluate various intervention strategies. As a first step to assess these strategies, consider adensity-independent model of the form

nt+1 = (jt+1ft+1 + at+1)nt

where nt is the density of females in year t, jt+1 is the fraction of juveniles that survive to yeart + 1, at+1 is the fraction of adults that survive to year t + 1, and ft+1 is the average numberof daughters produced per female in year t + 1. To tackle this issue, the Nature Conservancyhas giveng you the following following data from Stacey and Taper [1992]

j<-c(0.56, 0.64, 0.3, 0.4, 0, 0.38, 0.18, 0.25, 0.44);

a<-c(0.53, 0.68, 0.71, 0.38, 0.54, 0.69, 0.66, 0.49, 0.61);

f<-c(3.38, 1.27, 2.77, 2.17, 0.05, 4.0, 2.37, 0.5, 1.6)/2;

(a) Determine the r value for this population. Should the population be increasing or de-creasing in the long-term?

(b) How much do you need to increase juvenile survivorship to ensure the population willhave a positive r?

(c) How much do you need to increase adult survivroship to ensure the population will havea positive r?

(d) If there is only enough money to support one conservation effort, discuss which one youwould support and why.

Part I

Environmental Stochasticity

CHAPTER 2

Random matrix models

Not only the adult, but the whole life cycle will be considered the organism. Thisis an ancient notion, for philosophers have often pointed out that an individual con-ventionally means an organism in a short instant of time...For example, if we referto a “dog” we usually picture in our minds an adult dog momentarily immobilizedin time as though by a photographic snapshot...[But] is the dog not a dog from themoment of fertilization of its egg, through embryonic and fetal development, throughbirth and puppyhood, through adolescence and sexual maturity, and finally throughsenescence? – John Tyler Bonner (1965)

1. Random matrix products

Consider a population consisting of individuals in k different states or stages (e.g. age, geographicallocations, behavioral type). Let nt = (n1,t, . . . , nk,t) be the vector of densities of individuals in the kstates. Each time step, surviving individuals for one state contribute to other states by transitioningto a different state (e.g. growing older, dispersing to another patch, switching behavioral states) or byproducing offspring. Let At+1 = (sij,t+1) be a k×k matrix that captures this contributions. Specifically,the entry Aij,t+1 in the i-th row and j-th column of At is the contribution of individuals of state i toindividuals in state j. With this notation, we get the population dynamics are given by

(2.1) nt+1 = ntAt+1

As Aij,t correspond to contributions from one state to another, we assume that these entries are non-negative (we discuss the more general case in Section ??). 1

The solution of (2.1) is given by iteration:

nt = n0A1A2 . . . At

What can we say about the long-term behavior of this solution? We consider two cases: a constantenvironment and a fluctuating, but stationary, environment.

1.1. Temporally Homogeneous Case. An important special case occurs when the At are con-stant in time i.e. At = A for all t. We say A is primitive if there exists t such that At only has positiveentries. Namely, after t time steps, any subpopulation contributes positively to any subpopulation.Most (but not all) matrix models in demography are primitive. A classical result for primitive matricesis as follows:

1We have chosen to write down models using left rather than right multiplication which is the conventional choicein many ecological modelling texts. Here are five reasons why:

(1) The i-j entry of At corresponds to the contribution of state i to state j.(2) Equations read naturally from left to right: future population state equals the current population state pro-

jected forward by At+1.(3) Row vectors display nicer in a paragraph than column vectors(4) When plotting solutions in time with matplot, the t-th row is Nt.(5) This is how probabilists do it.

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1. RANDOM MATRIX PRODUCTS 15

Theorem 2.1. (Perron-Frobenius) If A is a non-negative, primitive matrix, then there exists a pairof unique, positive row vectors v and w, and a unique positive real number λ such that

•∑

i vi = 1 and∑

i viwi = 1, and• if n0 is nonegative and non-zero, then

limt→∞

nt/λt = (n0w

T )v

where T denotes the transpose of a vector. Loosely, we have the approximation

nt ≈ λt(n0wT )v

λ is known as the dominant eigenvalue of A with associated left and right eigenvectors v and wT

i.e. vA = λv and AwT = λwT . Each of these quantities has a biological interpretation. v is the stablestate distribution of the population as it describes the fraction of the population in each state in thelong-term. w is the vector of reproductive values as it describes the contributions of the initial numberof individuals in each state to the long-term population size. Finally, λ is the long-term ”growth rate”in the sense that it determine the factor by which the population expands or shrinks per time step inthe long-term. About the reproductive values, Ronald Fisher wrote

We may ask, not only about the newly born, but about persons of any choosen age,what is the present value of their future offspring; and if the present value is calculateat the rate determined as before, the question has definite meaning – To what extentwill persons of this age, on average, contribute to the ancestry of future generations?The question is one of some interest, since the direct action of Natural Selection mustbe proportional to this condition. – Ronald A. Fisher (1930)

Example 2.1. To illustrate the use of the Perron-Frobenius theorem, lets consider a matrix modelof loggerhead turtles developed by Crouse et al. [1987]. Their model broke the population into 7 stages:hatchlings, yearlings, juveniles, sub-adults, first time reproducers, remigrants, mature adults. Thematrix for their model is given by

A =

0 0.6747 0 0 0 0 00 0.737 0.0486 0 0 0 00 0 0.6610 0.0147 0 0 00 0 0 0.6907 0.0518 0 0

127 0 0 0 0 0.8091 04 0 0 0 0 0 0.809180 0 0 0 0 0 0.8089

The top row of Figure 1 shows the stable state distribution and reproductive values. On the leftof the bottom row, the dynamics of fraction ni,t/

∑j nj,t in each state converges to the stable state

distribution within 25 years. On the right hand side, the population density normalized by the growthrate,

∑i ni,t/λ

t, converges to the reproductive value of a single adult.

1.2. Stationary Environments. Now assume A1, A2, . . . is a stationary, sequence of randommatrices. Stationarity means that for all t the shifted sequence At, At+1, At+2, . . . and the originalsequence A1, A2, A3, . . . have the same distribution. That is

P[At ∈ A1, . . . , At+n ∈ An] = P[A1 ∈ A1, . . . , An ∈ An]

for all Borel sets A0, . . . ,An in Rk2 (i.e. the space of all k× k matrices). Ergodicity means, intuitively,the environmental sequence is not a mixture of two other stationary distributions. 2 An important

2More precisely, for any bounded Borel function H : Rk2 × Rk2 × . . . → R, P[H(A1, A2, . . .) = H(A2, A3, . . .)] = 1implies that H(A1, A2, . . .) is almost surely constant.


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Figure 1. Stable state distribution and reproductive values for the loggerhead model.Convergence of the dynamics of nt is shown for n0 = (0, 0, 0, 0, 0, 0, 1) i.e. one adult.

consequence of ergodicity is that if E[|h(A1)|] < ∞ for some function h : Rk2 → R, then the temporal

average of h, h(A1)+···+h(At)t

, converges to its expected value E[h(A1)] with probability one.This representation of the environmental dynamics allows for a diversity of possibilities as the

following examples illustrate.

Example 2.2. (Periodic environments) Let {B1, B2, . . . , Bn} be non-negative matrices correspond-ing to the population dynamics in different years or seasons. A0 is choosen with equal likelihood forthese n matrices and At+1 = B

i+1 mod n on the event that At = Bi. Then the environmental dynamicscorrespond to randomly choosing an initial environmental state and successive environmental stateschanging periodically. This leads to a stationary and ergodic sequence of matrices.

Example 2.3. (Independent and identically distributed environments) A1, A2, A3, . . . are a sequenceof independent and identically distributed random matrices.

Example 2.4. (Finite Markovian environments) Suppose the environmental dynamics are char-acterized by a finite number of matrices {B1, . . . , Bm} and the transition probabilities between thesematrices are given by the transition matrix P i.e. the i-j-th entry Pij of P is the probability of goingfrom the matrix Bi to the matrix Bj. If P is irreducible, there is a unique stationary distribution µ forP . Namely, there is a row vector µ such that µP = µ with

∑i µi = 1. Let A1, A2, . . . be the sequence

of random matrices defined by A1 equals Bi with probability µi and P[At+1 = Bj|At = Bi] = Pij. Thisrandom sequence of matrices is stationary and ergodic.


Example 2.5. (auto-regressive models) A standard model of a fluctuating environmental variable(e.g. precipitation or temperature) is of the form xt+1 = at+1xt + bt+1 where xt is the environmentalvariable at time t, at are i.i.d. with E[log at] < 0, and bt are i.i.d. with E[bt] < ∞. Asymptotically(as t → ∞) this sequence is stationary and ergodic. If the entries of At have some continuous depen-dence on xt (e.g. survivorship or fecundity as a function of precipitation or temperature), then theAt are asymptotically stationary and ergodic. We will discuss these models, and their multivariatecounterparts, more extensively in section XX.XX

Quite remarkably, there is a stochastic analog of the Perron-Frobenius Theorem. The version wepresent is a slight extension of a result due to Arnold et al. [1994]. A proof of this extension willbe given in section XX.XX. To state the theorem, we need to introduce two definitions. We say Atis primitive if there is a s such that the random product At+1At+2 . . . At+s has only positive entries.Define

m = mini,j

Bij and M = maxi,j

Bij

where B = A1A2 . . . As. Furthermore, define

log+ x = max{log x, 0}

to be the non-negative part of log x.

Theorem 2.2. (Random Perron-Frobenius Theorem) Let A1, A2, . . . be a primitive, stationary,ergodic sequence of random variables such that E[log+ 1

m] and E[log+M ] are finite. Then there exists

two stationary sequences of positive, random vectors vt and wt and stationary sequence of scalars λtsuch that

(1)∑

i vi,t =∑

i vi,twi,t = 1 for all i and t,(2) vtAt+1 = λt+1vt+1 for all t,(3) At+1w

Tt+1 = λt+1w

Tt for all t,

(4) for any n0 > 0 (i.e. all non-negative entries with one entry at least positive) and vector c > 0,

limt→∞

1

tlog ncT = E[log λ1] =: r, and

(5) there exists a τ > 0 such that for any n0 > 0,

limt→∞

log

∣∣∣∣ ntλ1λ2 . . . λt

− vt(n0wT0 )

∣∣∣∣ ≤ −τwith probability one.

The interpretation of this theorem is analogous to the Perron-Frobenius theorem. The stationarysequence of vectors vt correspond to the stable state distribution of the structured population. However,due to the temporal fluctuations, the stable-state distribution varies in time in a stationary manner.The final assertion of the theorem implies that the population distribution for any initial conditionapproaches the vt at an exponential rate. The λt describe how the population changes in size as itfollows the stable state distribution. In particular, in the long-term, the log population size growslinearly like E[log λ1]t. The wt describe the time varying reproductive values of individuals. The finalassertion of the theorem puts all the pieces together and implies that

nt ≈ (v0wT0 )λ1 . . . λtvt ≈ (v0w

T0 )ertvt for t sufficiently large

where r is the stochastic per-capita growth rate.The next example illustrates one of the earliest ecological applications of this random Perron-

Frobenius theorem.


Figure 2. Graphic depiction of the Silva et al. [1991] model for the neotropical, perennialgrass Andropogon semiberbis. There are 4 states (the nodes): 1 tiller (state 1), 2 −−10 tillers (state 2), 11 − −20 tillers (state 3), > 20 tillers (state 4). Arrows indicatecontributions of one stage to another. Burn populations in (a) and unburnt populationsin (b).

Example 2.6. (Burning grasses) Silva et al. [1991] examined the effect of temporal fluctuationsin fires on the persistence of tropical savanna grass Andropogon semiberbis. For this species, fireshelp remove “necromass” (dead biomass) and, thereby, improve survival, growth, and reproduction ofindividuals. Silva et al. [1991] were interested on how the frequency of fires influences the persistence ofthis species. They divided this size-structured population into 4 stages: individuals with 1 tiller (stage1), individuals with 2–10 tillers (stage 2), individuals with 11–20 tillers (stage 3), and individuals with> 20 tillers (stage 4). Based on two year experiments in burnt and unburnt plots, they estimate thevital rates under these two sets of conditions (Figure 2). The transpose of the following matrix wasused for the burnt years:

B=matrix(c(0.08,1.63,2.42,4.4,

0.21,0.64,0.35,0.16,

0,0.19,0.43,0.24,

0,0.03,0.23,0.48),4,byrow=TRUE)

and the transpose of the following matrix was used for the unburnt (or protected) years:

U=matrix(c(0,0.706,0.391,3.59,

0.018,0.158,0.136,0.093,

0,0.08,0.07,0.21,

0,0,0.01,0.07),4,byrow=TRUE)

To model the stochastic switching between burnt and unburnt years, Silva et al. [1991] assumedthat the probability of going from a burnt year to an unburnt year is p, from a burnt year to a burntyear is 1− p, from an unburnt year to a burnt year is q, and from an unburnt year to an unburnt yearis 1− q. Under these assumptions, the frequency of burnt years in the long term is

q

q + p


2 4 6 8 10 12 14

0.0

0.4

0.8

year

freq

uenc

y of

type

s

1 tiller2−10 tillers11−20 tillers>20 tillers

Figure 3. Convergence to the stable state distribition vt for 100 random initial condi-tions n0 for Silva’s model of Andropogon semiberbis. Parameters: p = q = 0.5

We can model these stochastic transitions between burnt years (state 1) and unburnt years (state 2)using the transition matrix

P =

(1− p pq 1− q

)where Pij is the probability of the environment going from state i to state j.

Figure 3 illustrates the convergence of nt/∑

i ni,t to the stable stage distribution for 50 randominitial conditions.

Figure 4 plots r for different p and q values against the fraction pq+p

of burnt years. This figure

demonstrates that (i) the critical burning frequency is approximately 85% and (ii) temporal correlationshave little effect on r.

Now suppose that q = 0.9 = 1 − p (i.e. every year there is a 90% changes of a burning) in whichcase r ≈ 0.1. Figure 5 plots the distribution of vt (left) and wt (right) values for each stage (stage 1on top, stage 4 on bottom). This figure illustrates that in the long term, most individuals are in thefirst stage, the fewest individual are in the 4th stage, and the greatest coefficient of variation occurs inthe 2nd stage. Intuitively, stage 4 has the greatest reproductive value due to these individuals havinghaving higher survivorship and fecundity than other stages, while stage 1 has the least reproductivevalue.

Example 2.7. (Well-mixed populations) Many species live in habitats that appear as discretepatches across the landscape e.g. butterflies in meadows, water fleas in ephemeral ponds, big hornsheep on mountain tops, annual plants on soil patches. Assume there are k habitat patches and thedensity of an annual population in the i-th habitat patch is ni,t in year t. Let Ri,t+1 be the numberof offspring produced by an individual living in patch i that make it to year t + 1. Assume offspringdisperse before reproducing and the fraction dispersing from patch i to patch j is Dij. Under these


0.0 0.2 0.4 0.6 0.8 1.0

−1.

00.

0

fraction of good years

r

Figure 4. Stochastic growth rates r for the burning grass model for a 20 by 20 grid ofp and q values.

0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5

0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5

0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5

0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5

Figure 5. Stable stage distributions (left) and reproductive values (right) when p =0.9 = 1− q.

2. SENSITIVITY OF r 21

assumptions, we get the following matrix model

nt+1 = ntRt+1D where Rt+1 = diag(R1,t+1, . . . , Rk,t+1).

In general finding r for this model is quite challenging. However, in the special case that the populationsare well-mixed i.e. the probability of going to patch j is independent of where an offspring is comingfrom, then finding r is straight forward. In this case, there are dj such that Dij = dj for all i and j.This implies that the beginning of each year, the fraction of the population in patch j is dj. Hence,the stable stage distribution is given by vt = (d1, d2, . . . , dk) for all time. Thus

vt+1 = λt+1vt

(d1, . . . , dk)Rt+1D = λt+1(d1, . . . , dk)

(d1R1,t+1, . . . , dkRk,t+1)D = λt+1(d1, . . . , dk)∑i

diRi,t+1(d1, . . . , dk) = λt+1(d1, . . . , dk)

andλt+1 =

∑i

diRi,t+1.

Thus,

r = E[log λt] = E[log∑i

diRi,t]

Exponentiating r, we get er is the geometric mean of the spatial mean of the fitnesses Ri,t. Thisexpression for “metapopulation” fitness was first discovered by Metz et al. [1983] and rediscoveredindependently in the two-patch case by Jansen and Yoshimura [1998]. Provided this “metapopulationfitness” is greater than one, the population grows at an exponential rate. If it is less than one, thenthe metapopulation tends toward extinction at an exponential rate.

As log is a concave function, applying Jensen’s inequality twice to this expression (with respect tothe spatial average and the temporal average) yields∑

i

di logE[Ri,t] > r >∑i

diE[logRi,t]

The first inequality implies that r > 0 requires that the arithmetic mean E[Ri,t] is greater than one inat least one patch. As the geometric means are less than the arithmetic means, the second inequalityimplies that r can be > 0 even if each patch, in and of itself, can not sustain the population i.e.E[logRi,t] < 0. For example, if the fitness in patch is log normal with log-mean µ, log variance σ2 anddi = 1

k, then E[Ri,t] = exp(µ+ σ2/2) and E[logRi,t] = µ. In the limit of many patches, the law of large

numbers implies that

limk→∞

E[log1

k

∑i

Ri,t] = E[log eµ+σ2/2] = µ+ σ2/2

Hence, if µ < 0, µ+ σ2 > 0, and k is sufficiently large, then r > 0 despite all patches being unable, inand of themselves, to support a population. Figure 6 illustrates this phenomena where there is criticalnumber of habitat patches above which the metapopulation growth rate r is positive.

2. Sensitivity of r

While r = E[log λ1] can be easy to compute for unstructured populations due to the commutativityof scalar products, explicitly computing r for structured populations is often analytically intractable.Two notable exceptions are Roerdink [1987]’s formula for r for model of biennial populations, randommatrices that share a reproductive value or share a stable structure [Tuljapurkar, 1986, 1990], see, e.g.,Example 2.7. One useful approach to getting some insights into how r depends on the nature of the


5 10 15 20−0.

100.

00

# patches kstoc

hast

ic g

row

th r

ate

r

Figure 6. Stochastic growth rates r for the well-mixed metapopulation model with allpatches being stochastic sinks. Lower dashed line corresponds to the stochastic growthrate within a patch and the upper dashed line corresponds to the expected fitness in apatch

stochastic forcing is study small perturbations of random matrix products for which r is understood.For example, small random perturbations of a constant matrix model, or adding dispersal limitationto the metapopulation model in Example 2.7. What do we mean by small random perturbations?Suppose that you know the value of r for the random matrix model nt+1 = ntAt+1 and you wantto know what happens when you replace At+1 by Ct+1(ε) = At+1 + εBt+1 where C1(ε), C2(ε), . . . , isa stationary, ergodic sequence of matrices. If Ct(ε) satisfy the assumptions of the Random Perron-Frobenius Theorem, then the stochastic growth rate r(ε) exists as a function of ε. Given that oneunderstands r(0), wouldn’t it be great if one could write down an Taylor’s expansion of the form

r(ε) = r(0) + r′(0)ε+1

2r′′(0)ε+ . . .

Remarkably, such a Taylor expansion is possibly provided that r(ε) is an analytic function. So when isr analytic? (Feel free to skip the rest of this paragraph if you are interested in this technical detail.) Asufficient condition for analyticity is that there is common underlying stochastic process driving bothsequences of matrices. Namely, there is a stationary, ergodic sequence ξ1, ξ2, . . . in separable metricspace X (e.g. a finite set of points {1, 2, ..n} or Rn for some n) and continuous functions A,B : X → Rk2

such that At = A(ξt) and Bt = B(ξt). One can view the ξt as describing the environmental state ofthe system (e.g. the climatic conditions) and the functions A,B map environmental conditions todemographic effects i.e. vital rates. For example, if there are a finite number of environmental states{1, . . . , n}, then ξ1, ξ2, . . . might be given by an irreducible Markov chain on these states and A(i)and B(i) are matrices of vital rates when the system is in the i-th environmental state. Or ξt aremultivariate numerals and aij(ξ) describe how the vital rates depend on the value of the multivariatenormal. Under this additional assumption r(ε) is analytic and we can compute derivatives.

As the process of finding the first derivative r′(0) is fairly straight-forward and provides good practicewith understanding the key components of the Random Perron-Frobenius theorem, we go through thedetails for computing it. To this end, let wt, vt, and λt be as given by the Perron-Frobenius Theorem


for the unperturbed matrix model nt+1 = ntAt+1. Let gt(ε) = E[log v0C1(ε) . . . Ct(ε)wTt ]. The Perron-

Frobenius theorem applied to Ct(ε) implies that

r(ε) = limt→∞

gt(ε).

Notice that

v0C1 . . . CtwTt = v0(A1 . . . At)w

Tt + ε

t∑s=1

v0A1 . . . As−1BsAs+1 . . . AtwTt +O(ε2)

= λ1 . . . λt + ε

t∑s=1

λ1 . . . λs−1λs+1 . . . λtvs−1BswTs +O(ε2)

= λ1 . . . λt

(1 + ε

t∑s=1

vs−1BswTs /λs

)+O(ε2)

As log(1 + x) = x+O(x2), we get that

log(v0C1 . . . CtwTt ) =

t∑s=1

log λs + εt∑

s=1

vs−1BswTs /λs +O(ε2)

Taking expectation, dividing by t, and taking the limit at t→∞ yields

E[log λ1] + εE[v0B1wT1 /λ1] +O(ε2).

Hence,

(2.2) r′(0) = E[v0B1w1/λ1]

and

(2.3) r(ε) = r(0) + εE[v0B1w1/λ1] +O(ε2).

Example 2.8. (Perturbations of deterministic models) If At = A is a constant matrix model, thenv0 = v and w1 = w are the left and right eigenvectors normalized so that

∑i vi =

∑i viwi = 1, and

λt = λ is the associated eigenvector. In this case, r′(0) = vE[B1]wT/λ. If B1 has all zeros except a

value of one in the i–j-th entry, then r′(0) = viwj/λ. In words, the increase in r due to an increase inthe contribution of individuals in stage i to stage j is proportional to the long-term frequency of stagei individuals (i.e. vi) and the reproductive value of stage j individuals (i.e. wj). This corresponds tothe sensitivity of r to increases in the i–j-th entry of the unperturbed matrix A.

For example, Figure 7 shows all of the sensitivities of r ≈ −0.05653757 for the loggerhead modelfrom Example ??. Notice that the extremely large sensitivities corresponds to the transitions fromyearlings (stage 2) to adults (stages 5 through 7). Hence, if even a small percentage of yearlings wereable to become mature adults within a year, there would be a large increase in the population rate r.However, this change isn’t biologically feasible, consistent with the fact that the corresponding entriesin A are zero. Hence, to look at “biologically feasible” sensitivities, one should only consider the onesthat correspond to non-zero entries of the matrix. Alternatively, one can look at the elasticity of r tochanges in A i.e. the relative change in r due to a relative change in A. These are given by wjviAij/λand plotted in the right hand side of Figure 7. These elasticities suggest that a relative increase insurvival of sub-adults (stage 3) would produce the largest increase in r.

Example 2.9. (Should I stay or should I go?) Lets consider a two patch version of the metapopu-lation matrix model from Example 2.7 with d1 = d2 = 1/2:

At =

(R1,t 0

0 R2,t

)(12

12

12

12

).


020

4060

80

0.0

1.0

2.0

3.0

Figure 7. Senstivities for r for the loggerhead model on the left. Elasticities on the right.Each group of bars corresponds to the sensitivities/elasticities for the corresponding rowof the matrix.

For this model, vt = (1/2, 1/2) for all t, λt = (R1,t + R2,t)/2, and (check for yourself!) wt =(R1,t+1, R2,t+1)/λt+1.

To understand the effect of dispersal limitation on the stochastic growth rate, we consider thematrix model given by

Ct(ε) =

(R1,t 0

0 R2,t

)(12

+ ε 12− ε

12− ε 1

2+ ε

)= At + ε

(R1,t −R1,t

−R2,t R2,t

)︸︷︷︸

=:Bt

i.e. an additional fraction ε ≥ 0 of individuals stay in their natal patch. The effect of this additionalpatch fidelity on the stochastic growth rate is determined by

r′(0) = E[vBtw

Tt

λt

]= E

[R1,t −R2,t

λt(1/2,−1/2)wTt

]=

1

2E[R1,t −R2,t

λt

R1,t+1 −R2,t

λt+1

]= 2E

[R1,t −R2,t

R1,t +R2,t

R1,t+1 −R2,t+1

R1,t+1 +R2,t+1

]If we define Wi,t =

Ri,tR1,t+R2,t

as the relative fitness of patch i in year t, then our approximation is given

byr(ε) ≈ r(0) + 2εE[(W1,t −W2,t)(W1,t+1 −W2,t+1)]

If the Wi,t are identically distributed and independent in space, then this approximation has twoimplications. If the difference in relative fitness is positively auto-correlated in time, then r′(0) > 0and greater site fidelity increases the stochastic growth rate. If the relative fitness is negatively auto-correlated in time, then r′(0) < 0 and greater site fidelity decreases the stochastic growth rate. Hence,this analysis provides an answer to a query posed by the Clash in the 1980s:


0.00 0.05 0.10 0.15 0.20 0.25

−0.

010.

010.

03

site fidelity ε

stoc

hast

ic g

row

th r

ate

r

Figure 8. Stochastic growth rates r and the first order approximation for the two patchmodel. Temporal correlations for three curves are 0.5, 0,−0.5.

Should I stay or should I go now?Should I stay or should I go now?If I go there will be troubleAn’ if I stay it will be doubleSo come on and let me know

Namely, go if relative fitness differences are negatively auto-correlated, and stay if they are positivelyauto-correlated. Figure 8 illustrates how well the approximation (lines) works. A more general formulafor any number of patches is derived in [Schreiber, 2010].

The approximation (2.3) is useful for examining what happens if the random perturbation Bt hasnon-zero mean. However, often stochastic demographers are interested in what effect does just increasesvariation in some of the vital rates (i.e. entries of At) have on the long-term growth rate r. In thiscase, one would want E[Bt] = 0. If the Bt are independent of the vt, wt for At, then r′(0) = 0 and oneneeds to consider higher order correction terms. In the exercises, you are asked to verify the followingclassic result from stochastic demography [Tuljapurkar, 1990].

Theorem 2.3. Variance sensitivity Assume A1, A2, . . . are i.i.d., B1, B2, . . . are i.i.d., and the Atsand Bts are independent of one another. If E[Bt] = 0 and r(ε) is the stochastic growth rate of At+εBt,


0.0

1.0

2.0

−4

−3

−2

−1

0

Figure 9. Sensitivity of r to increases in the mean value of the vital rates (left) andvariation in the vital rates (right). Groups of bars (from left to right) corresponds tocontributions from stages 1, 2, 3, and 4, respectively. Parameters: 1− p = q = 0.9

thenr′(0) = 0 and r′′(0) = −Var

[v0B1w

T1 /λ1

].

Hence,

(2.4) r(ε) = r(0)− 1

2Var

[v0B1w

T1 /λ1

]ε2 +O(ε3)

A key implication of this result is that temporally uncorrelated fluctuations in demographic ratesdecrease the per-capita growth rate r of populations. However, as suggested by Example 2.9, temporalauto-correlated fluctuations can increase population growth rates.

Example 2.10. (Tuljapurkar’s approximation) An important special case of (2.4) is when At is aconstant matrix A. Then vt = v, wt = w, and λt = λ are constant. Define Sij = viwj/λ to be thesensitivity of r = log λ to changes in the i–j-th entry of A. If Ct(ε) = A+ εBt where Bt are i.i.d. withmean 0, then (try this for yourself!)

r(ε) ≈ r − 1

2λ2

∑ij,k`

SijSk`Cov[bij,t, bk`,t].

Hence r is most sensitive to vital rates that exhibit the greatest variation i.e. the largest S2ijVar[bij,t]

terms. Negative correlations between vital rates can reduce, but never eliminate, the negative effect ofenvironmental fluctuations on the population growth rate.

Example 2.11. (Burning grasses revisited) Let E(ij) be the 4 × 4 matrix with a 1 in the i-jthentry and zeros elsewhere. To see the effect of increasing the mean value of any of the entries of At,we consider Bt = E(ij) for all i and j values. The left hand side of Figure 9 plots r′(0) for all of theseperturbations. The first block bars corresponds to contributions from stage 1, the second group of barscorresponds to contributions from stage 2, etc. These sensitivities indicate that increasing the fecundityof stage 4 individuals provides the biggest boost to r. To see the effect of increasing the variation in anyof the entries, we use the perturbation Bt = ±E(i, j) where ± occur with equal probabilities. Thesesensitivities are plotted on the right hand side of Figure 9. These bar plots show that variation in thefecundity of the stage 4 individuals has the most negative impact on the long-term growth rate r.

4. RANDOM AFFINE MODELS 27

3. General random products

It will be useful to consider random matrix models nt+1 = ntAt+1 where At are allowed to havenegative as well as positive entries. For example, these type of matrices arise when linearizing adeterministic model around an equilibrium. For this general class of models, one can say somethingabout the “stability” of the zero solution. Assume At are stationary and ergodic and define ‖A‖ =maxi,j |Aij|.

Theorem 2.4. (Stability of random matrix products) If E[log+ ‖A1‖] <∞, then there exists a realnumber r (possibly −∞) such that

limt→∞

1

tlog ‖A1 . . . At‖ = r

with probability one. In particular, if r < 0, then

limt→∞

nt = 0 with probability one

for all n0. Alternatively, if r > 0 and n0 is a random variable uniformly distributed on the unit sphere,then

limt→∞

∑i

|ni,t| =∞ with probability one.

r is often called the dominant Lyapunov exponent. As with the projection matrices, r < 0 impliesstability of the origin as with probability one all solutions converge exponentially fast to the origin.Alternatively, r > 0 implies that “most” initial conditions result in nt going to infinity at an exponentialrate.

4. Random affine models

The random matrix models from the previous sections are good described closed populations whoremain at sufficiently low densities that density-dependent feedbacks are negligible. Many populationsare open and experience immigration. To account for this immigration, we need to allow for randominputs which often uncorrelated to local densities. Hence, we are interested in models of the form

(2.5) nt+1 = ntAt+1 + bt+1

where are At and bt are given by ergodic, stationary sequences of random matrices and vectors, respec-tively. When the At and bt are non-negative matrices and vectors, bt correspond to random inputs ofindividuals in all the stages to the focal population i.e. the population being modeled. More generally,At and bt may have positive and negative entries e.g. they come from the linearization of a density-dependent model (see Chapter XX.XX). Model (2.5) is called a generalized auto-regressive process. Forthe rest of the section, I assume that the At and bt are i.i.d. I still need to check which ofthe results extend to the stationary case.

Iterating (2.5) for t time steps and starting with n0 yields

nt = n0A1 . . . At + bt + bt−1At + . . . b1A2 . . . At

Let r be dominant Lyapunov exponent associated with the product A1 . . . At. If r < 0, then the termn0A1 . . . At goes to zero with probability one. Hence, asymptotically, we are only left with the term

bt + bt−1At + . . . b1A2 . . . At

By our assumption of the sequences being i.i.d., the distribution of this term is the same as thedistribution of

b1 + b2A1 + b3A1A2 · · ·+ btA1 . . . At−1


log10 density

0 1 2 3 4 5 6

log10 density

0 1 2 3 4 5 6

log10 density

0 1 2 3 4 5 6

Figure 10. Approximate stationary distributions for the simple immigration model withσ2 = 0.01, 0.02, 0.1, r = −0.05, and I = 1.

Hence, provided the limit is well-defined with probability one, we would expect that nt converges indistribution to

(2.6) n = limt→∞

b1 + b2A1 + b3A1A2 · · ·+ btA1 . . . At−1

As the distribution of this limiting expression unchanged by multiplying by a new A and adding a newb, it is stationary.

To state a theorem that justifies this heuristic, define a subspace V of Rk to be invariant for (2.5) ifP[X1 ∈ V |X1 = x] = 1 whenever x ∈ V . Namely, if you start in the subspace you stay in the subspacefor all time with probability one. For most of the models we consider, the only invariant subspace isRk itself.

Theorem 2.5. Assume that E[log+ ‖A1‖] and E[‖b1‖] are finite, Rk is the only invariant subspacefor (2.5), and the dominant Lyapunov exponent r for A1, A2, . . . is negative. Then (2.6) exists almost-surely. Moreover, for any n0, nt converges in distribution to n i.e. for any Borel set A ⊂ Rk,

limt→∞

P[nt ∈ A] = P[n ∈ A]

and

limt→∞

1

t

t∑s=1

h(ns) = E[h(n)] with probability one

for any continous function h : Rk → R such that E[|h(n)|] <∞.

Example 2.12. (A simple immigration model) Let nt be the density of an unstructured population(i.e. k = 1) for which the fitness of an individual is Rt+1 and It+1 immigrates in by time t + 1.Then nt+1 = ntRt+1 + It+1. Provided that r = E[logRt] < 0 and E[It] is finite, then nt convergeswith probability one to a stationary distribution. Consider the special case where Rt is log-normallydistributed with log mean r and log variance σ2 and It = I for all time (i.e. constant immigration).Then r < 0 ensures the existence of a stationary distribution. Figure 10 illustrates these distributionsfor r = −0.05, σ2 = 0.01, 0.02, 0.1, and I = 1. For σ2 = 0.1, the distribution is much broader thanthe smaller σ2 values; populations reaching densities of a million quite frequently. We will see why thisvalue of σ2 is special shortly.


Example 2.13. Higher order auto-regressive processes AR(k) Auto-regressive models of order kare used extensively in modelling weather, stock markets, etc. If xt is the scalar variable of interest(e.g. temperature), then AR(k) models are of the form

xt+1 = a+k∑s=1

aixt+1−s + ηt+1

where a, bi are constants and η1, η2, . . . are i.i.d. with zero mean i.e. a white noise processes. Hence,for these processes, the state of the system in the next time step depends on the state of the systemfor the previous k time steps.

To turn this model into the type we have been talking about, define ni,t = xt+1−i for i = 1, . . . , k.Then we get

nt+1 = nt

b1 1 0 . . . 0 0b2 0 1 . . . 0 0...

......

......

...bk−1 0 0 . . . 0 1bk 0 0 . . . 0 0

+

a+ ηt+1

0...00

If the |bi| < 1, then this auto-regressive process has a stationary distribution.

Figure 11 (upper panels) shows the daily average temperature at Bodega Marine Laboratory during2014. Plotting tomorrow’s temperature against today’s temperature demonstrates a fairly linear rela-tion which suggest a first order auto-regressive process could be used to model this data. In fact, AICmodel selection yields a third auto-regressive model with coefficients b1 = 0.7898, b2 = −0.0595, b3 =0.1711, a = 25.02924 and white noise with variance σ2 = 1.284. Simulating this AR(3) process yieldsthe time series shown in the lower panels of Figure 11.

Example 2.14. (Stochastic ferns) Random affine models can be used to generate all types ofsurprisingly fun figures. For example, suppose that At and bt are given by(

0.4 0.06−0.3733 0.6

)and

(0.3533 0

)with probability 0.2933 and (

−0.8 0.1371−0.1876 0.8

)and

(1.1 0.1

)with the complementary probability. As there are only a finite number of non-zero matrices and vectorsbeing choosen, E[log+ ‖At‖],E[‖b‖] are finite. Furthermore, as ‖At‖ ≤ 0.8, we get that

r = limt→∞

1

tlog ‖A1 . . . At‖ ≤ log 0.8 < 0

with probability one. Hence, iterating this random system should converge to a stationary distribution.Indeed, it converges to a surprisingly cool fern as shown in Figure 12.

Given a random affine model with a unique stationary distribution n, what can say about thedistribution of n? Using independence and stationarity, we can fairly easily compute the mean andco-variance of this distribution, assuming that they exist. By stationarity and independence, if E[n]exists, then it satisfies

E[n] = E[n]E[A1] + E[b1]

(I − E[A1])E[n] = E[b1].

If the spectral radius of E[A1] is less than one (this is the necessary condition for the expectationexisting), then I − E[A1] is invertible and

(2.7) E[n] = (I − E[A1])−1E[b1]


0 100 200 300 400

−5

515

25

day

tem

pera

ture

(C

)

8 10 12 14 16 18

812

16

temp. today

tem

p. to

mor

row

0 100 200 300 400

−5

515

25

day

sim

ulat

ed te

mp

6 8 10 14 18

610

1418

temp. today

tem

p. to

mor

row

Figure 11. Actual (top) and simulated (bottom) daily average temperatures at BodegaMarine Lab in 2015. Simulated data is from an AR(3) model whose parameters aredescribed in the main text.

To compute the co-variance matrix provided, we begin by assuming the expectation exists anddefine y = n− E[n] and β1 = E[n]A1 + b1 − E[n]. Then (check the details for yourself!) E[β1] = 0 andE[y] = 0. Stationarity and independence imply

E[yT y] = E[(yA1 + β1)

T (yA1 + β1)]

= E[AT1 ]E[yT y]E[A1] + E[βT1 β1]

Cov[n] = E[AT1 ]Cov[n]E[A1] + Cov[β1]

To solve this system of linear equations, we can take advantage of two matrix operations: the vecoperation and the Kronecker product ⊗. The vec operation vec(A) takes a matrix A and createsa long vector by concatenating the column vectors. Two key properties of the vec operation arevec(A+B) = vec(A) + vec(B) (obvious), and vec(ABC) = (CT ⊗A)vec(B) (less obvious). Using thesetwo properties, we get

vec(Cov[n]) = vec(E[AT1 ]Cov[n]E[A1] + Cov[β1]

)= vec

(E[AT1 ]Cov[n]E[A1]

)+ vec (Cov[β1])

=(E[A1]

T ⊗ E[A1]T)

vec(Cov[n]) + vec (Cov[β1])(I − E[A1]

T ⊗ E[A1]T)

vec(Cov[n]) = vec (Cov[β1])


Figure 12. Fern from a random affine model.

Provided that the spectral radius of E[A1]T ⊗E[A1]

T is less than one, then we can take the inverse andget the following formula for the co-variance matrix:

(2.8) vec(Cov[n]) =(I − E[AT1 ]⊗ E[AT1 ]

)−1vec(Cov[β1])

Example 2.15. (The simple immigration model revisited) Let nt+1 = ntRt+1 + I where R1, R2, . . .is a log-normal sequences of i.i.d. random variables with log mean r and log variance σ, and I ≥ 0corresponds to constant rain of immigration. This processes has a stationary distribution provided thatr = E[logRt] < 0. Provided that E[Rt] = exp(r + σ2/2) < 1, the mean of this stationary distributionis given by (verify this for yourself!)

E[n] = I/(1− exp(r + σ2/2)).

As β1 = E[n]R1 + I − E[n], Var[β1] = E[n]2Var[R1] and Var[R1] = (eσ2 − 1)e2r+σ

2, the variance of n is

given by

Var[n] = E[n](eσ2 − 1) exp(2r + σ2)/(1− exp(2r + σ2))

As E[logRt] = r < 0 doesn’t imply that E[Rt] = exp(r + σ2/2) < 1, we get this immigration modelhas a stationary distribution with infinite moments whenever r < 0 ≤ r− σ2/2 e.g. −r = σ2 = 0.05 asillustrated in Figure 10.

Example 2.16. (AR models revisited) Consider the AR(k) model described in Example 2.13. UP-COMING

Example 2.17. (Open two stage population) Consider a population with two stages, juveniles andadults. Juveniles survive with probability 0.1 in which case they become adults. Adults survive withprobability 0.5 to the next year. New juveniles only arive through immigration. Let It be the number


of juviniles that arrive in year t. Then get

nt+1 = nt

(0 0.50 0.1

)+

(It+1

0

)Assume that log It is normally distributed with µ and variance σ2. Then

E[A1] =

(0 0.50 0.1

)and E[b1] =

(exp(µ+ σ2)

0

)Therefore

A=rbind(c(0,0.1),c(0,0.5))

mu=0.1

sigma=0.1

b=c(exp(mu+sigma^2/2),0)

Covb=rbind(c((exp(sigma^2)-1)*exp(2*mu+sigma^2),0),c(0,0))

require(fBasics)

## Loading required package: fBasics

## Loading required package: timeDate

## Loading required package: timeSeries

## Warning: package ’timeSeries’ was built under R version 3.2.3

##

##

## Rmetrics Package fBasics

## Analysing Markets and calculating Basic Statistics

## Copyright (C) 2005-2014 Rmetrics Association Zurich

## Educational Software for Financial Engineering and Computational Science

## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.

## https://www.rmetrics.org --- Mail to: [email protected]

EXERCISES 33

Exercises

(1) Verify the expression for the reproductive values in Example 2.9(2) Let d1, d2 ≥ 0, d1 + d2 = 1,

At =

(R1,t 0

0 R2,t

)(d1 d2d1 d2

)and

Ct =

(R1,t 0

0 R2,t

)(d1 − ε d2 + εd1 d2

)Find r′(0). Interpret the result in terms of the stochastic growth rate of a two patch model.

(3) Prove that (2.4) holds.(4) Verify the variance sensitivity formula in Example 2.10(5) Verify the expressions for E[n] and Var[n] in Example 2.15(6) Find E[n] and Cov[n] for Example 2.14.(7) For the auto-regressive model of rainfall in Example 2.13, let n = (n1, n2, n3) be the stationary

distribution. Find E[n] and Cov[n].(8) Desert tortoise, Gopherus agassizii, was listed as endangered in 1989, and a draft recovery

plan was issued in 1993. Declines are occurring throughout the range of the species, butthey appear to be particularly severe in the Western Mojave. Direct human impacts on thetortoise include habitat degradation and habitat loss, hunting (up to 14% of mortality in someareas), and getting run over by cars or off-road vehicles. Indirect impacts on the populationsinclude habitat degradation by sheep or cattle grazing, predation by ravens (which attackyearlings and juveniles, and are associated with human activity), and an upper respiratorytract infection that may have been introduced by release of pet tortoises into the wild. Thisgoal of this homework is to compare two or more management scenarios (e.g. reducing humandisturbance, removing ravens) and determine which will have a greater effect in conserving thedesert tortoise populations. Human disturbance mainly affects larger individuals, while ravenpredation is limited to smaller ones. To achieve this goal, you need to develop a stage-structuredmodel using the following data and analyze this model using the techniques developed in class.

In 1978, K. H. Berry classified Desert Tortoise sizes as follows

class name maximum carpacae length (mm)0 Yearling1 Juvenile 1 <602 Juvenile 2 60–993 Immature 1 100–1394 Immature 2 140–1795 Subadult 180–2076 Adult 1 208–2397 Adult 2 >240

Government reports and previously unanalyzed mark-recapture data at 8 Bureau of LandManagement permanent study plots in the Western Mojave provide the following informationabout growth, survival, and reproduction for the turtle populations. Growth is defined as theprobability, conditional on survival, of moving from one size class to the next largest one; ofthe >1400 individuals transitions in the data, no tortoises either shrank or were estimated togrow more than one size class in a year

EXERCISES 34

Demographic rate Mean SDGrowth of stage 2 0.208 0.268Growth of stage 3 0.280 0.158Growth of stage 4 0.287 0.261Growth of stage 5 0.269 0.187Growth of stage 6 0.018 0.037

Survival of stage 2 0.716 0.232Survival of stage 3 0.839 0.176Survival of stage 4 0.785 0.147Survival of stage 5 0.927 0.071Survival of stage 6 0.867 0.129Survival of stage 7 0.860 0.123

Since no data are available on the demographic rates for yearlings and size class one tortoises,you should make the optimistic assumption that they are the same as those for size class 2individuals.

For the Mojave, there were no direct observations of individual fecundity (e.g. egg orhatchling production). Instead, you can use four estimates for fecundity that were computedin various ways:

Yearling production Low Med-low Med-high HighYearling production of stage 5 0.042 0.42 1.3 2.22Yearling production of stage 6 0.069 0.69 1.98 3.38Yearling production of stage 7 0.069 0.69 2.57 4.38

That amount of variation is certainly not ideal but it’s not necessarily catastrophic, becausethe model predictions that you care about may not be affected even by that much parameteruncertainty.(a) Write down a matrix model for each of the low and high fecundity estimates. Compute

the dominant eigenvalue, stable age distribution, and the reproductive values for each ofthese models. Discuss the similarities and differences of these quantities for the low andhigh fecundity scenarios.

(b) To see the effect of stochasticity in this system, assume that the conservation biologistshave implemented a strategy that increases survivorship of all stages by 5.5%. Further-more, assume the probability of going from a low fecundity year to a high fecundity yearis p and the probability of going from a high fecundity year to a low fecundity year is q.

(i) High and low years are uncorrelated when p = 1 − q. Assuming that p = 1 − q,compute and plot the stochastic growth rate as p goes from 0 to 1. Discuss for whatfrequencies of low fecundities the population can persist.

(ii) A frequency of 50% low fecundity years occurs whenever pp+q

= 1/2. Equivalently

p = q. Assuming p = q, compute and plot the stochastic growth rate as function p.Discuss how varying p corresponds to varying the temporal correlation of the envi-ronmental states and discuss how this temporal correlation influences the stochasticgrowth rate and thereby persistence.

CHAPTER 3

Nonlinear single species models

Some negative feedback between r and population density (that is, density depen-dence) is a necessary (but not sufficient) condition for population regulation. – Pe-ter Turchin [1999]

1. Unstructured models

Our study of non-linear, single species models begins with unstructured models where nt is thedensity of the population at time t. To simultaneously account for density-dependent feedbacks andenvironmental fluctuations, let E be a separable metric space (e.g. Rn for some n) that represents allpossible environmental states. For example, if the important environmental variables are temperatureand precipitation, then E might be R× [0,∞). Alternatively, if we model the fluctuations in n demo-graphic parameters directly without explicitly describing their dependence on environmental factors,then E is some subset of Rn. We will write ξ ∈ E for a particular environmental state.

Let f(n, ξ) be the fitness of an individual when the population density is n and the environmentalstate is ξ. Let ξ1, ξ2, · · · ∈ E be a sequence of i.i.d. random variables determining the environment.1

The population dynamics are given by

(3.1) nt+1 = ntf(nt, ξt+1).

1.1. Negative density-dependence. Many population experience negative feedbacks of densityon survival and reproduction. For example, as densities get higher, there might be fewer resourcesper individual which lowers survivorship and reproductive output of individuals. To account for thesenegative feedbacks, we assume that f(n, ξ) is a decreasing function of n. We will relax this assumptionsubstantially when examining structured population models later in this chapter.

For models of this type, we may be interested in when populations are regulated : their densitiestend remain bounded away from zero and infinity. To identify when regulation occurs, we can uselinearizations of the population dynamics when the population is at either low or high populationdensities. In the presence of negative density dependence, the per-capita growth rate of the populationis greatest at low densities. Hence, for the population to tend away from extinction, it should increasewhen at low densities. So,consider that n0 ≈ 0 but not zero. Then as long nt remains at low densities,we can approximate the dynamics by

nt+1 ≈ ntf(0, ξt+1)

We know from the previous chapter that for this stochastic linear difference equation r = E[log f(0, ξ1)]determines whether the population has a tendency to increase (i.e. r > 0) or decrease (i.e. r < 0).

To know whether the population tends to decrease at high densities, we can define

f(∞, ξ) = limn→∞

f(n, ξ)

for all ξ ∈ E . This limit is well defined (possibly 0) as f is a decreasing function of n for all ξ. To avoidan absurd heuristic, lets assume that this limit is strictly positive for all ξ. If nt is sufficiently large,

1Need to see if we can extend the theorems in this section to the stationary case

35

1. UNSTRUCTURED MODELS 36

then we can approximate the dynamics by the linear stochastic difference equation

nt+1 ≈ ntf(∞, ξt+1).

In this case, we expect that

s := E[log f(∞, ξ1)](possibly −∞) determines whether the population continues to increase (i.e. s > 0) or starts to decrease(i.e. s < 0).

Based on these linear approximations, we have three scenarios. First, if r < 0, then under the bestconditions (i.e. low densities) the population is decreasing and, consequently, we expect the populationto tend toward extinction. Second, if s > 0, then under the worst conditions (i.e. high densities) thepopulation is increasing and, consequently, we expect the population to grow without bound. Finally,if r > 0 and s < 0, then the population tends to increase when rare and decrease when very common.Hence, in this case, we expect the population to persist in manner that doesn’t cause madness. Indeed,the following theorem justifies these conclusions.

Theorem 3.1. Schreiber [2012] Assume f(n, ξ) is a positive decreasing function in n for all ξ ∈ Eand E[log+ f(0, ξt)] <∞. Then

Extinction: if r = E[log f(0, ξ1)] < 0, then limt→∞ nt = 0 with probability whenever n0 ≥ 0,Unbounded growth: if s = limn→∞ E[log f(n, ξ1)] > 0, then limt→∞ nt = ∞ with probability

whenever n0 > 0, andStochastic persistence: if r > 0 and s < 0, then for all ε > 0 there exists M > 0 such that

lim inft→∞

#{1 ≤ s ≤ t : 1/M ≤ ns ≤M}t

≥ 1− ε with probability one

whenever n0 > 0.

Stochastic persistence, the final case, implies that the typical trajectory spends most of its time ina sufficiently large compact interval excluding the extinction state 0. In the words of Peter Chesson[1982]

This criterion requires that the probability of observing a population below any givendensity, should converge to zero with density, uniformly in time. Consequently itplaces restrictions on the expected frequency of fluctuations to low population levels.Given that fluctuations in the environment will continually perturb population densi-ties, it is to be expected that any nominated population density, no matter how small,will eventually be seen. Indeed this is the usual case in stochastic population modelsand is not an unreasonable postulate about the real world. Thus a reasonable per-sistence criterion cannot hope to do better than place restrictions on the frequencieswith which such events occur.

The proofs for the first two cases of Theorem 3.1 are fairly straightforward. Consequently, we discussthem briefly. Suppose that r < 0. As f(n, ξ) is a decreasing function of n, we have f(0, ξ) ≥ f(n, ξ)for any n ≥ 0. Hence,

nt+1 ≤ ntf(0, ξt+1)

for all t. Iterating this inequality yields

nt ≤ n0f(0, ξ1) . . . f(0, ξt)

Hence, the law of large numbers implies

lim supt→∞

1

tlog nt ≤ lim

t→∞

1

t

t∑s=1

log f(0, ξs) = r < 0 with probability one.


0 20 60 100

510

1520

2530

N

S

0 10 30 50

01

23

45

σ = 0

timede

nsity

0 10 30 50

01

23

45

σ = 0.1

time

dens

ity

Figure 1. Grain beetle dynamics. Left panel shows fitting of the survivorship curvea/(1 + bnc) to survivorship data of the flour beetle species Tribolium castaneum: a =0.8, b = 0.0148920, and c = 4.2470522. The right hand panels show simulations withr = µ+ log 0.8 = 0.1 and σ = 0 (middle), σ = 0.1 (right)

Thus nt converges to zero exponentially fast with probability one. In the exercises, you are asked toprove the complementary result in the case that s > 0.

To illustrate Theorem 3.1, we apply it to stochastic versions of the Ricker model, a seed-bank modelfor annual plants, and a model of checkerspot butterflies.

Example 3.1. (Generalized Beverton-Holt model) Bellows [1981] used a model of the form f(n, ξ) =λ a/(1 + bnc) where ξ = (λ, a, b, c) to described the demography of various species of grain beetles. Inthis model, a > 0 is the probability of an individual surviving to adulthood and reproducing, λ > 0corresponds to the low-density reproductive output of a reproducing individual, b > 0 measures thestrength of competition, and c > 0 determines the strength of density-dependence (e.g. b ≤ 1 is under-compensatory and b > 1 is over-compensatory). Suppose λt are log-normally distributed with log meanµ and log standard deviation σ, but a, b and c are constant. The low-density per-capita growth rate isr = µ+ log a. The high-density per-capita growth rate is (check for yourself!) s = −∞.

Figure 1 shows a fitting of the survivorship curve a/(1 + bnc) to survivorship data of the flourbeetle species Tribolium castaneum: a = 0.8, b = 0.0148920, and c = 4.2470522. Hence, provided thatµ > − log 0.8, the corresponding population dynamics model would exhibit persistence. The right handpanels of Figure 1 show simulations with r = µ+ log 0.8 = 0.1 and σ = 0 (middle), σ = 0.1 (right).

Example 3.2. (An annual plant model) Consider an annual plant where n is the density of seeds inthe ground. Each year a fraction g of seeds germinate. Of the seeds that don’t germinate, a fraction ssurvive to the next year. At low densities, germinating seeds produce a yield of Y seeds. Competitionamong st germinating individuals reduces yield by a factor of 1/(1 +agn) where a > 0 is a competitioncoefficient. Then our model is given by

f(n, ξ) =gY

1 + agn+ (1− g)s where ξ = (a, g, s, Y )

The low-density and high density per-capita growth rates are given by

r = E[log f(0, ξ1)] = E[log(g1Y1 + (1− g1)s1)] and s = limn→∞

E[log f(n, ξ1)] = E[log(1− g1)s1] < 0

As the latter quantity s is always negative, this model always exhibits bounded dynamics. Persistencerequires that r is positive.


0 20 40 60 80 100

030

00

years

seed

den

sity

Figure 2. Annual plant dynamics with Yt = 20 with probability 0.5 and 0 otherwise,g = 0.2, s = 0.9, and a = 0.01.

Consider the special case in which only Y fluctuates between Ymax and 0 with probabilities p and1− p respectively and the remaining parameters are constant. One could view this as each year beingeither a drought year or high rainfall year. Under these conditions, we get

r = p log(gYmax + (1− g)s) + (1− p) log((1− g)s).

Figure 2 illustrates the dynamics from this model when r > 0.

Now lets look at an application of these models to a real ecological puzzle.

Example 3.3. (Extinction of checkerspot butterflies) In the 1990s, two populations of checkerspotbutterflies went extinct in Northern California. The population densities for one of these populationsis shown in the left hand side of Figure 3. These extinctions were observed to coincide with a changein precipitation variability in the 1970s. As shown in the right hand side of Figure 3, the standarddeviation in precipitation was approximately 50% higher after 1971 than before 1971.

To evaluate whether this shift in precipitation variability may have caused the extinction of thecheckerspots, McLaughlin et al. [2002] developed a stochastic difference equation of the following type

nt+1 = nt exp(a− bnt + cξ−2t+1)

where ξt is precipitation in year t. Using linear regression on a log-scale yields a model whose fit forone-year predictions are shown as red diamonds in Figure 3.

The model with random draws from the pre-1971 precipitation data yields r = E[a+ cξ−21 ] = 0.04.Hence, the population is expected to persist with this form of climatic variability as illustrated inthe simulations on the left hand side of Figure 5. The model with random draws from the post-1971precipitation data yields r = −0.049. Hence, the population is expected to tend toward extinction withthis form of climatic variability as illustrated in the simulations on the right hand side of Figure 5.

Running the two scenarios and comparing extinction risk.

1.2. Positive density-dependence. Lets now assume that f(n, ξ) is an increasing function ofdensity i.e. f exhibits positive density dependence. Common causes of this positive density-dependentfeedback include predator saturation, cooperative predation, increased availability of mates, and con-specific enhancement of reproduction [Courchamp et al., 1999, Stephens et al., 1999, Gascoigne andLipcius, 2004, Courchamp et al., 2008, Gascoigne et al., 2009, Kramer et al., 2009].


1960 1970 1980 1990

110

1000

year

dens

ity

datamodel

1930 1950 1970 1990

100

300

500

700

year

prec

ipita

tion

(mm

) SD=105 SD=157

Figure 3. Checkerspot population dynamics (left) and precipitation (right). Model fitfor population dynamics as red diamonds.

0 50 150

110

0

pre−1971 rainfall

year

dens

ity

0 50 150

110

0

post−1971 rainfall

year

dens

ity

Figure 4. Simulated checkerspot population dynamics with pre-1971 precipitation data(left) and post-1971 precipitation data (right)

To understand the dynamics for models with positive density-dependence, we can, once again, definethe low-density and high-density per-capita growth rates, r and s, as in the case of negative density-dependence. However, unlike the negative density-dependent case, we have s > r. The signs of r and sdetermine three scenarios. If r > 0, then the population tends to increase under the worst conditionsand therefore the population should increase without bound. If s < 0, then the population tends todecrease under the best conditions and, consequently, it should asymptotically go extinct. Finally, ifr < 0 and s > 0, then depending on initial conditions we may expect that asymptotic extinction orblow might happen.


0 50 150

110

0pre−1971 rainfall

year

dens

ity

0 50 150

110

0

post−1971 rainfall

year

dens

ityFigure 5. Simulated checkerspot population dynamics with pre-1971 precipitation data(left) and post-1971 precipitation data (right)

To state a theorem about this trichotomy, we say that {0,∞} is accessible from a set B ⊂ (0,∞)if for any M > 0, there exists γ > 0 such that

P [{ nt ∈ [0, 1/M ] ∪ [M,∞) for some t ≥ 0} | n0 = x] > γ

for all x ∈ B. Intuitively, accessibility of 0 and ∞ from B means that if n0 lies in B then there is apositive probability that it gets arbitrarily small or arbitrary large after enough time.

Theorem 3.2. Assume f(n, ξ) is an increasing function of n for all ξ ∈ E. Define s = limn→∞ E[log f(n, ξ1)]and r = E[log f(0, ξ1)]. Then

Extinction: if s < 0, then limt→∞ nt = 0 with probability one whenever n0 ≥ 0.Unbounded growth: if r > 0, then limt→∞ nt =∞ with probability one whenever n0 > 0.Conditional persistence: if r < 0 and s > 0, then for any 0 < δ < 1, there exist m,M > 0

such that

P[

limt→∞

nt =∞∣∣∣n0 = x

]≥ 1− δ and P

[limt→∞

nt = 0∣∣∣n0 = y

]≥ 1− δ,

for all x ∈ [M,∞) and all y ∈ (0,m].Moreover, if {0,∞} is accessible, then

P[{

limt→∞

nt = 0 or ∞} ∣∣∣n0 = x

]= 1

for all x ∈ (0,∞).

The final statement of the theorem about conditional persistence has three implications. First,if the initial population density is sufficiently small, then extinction is highly likely. Second, if theinitial population density is sufficiently large, then unbounded growth is highly likely. Third, if theenvironmental stochasticity pushes population densities eventually to either high or low densities (i.e.accessibility of {0,∞}), then extinction or growth without bound occurs with probability one.

While models with positive density-dependence aren’t realistic in the long term (negative density-dependence must ultimately kick in), they are still useful as means to understand how positive density-dependence influences the establishment of populations. The next example illustrates their use froman empirically inspired study.


0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.5

1.0

1.5

2.0

Allee threshold C

SD

of n

oise

σ

Figure 6. Probability of n100 > 1000 for nt+1 = nt exp(0.1(nt − C) + ξt+1 where ξt arenormal with mean 0 and standard deviation σ and n0 = 5.

Example 3.4. (Allee effects in Gypsy moths) Liebhold and Bascompte [2003] used models withonly positive density-dependence to examine numerically the joint effects of Allee effects, environmentalstochasticity, and externally imposed mortality on the probability of successfully exterminating aninvasive species such as the gypsy moth. Their fitness function was

f(x, ξ) = exp (γ(x− C) + ξ)

were C is the deterministic Allee threshold, γ is the “intrinsic rate of natural increase,” and ξ arechoosen to be normal random variables with mean 0.

By assuming that ξt are normally distributed, {1,∞} is accessible from any initial positive density.Furthermore, as r = E[−γC + ξt] = −γC < 0 and s = +∞ for this model, our results imply bothextinction and unbounded growth occur with positive probability and, thereby, provide a rigorousmathematical foundation for Liebhold and Bascompte [2003]’s numerical analysis. Figure 6 recreatesone of the figures from their paper. The paper estimates the probability of establishment (i.e. nt →∞)for a range of threshold values and noise levels. Quite interestingly, this figure illustrates that higherlevels of environmental stochasticity can increase the probability of establishment when the initialdensity lies below the threshold value (i.e. values of C ≥ 5 in Figure). Conversely, environmentalstocahsticity reduces the probability of establishment for initial densities greater than the thresholddensity (i.e. values of C < 5 in the Figure).

2. STRUCTURED MODELS 42

0 10 20 30 40 50

050

100

150

σ = 0.01

time

dens

ity

0 10 20 30 40 50

050

100

150

σ = 0.05

timede

nsity

Figure 7. Simulations of the sterile insect release model with µ = 0.1 and S = 100.Dashed line indeicates the density at which the geometric mean of the fitness is one.

Example 3.5. (Sterile insect releases) Let nt be the density of females of an insect pest, S thenumber of sterile males, and ξt the number of daughter produced by a mated male. Lets assume ξt arelog-normally distributed with log mean µ and log standard deviation σ. If we assume there is a 50-50sex ratio and mating is panmetic, then

nt+1 = ntξt+1nt

nt + S

where ntnt+S

is the probability of a female mating a non-sterile male. Then f(n, ξ) = ξn/(n + S),

r = E[log 0] = −∞ and s = E[log ξt] = µ. As ξt are log normally distributed, {0,∞} are accessible.Thus, if µ > 0, there is a positive probability of converging to 0 or ∞ for any positive initial densityn0.

For this model, geometric mean of fitness at density n is eµn/(n + S). When this mean equals1, the population has no inherent tendency to increase or decrease. Indeed, in the absence of noise,this critical density, n∗ = S/(eµ − 1), is an equilibrium of the deterministic model. In the absence ofnoise, nt converges to zero whenever n0 < n∗. In the presence of noise, however, nt may grow withoutbounded even if n0 < n∗. Hence, environmental stochasticity can rescue populations expected to goextinct (Figure 7).

2. Structured models

To account for population structure, let nt = (n1,t, n2,t, . . . , nk,t) be vector of densities of individualsin k different stages or states at time t. Let A(n, ξ) be k×k non-negative matrices that vary continuouslywith n and ξ. Then the population dynamics are given by

nt+1 = ntA(nt, ξt+1)


where ξ1, ξ2, ξ3, . . . are an ergodic stationary sequence of random variables on a separable metric spaceE . As in the structured case, we will be interested in the conditions that ensure a population is regulatedi.e. tends to stay bounded away from extinction and infinitely large population densities.

2.1. Bounded population dynamics. Unlike the unstructured case, there is no unique way todefine the population growth rate at infinity i.e. there are many ways that to approach an infinitepopulation size. Hence, we need another approach to ensure that populations tend to remain bounded.To this end, we introduce functions, V : Rk

+ 7→ R, that become arbitrarily large for arbitrarily largepopulation sizes, and such that log V on average decreases when the population gets large. For thefirst requirement, we require that V is proper i.e lim∑

i ni→∞ V (n) = ∞. For example, any weighedcombination of the population densities, V (n) =

∑iwini is proper with wi > 0, is proper. The following

theorem clarifies what we mean by log V decreasing on average for large population sizes.

Theorem 3.3. Benaım and Schreiber [2009] If there exists a proper function V : Rk+ 7→ R and

non-negative, random variables α, β : E → R such that

(1) V (nA(n, ξ)) ≤ α(ξ)V (n) + β(ξ);(2) E[logα] < 0;(3) E[log+ β] <∞

then there exists C > 0 such that

(3.2) lim inft→∞

P[∑i

ni,t ≤M ] ≥ 1− C

M

and

(3.3) lim inft→∞

#{1 ≤ s ≤ t :∑

i ni,t ≤M}t

≥ 1− C

Mwith probability one.

The condition (3.2) corresponds to what Chesson and Ellner call stochastically bounded from above [Ches-son, 1978, 1982, 1984, Chesson and Ellner, 1989, Ellner, 1984].

Example 3.6. (Metapopulations revisited) Consider a k-patch model to one where offspring disperseand adults are sedentary (i.e. don’t disperse) and iteroparous (i.e. potentially reproduce multipletimes in their life time). Let ni,t be the population density in patch i, with i = 1, 2, . . . , k. Letξi,t+1 exp(−aini,t) be the number of offspring produced per in patch i in year t. A fraction Dij ofthe offspring in patch i disperse to patch j. After reproducing, adults survive to the next year withprobability si in patch i. Hence, the population dynamics are given by

nt+1 = nt diag(ξ1,t+1 exp(−a1n1,t), . . . , ξk,t+1 exp(−aknk,t))D + diag(s1, . . . , sk)︸︷︷︸=A(nt,ξt+1)

To check stochastic boundedness, let V (n) =∑

i ni be the total population size. For non-negative x,the maximum of x exp(−ax) is 1/(ae) at x = a. Therefore,

V (nA(n, ξ)) =k∑i=1

niξi exp(−aini) + sini

≤k∑i=1

ξi/(aie) + maxi{si}V (n)

Define α(ξ) = maxi{si} and βi(ξ) =∑

i ξi/(aie). As E[logα(ξ)] = log maxi si < 0, Theorem 3.3 appliesprovided that E[log+∑

i ξi/(aie)] <∞. This latter requirement is true for many distributions used inecological models e.g. log-normally or gamma distributed ξi,t or any finite distribution of ξi values.


Example 3.7. (A stochastic Leslie matrix model) Let ni be the density of individuals in age classi. Let si < 1, for i = 2, . . . , k, be the probability of individuals of age i − 1 surviving to age i. Now,suppose that “size” of an individual of age i is wi > 0, and the amount of resources acquired by anindividual of age i is proportional to wi. Assuming that resource availability decreases with the total“size”

∑iwini of the population, we can model the per-capita fecundity of individuals of age i by

wiξ/(1 +∑

iwini). Under these assumptions, we have

A(n, ξ) =

w1ξ1+

∑i wini

s2 0 0 . . . 0w2ξ

1+∑i wini

0 s3 0 . . . 0...

...... 0 . . .

...wk−1ξ

1+∑i wini

0 0 . . . 0 skwkξ

1+∑i wini

0 0 . . . 0 0

Lets consider the case that the ξi are log-normally distributed with log means µi and log standarddeviations σi. Define V (n) =

∑iwini, s = max2≤i≤k si. Then

V (nA(n, ξ)) = w1ξk∑i=1

wini1 +

∑iwini

+k∑i=2

wisini−1

≤ w1ξV (n)

1 + V (n)+ sV (n)

Suppose there is v∗ > 0 such that E[log(w1ξt/(1 + v∗) + s)] < 1 e.g ξt is log-normally distributed oris drawn from a finite, discrete distribution. Theorem 3.3 applies for α(ξ) = w1ξ/(1 + v∗) + s andβ(ξ) = w1ξ. (verify yourself!)

2.2. Stochastic persistence and extinction. To understand when populations are likely topersist, we can examine the population growth rate when rare. Namely, if n0 ≈ 0 but not zero, thennt+1 ≈ ntA(0, ξt+1) which implies

nt ≈ n0A(0, ξ1)A(0, ξ2) . . . A(0, ξt)

provided that n0, n1, . . . , nt−1 remain close to zero. For this linear approximation, the Random PerronFrobenius Theorem in Chapter 2 implies that the population tends grow exponentially at a rate r whichsatisfies

r = limt→∞

1

tlog ‖A(0, ξ1) . . . A(0, ξt)‖ with probability one

where ‖A‖ = maxij |Aij|. Hence, if r > 0, then the population increases when rare and we expect itto persist. Indeed, one can prove the following theorem which generalizes results from [Benaım andSchreiber, 2009, Roth and Schreiber, 2014].

Theorem 3.4. 2 Assume

• A(n, ξ) depend continuously on n and ξ and are primitive (i.e. there exists s > 0 such thatA(n1, ξ1) . . . A(ns, ξs) has strictly positive entries with probability one for any n1),• the conditions of Theorem 3.3 hold for some choice of V , α, and β, and• E[log+ ‖A(0, ξ1)‖] <∞ so that r is well-defined.

If r > 0, then nt+1 = ntA(nt, ξt+1) is stochastically persistent and bounded: for all ε > 0 there existsM > 0 such that

lim inft→∞

#{1 ≤ s ≤ t : 1/M ≤ mini ni,s and maxi ni,s ≤M}t

≥ 1− ε with probability one

2This ”theorem” is currently a conjecture. Special cases of this theorem are proved in [Benaım and Schreiber, 2009,Roth and Schreiber, 2014].

3. ERGODICITY AND STATIONARY DISTRIBUTIONS 45

0 50 150 250

020

040

0d = 0

time

patc

h de

nsiti

es

0 50 150 250

020

040

0

d = 0.25

timepa

tch

dens

ities

Figure 8. Persistence of coupled sink patches.

whenever n0 > 0. If r < 0, then the extinction state is a stochastic attractor: for all ε > 0, there existsδ > 0 such that

P[

limt→∞

nt = 0∣∣∣n0

]≥ 1− ε

whenever∑

i ni,0 ≤ δ. Furthermore, if {0} is accessible (i.e. for all δ > 0 and n0 > 0, P[∑

i ni,t ≤δ for some t|n0] = 1), then limt→∞ nt = 0 with probability one.

In the case of r < 0 without additional assumptions (e.g. accessibility of 0), asymptotic extinction ishighly likely only if the initial, total population density is sufficiently small. Accessibility ensures thatpopulation densities get sufficiently low, sufficiently frequently, that the process tends exponentiallytoward the origin. In the exercises, you are asked to show that r < 0 for models with only negativedensity-dependence (with or without the accessibilty assumption) imply that limt→∞ nt = 0 withprobability one.

Example 3.8. (Return of the metapopulation model) Consider the k patch model presented inExample 3.6. This model satisfies the assumptions of Theorem 3.4, for example, if the ξi,t are log-normally distributed. Moreover, as the entries of A(n, ξ) are non-increasing functions of the entries ofn, r < 0 implies that the meapopulation converges exponentially to extinction for all initial conditions.In Example 2.9, we found a first order approximation of r for the case of k = 2 patches and non-overlapping generations i.e. si = 0 for all i. This approximation showed that r could be positiveeven if the expected fitness in every patch was less than one. Specifically, this occurred when patcheswere sufficiently uncorrelated in space, positively auto-correlated in time, and partially connected bydispersal. Schreiber [2010] shows that similar conclusions hold for any number of patches. Figure 8illustrates this phenomena for a system with 10 patches: convergence to extinction when there is nodispersal (left), and stochastic persistence when one quarter of the individuals disperse (right).

3. Ergodicity and stationary distributions

When r > 0, Theorem 3.4 implies that fraction of time spent at arbitrarily low densities is arbitrarilysmall. This theorem, however, does not ensure that there is a unique positive stationary distribution.For this stronger conclusion, we need either additional structure on the nonlinearities or on the thenoise of the system.


3.1. Irreducibility in i.i.d. environments. Irreducibility occurs when for all initial conditions,there is a tendency to visit the same set of states. More precisely, given ε > 0, we say that {nt} isirreducible over (ε,∞)k if there exists a probability measure Φ on (ε,∞)k such that for all n0 ∈ (ε,∞)k

and every Borel set A ⊂ (ε,∞)k there exists T ≥ 1 (depending on n0 and A) such that

P[nT ∈ A|n0] > 0

whenever Φ[A] > 0.Under the assumption of irreducibility and an i.i.d. environment, r > 0 implies the existence of a

unique stationary distribution.

Theorem 3.5. (Uniqueness) Assume that {nt} is irreducible over (ε,∞)k for all sufficiently smallε > 0. If r > 0, then there exists a unique invariant probability measure π such that π({0}) = 0 and

limt→∞

1

t

t∑s=1

h(ns) =

∫h(n)µ(dn) with probability one

for n0 > 0 and all continuous h : Rk+ → R with

∫|h(n)|µ(dn) <∞.

Theorem 3.5 ensures the asymptotic distribution of one realization of the population dynamics isgiven by the positive stationary distribution π. Hence, π provides information about the long-termfrequencies that a population trajectory spends in any part of the population state space.

To gain information about the distribution of nt across many realizations of the population dy-namics, we need a stronger irreducibility condition. This stronger condition requires that after a fixedamount of time independent of initial condition, any positive population state can move close to anyother positive population state. More precisely, we say that {nt} is strongly irreducible over (ε,∞)k ifthere exists a probability measure Φ on (ε,∞)k, an integer T ≥ 1 and some number 0 < h ≤ 1 suchthat for all n0 ∈ (ε,∞)k and every Borel set A ⊂ (ε,∞)k

P[nT ∈ A|n0] ≥ hΦ(A).

To state the next result given probability measures µ, ν on (0,∞)k define

‖µ− ν‖ = supB|µ(B)− ν(B)|

where the supremum is taken over all Borel sets B ⊂ (0,∞)k.

Theorem 3.6. (Convergence in distribution) In addition to the assumptions of Theorem 3.5 assumethat {nt} is strongly irreducible over (ε,∞)k for all sufficiently small ε > 0. Then the distribution ofnt converges to π as t→∞ whenever n0 ∈ (0,∞)k; that is

limt→∞‖P[nt ∈ ·|n0]− π‖ = 0 for all n0 ∈ (0,∞)k.

Example 3.9. (The LPA model) To study the dynamics of flour beetles (genus Tribolium), ? builta stochastic, nonlinear, stage-structured model. For these pests of grains, individuals are in one ofthree stages: larvae (Lt), pupae (Pt), and adults (At). In each week, each adult, on average, produceb eggs of which a fraction exp(−c1At − c2Lt) escape cannabilism and become larvae. Each week, µ1

of larvae survive and become pupae, and a fraction exp(−c3At) of pupae survive to become adults.Adults survive to the next week with probability µ2. This yields the deterministic model

Lt+1 = bAt exp(−c1At − c2Lt)Pt+1 = Lt(1− µ1)

At+1 = Pt exp(−c3At) + (1− µ2)At


0.0 0.2 0.4 0.6 0.8 1.0

050

100

adult mortality

larv

al d

ensi

ty

0.0 0.2 0.4 0.6 0.8 1.0

050

150

adult mortality

Figure 9. LPA bifurcation diagram with and without noise.

To account for stochastic fluctuations, ? multiplied each of the equations by exp(ξi,t+1) where ξt =(ξ1,t, ξ2,t, ξ3,t) is a multivariate normal random variable with mean 0 and covariance matrix Σ. Thisleads to a stochastic difference equation of the form

(Lt+1 Pt+1 At+1

)=(Lt Pt At

) 0 (1− µ1) exp(ξ2,t+1) 00 0 exp(−c3At + ξ3,t+1)

b exp(−c1At − c2Lt + ξ1,t+1) 0 (1− µ2) exp(ξ3,t+1)

In the exercises, you are asked to verify that (1) these equations are stochastically bounded, (2) theyare strongly irreducible, and (3) there exists a critical b below which the population goes extinct andabove which it persists.

3.2. Compensatory negative density-dependence. Independent of the noise structure, thereexist assumptions on the nonlinearities ensuring the existence of a unique stationary distributions.Biologically, these assumptions corresponds to the populations experiencing compensating, negativedensity-dependence.

The first of the additional assumptions is purely technical, but is meet for most models.

A1: Smoothness: The map (n, ξ) → A(n, ξ) is Borel, Fξ(n) = nA(n, ξ) is twice continuousdifferentiable for all ξ, n ∈ Rk

+, and

E

(sup‖n‖≤1

log+(‖Fξ(n)‖+ ‖DFξ(n)‖+ ‖D2Fξ(n)‖

))< +∞

To account for competition, we assume all of the entries should be non-increasing functions of thepopulation densities. Moreover, for each population state, we assume one of its entries is a decreasingfunction of its density or the density of one the other population states.

A2: Negative density-dependence: The matrix entries Aij(n, ξ) satisfy

∂Aij∂nl

(n, ξ) ≤ 0

for all n and ξ. Moreover, for each i there exists some j and l such that this inequality is strictfor all n and ξ.

Finally, we assume that the density-dependence is compensatory.

4. EXERCISES 48

A3: Compensatory negative density dependence: All entries of the derivative DFξ(n) ofFξ(n) are non-negative for all n and ξ.

This assumption ensures that the population dynamics are monotone. In other words, if n ≥ n, thenFξ(n) ≥ Fξ(n).

Theorem 3.7. (Benaım and Schreiber [2009]) If assumptions A1-A3 hold and r > 0, then thereexists a unique stationary distribution π, such that π[{0}] = 0,

limt→∞

1

t

t∑s=1

h(ns) =

∫h(n)π(dn) with probability one

andlimt→∞‖P[nt ∈ ·|n0]− π‖ = 0

for n0 > 0 and all continuous h : Rk+ → R with

∫|h(n)|π(dn) <∞.

Example 3.10. (Revenge of the Leslie matrix models) Verify for yourself that Example 3.7 satisfiesthe assumptions of the Theorem.

4. Exercises

(1) Prove the second statement of Theorem 3.1.(2) Construct an example of a model where f(n, ξ) is strictly decreasing with n but limt→∞ nt =∞

with probability one whenever n0 > 0.(3) Consider nt+1 = ntA(nt, ξt+1) which satisfies the first and third assumption of Theorem 3.4.

(a) Assume Aij(n, ξ) are non-increasing functions of the coordinates of n. Prove if r < 0,then limt→∞ nt = 0 with probability one.

(b) Assume Aij(n, ξ) are non-decreasing functions of the coordinates of n. Prove if r > 0,then limt→∞ nt =∞ with probability one.

CHAPTER 4

Multispecies models

The diversity of the plankton [is] explicable primarily by a permanent failure to achieveequilibrium as the relevant external factors changes. – G.E.Hutchinson [1961]

Nearly all species interact with other species. These interactions can be classified by the net effect ofeach species on the average fitness of the other species. When the net effects are mutually harmful, thespecies are exhibiting competitive interactions. Mutually beneficial interactions occur for mutualists.Finally, if one species benefits while the other is harmed, then the species are likely predators and theirprey, or pathogens/parasites and their hosts. Intuitively, we expect competitive interactions to inhibitthe persistence of one or both of the competitors, mutualistic interactions to facilitate persistenceof each species, and predator/prey or host/parasite interactions to facilitate persistence of one andinhibit persistence of the other. A basic issue of practical and theoretical interest is unraveling howenvironmental fluctuations interact with these nonlinear feedbacks to determine the composition ofecological communities and their dynamics. We examine some of these issues in this chapter.

1. The lottery model

To get a feeling for some basic principles underlying the analysis of multi-species models, we beginwith the lottery model. Introduced by Chesson and Warner [1981], this model examines the dynamicsof two species competing for space. Space that is so limiting that it is usually fully occupied by thecompeting species. Let ni,t denote the fraction of space occupied by species i. At the beginning of yeart+ 1, each individual of species i produces ξi,t+1 offspring and a fraction δ (independent of species forsimplicity) of the reproducing adults die, thereby freeing up space. Each offspring is as likely as theother to “capture” one of the freed up locations. The fraction of freed up space captured by species1 is equal to the fraction of the total offspring (ξ1,t+1n1,t+1 + ξ2,t+1n2,t+1) produced by the communitythat belong to species 1:

ξ1,t+1n1,t+1

ξ1,t+1n1,t+1 + ξ2,t+1n2,t+1

Under these assumptions, the competitive dynamics are

(4.1)

n1,t+1 = (1− δ)n1,t + δξ1,t+1n1,t+1

ξ1,t+1n1,t+1 + ξ2,t+1n2,t+1

n2,t+1 = (1− δ)n2,t + δξ2,t+1n2,t+1

ξ1,t+1n1,t+1 + ξ2,t+1n2,t+1

While this models appears to be two dimensional, in fact, it is one dimensional as n1,t+n2,t = 1 (Fig. 1).We begin by examining the dynamics of this model in two special cases. First, lets assume that

ξi,t = ξi are constant and positive. Then (see the exercises) one can prove that the species with thelarger ξi value, say species 1, excludes the other i.e.

limt→∞

nt = (1, 0)

whenever n1,0 > 0.For the second case, we assume ξi,1, ξi,2, . . . are ergodic stationary sequences of positive random

variables, and δ = 1 i.e. all individuals die after giving birth (semelparity). If we define xt = n1,t/n2,t

49

1. THE LOTTERY MODEL 50

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.4

0.8

n1

n 2

Figure 1. State-space for the lottery model n1 +n2 = 1. (1, 0) and (0, 1) correspond toequibria of the model due to the no cats, no kittens principle.

(assuming n2,t > 0), then yt satisfies the linear, stochastic difference equation

xt+1 =ξ1,t+1

ξ2,t+1

xt

Hence,

log xt = log x0 +t∑

s=1

logξ1,sξ2,s

Provided that E[log+ ηt] < 0, the Birkhoff Ergodic Theorem implies

limt→∞

1

tlog xt = E[log ξ1,t − log ξ2,t]

with probability one. Hence, the species with the larger geometric mean of ξi,t displaces the otherspecies.

The first special case is an example of Gauses’ competitive exclusion principle: two species compet-ing for the same resource in the same way can not coexist at equilibrium. Due to the fact that manyspecies do coexist despite competing, ecologists have been studying coexistence mechanisms for nearlya century. As stated in the quote by Hutchinson at the beginning of this chapter, fluctuations in envi-ronmental conditions might allow for coexistence especially if different species are favored at differenttimes. The second special case of the lottery model shows that fluctuations, in and of themselves, areinsufficient to mediate coexistence if the populations have non-overlapping generations. This raises thequestion, can coexistence occur if there are fluctuations and overlapping generations?

To answer this question, we can, as in the single species models, look at what happens when eachspecies becomes rare. When species 1 is rare, we have n1,t ≈ 0 and n2,t ≈ 1 and we can approximate

1. THE LOTTERY MODEL 51

the dynamics of species 1 by the stochastic linear difference equation

n1,t+1 ≈ (1− d)n1,t + dξ1,t+1ntξ2,t+1

=

((1− d) + d

ξ1,t+1

ξ2,t+1

)n1,t

Hence, the per-capita growth rate of species 1 when rare is

r1 = E[log

(1− d+ d

ξ1,tξ2,t

)]Similarly, the per-capita growth rate of species 2 when rare is

r2 = E[log

(1− d+ d

ξ2,tξ1,t

)]Intuitively, we would expect that if r1 > 0 and r2 > 0 (i.e. both species tend to increase when rare),then the species should coexist. Alternatively, if r1 < 0 or r2 < 0, then the species with the negativeri are prone to extinction whenever they become rare. Indeed, one can prove the following theoremusing methods from the single species models (see Exercises). Chesson [1982] provided the first proofof several of the statements in this Theorem.

Theorem 4.1. If r1 > 0 and r2 > 0, then the species coexistence in the sense of stochastic persis-tence in distribution and empirical measures i.e. for all ε > 0 there exists δ > 0 such that

P[n1,t ∈ (δ, 1− δ)|n0] ≥ 1− εwhenever n1,0 ∈ (0, 1) and t is sufficiently large, and

lim inft→∞

#{1 ≤ s ≤ t : n1,t ∈ (δ, 1− δ)t

≥ 1− ε

with probability one whenever n1,0 ∈ (0, 1).If ri < 0, then for all ε > 0, there exists δ > 0 such that

P[ limt→∞

ni,t = 0|n0] = 1

whenever ni,0 ≤ δ.

But can r1 and r2 both be positive? In general understanding the sign of the ri can be challenging.However, in the special case that the ξi,t are log-normally distributed and d is small, explicit inter-pretable expressions are possible. To see why, let ri(d) denote ri as a function of d. These functions(at least for nice ξt) are analytic. We have (check for yourself!) ri(0) = 0 and

r′i(0) = E[ξi,tξj,t

]− 1 where j 6= i

Hence, if E[ξi,tξj,t

]> 1, then ri(d) > 0 for sufficiently small d.

Now, assume that ξi,t are log-normally distributed with log means µi, log variances σ2i and correlation

ρ. The first term of r′1(0), E[ξi,tξj,t

], is the expectation of a log-normal random variable with log mean

µ1 − µ2 and log variance σ21 − 2ρσ1σ2 + σ2

2. Hence,

r′1(0) = exp(µ1 − µ2 + σ21/2− ρσ1σ2 + σ2

2/2)− 1

This term is positive if

σ21/2− ρσ1σ2 + σ2

2/2 > µ2 − µ1

Similarly, r′2(0) is positive if

σ21/2− ρσ1σ2 + σ2

2/2 > µ1 − µ2

2. TWO SPECIES COMPETITION 52

0 50 150

0.2

0.6

time

dens

ities

0 50 150

0.0

0.4

0.8

timede

nsiti

es

Figure 2. Dynamics of coexistence and exclusion for the lottery model

Equivalently, both terms are positive if

σ21/2− ρσ1σ2 + σ2

2/2 > |µ1 − µ2|which occurs if (i) the variances are larger relative to the mean log fitness difference |µ1 − µ2| and (ii)if the correlation between the two species fitnesses isn’t too close to one.

Thus, our analysis has identified two of the key ingredient’s of the “storage effect” due to Chessonand Warner [1981]. Namely, coexistence occurs if each species has a period in which it outperformsthe other (i.e. the fitness variance is sufficiently high and the correlation is sufficiently low), and eachspecies can store the gains during these favorable periods until the next favorable period (i.e. d issufficiently low). The third key ingredient is “hidden” in this model but is uncovered in the nextsection. Figure 2 illustrates coexistence (left) and exclusion dynamics (right). Figure 3 illustrates howthe conditions for coexistence depend on d and σ2

i = σ2 for i = 1, 2. Consistent with our analysis,coexistence requires that σ2 is sufficiently large and d is sufficiently small.

2. Two species competition

And NUH is the letter I use to spell Nutches, Who live in small caves, known asNiches, for hutches. These Nutches have troubles, the biggest of which is the factthere are many more Nutches than Niches. Each Nutch in a Nich knows that someother Nutch Would like to move into his Nich very much. So each Nutch in a Nich hasto watch that small Nich or Nutches who haven’t got Niches will snitch. –Dr. Seuss,On Beyond the Zebra

The lottery model identifies that under certain conditions, temporal fluctuations can allow for coexis-tence. A key component of this coexistence mechanisms is that there are years (i.e. due to differentenvironmental conditions) where each species does better than the other species. As suggested by thequote of Dr. Seuss, this temporal partitioning of good years amongst the species can be viewed as each


−0.10

−0.05

0.00

0.05

0.1 0.2 0.3 0.4 0.5

0.2

0.4

0.6

0.8 0 0

log fitness variance σ2

frac

tion

dyin

g d

Figure 3. Regions of coexistence and extinction for the lottery model. Contours cor-respond to the values of the minimum of the low density, per-capita growth rates ri.Coexistence occurs for where this minimum is positive. Parameters: lognormal ξt withlog means 0.1, 0.2 and variances σ2

i = σ2 and correlation ρ = 0.

species having a niche in time. In the lottery model, however, the total community density n1 +n2 = 1remains constant for all time. While there are some competitive communities where this assumptionnearly holds (e.g. the forest communities on Barro Colorado Island), many communities total densitiesfluctuate in time and, consequently, are best described by truly multivariate models. We begin byexamining two species competition models to clarify some of the issues that arise in analyzing theirdynamics.

Let nt = (n1,t, n2,t) is the vector of densities of the two interacting species at time t. As with thesingle species models, each species has a fitness fi(n, ξ) that depends on the environmental state ξ ∈ Eand community state n = (n1, n2). In a fluctuating environment, the dynamics of these species aregiven by the coupled difference equations

(4.2)n1,t+1 = n1,tf1(nt, ξt+1)

n2,t+1 = n2,tf2(nt, ξt+1)

where fi are continuous, positive functions. We will assume throughout this section that ξ1, ξ2, . . . area sequence of independent and identically distributed random variables.

Lets assume the species are competing intraspecifically and interspecifically i.e. fi are decreasingfunctions of both n1 and n2. Let e1 = (1, 0) and e2 = (0, 1) be the standard basis elements. To focuson their interactions, lets assume that each species stochastically persists in the absence of the otherand the dynamics converge to a unique stationary distribution. Namely, for species 1 (and similarlyfor species 2), the single species model

(4.3) n1,t+1 = n1,tf1(n1,te1, ξt)


satisfies conditions of either Theorems 3.7 or 3.6 of Chapter 3. Thus, there is a positive, randomvariable n1 such that n1,t converges to n1 in distribution and empirically i.e.

limt→∞

E[h(n1,t)|n0] = E[h(n1)]

and

limt→∞

1

t

∑s=1t

h(n1,s) = E[h(n1)] with probability one

for all continuous functions h : [0,∞) → R such that E[|h(n1)|] < ∞. Let n2 be correspondingstationary distribution for species 2. In particular, we have ri(0) = E[log ai(0, ξt)] > 0 for each of thespecies.

To determine whether the species coexist or not, we examine what happens when one species rare,say species 2. If n2,t ≈ 0, then we can approximate the dynamics by the partially coupled system

n1,t+1 ≈ n1,tf1(n1,te1, ξt+1)

n2,t+1 ≈ n2,tf2(n1,te1, ξt+1)

where, recall, that e1 = (1, 0). The second equation is (approximately) a stochastic linear differenceequation and the per-capita growth rate of species 2 is given approximately by

limt→∞

1

t

t∑s=1

log f2(n1,se1, ξs+1)

whenever the limit exists. As n1,t converges empirically to n1 and ξt are ergodic, we would expect thatthis limit converges with probability one

E[log f2(n1e1, ξt)] =: r2(n1)

where ξt is independent of n1. Indeed, this fact is true (see, Section XX). If r2(n1) > 0, we expectspecies 2 to increase when rare. If r2(n1) < 0, then we expect species 2 to decrease when rare. We cansimilarly define r1(n2).

Intuitively, if each species tends to increase when rare i.e. r1(n2) > 0 and r2(n1) > 0, then weexpect the species to coexist. This “mutual invasibility” condition was introduced by Turelli [1978]who wrote

the theory is based on a heuristic analytical approximation that provides conditionsunder which a rare invading species can increase in the presence of a community ofestablished competitors.

Under the aforementioned assumptions and two additional assumptions 1, Chesson and Ellner [1989]proved the following result.

Theorem 4.2. (Coexistence of competitors) If r1(n2) > 0 and r2(n2) > 0, then the system stochas-tically persistent in distribution i.e. for all ε > 0 there exists δ > 0 such that

P[nt ∈ (δ,∞)2|n0] ≥ 1− εfor all n0 � 0 and t sufficiently large.

In fact, I think one can also prove that

lim inft→∞

#{1 ≤ s ≤ t : nt ∈ (δ,∞)2}t

≥ 1− ε

1Let e1 = (1, 0) and e2 = (0, 1). Then the additional assumptions are: (i) P[fi(xei, ξt) = 1] < 1 for all x > 0, and(ii) there are positive constants ci > 0 such that E[inf0<x<ci log fi(xei + njej , ξt)] > −∞ where nj has the stationarydistribution and is independent of ξt.


with probability one for ni,0 > 0 for i = 1, 2. Again, these statements mean there is a small probabilitythat any species will be at a low density, and the fraction of time any species spends at a low densityis small. It also worth noting that the results from Chapter 3 give sufficient conditions for stochasticboundedness of both species in terms of the proper functions V .

Example 4.1. (Exploitative competition and the storage effect) Consider two species competing fora single limiting resource (e.g. space, water, nutrients). Individuals from species extract the resource atan “intensity” ai. This intensity is determined by the length of time individuals extract the resource aswell as how efficient they are in extracting the resource. Assume the resource availability is a decreasingfunction of the total rate at which individuals of either species extract the resource. Namely, thereis a decreasing function, call it R, such that the resource level equals R(

∑j ajnj). For species i, the

resource extracted per individual is aiR(∑

j ajnj). Let bi be the number of offspring produced per unit

of extracted resource. Let si be the probability that an adult (i.e. not one of the new offspring) survivesto the next year. Under these assumptions, we get

fi(ξ, n) = aibiR(∑j

ajnj) + si where ξ = (a1, a2, b1, b2, s1, s2)

For various choices of R (e.g. R(x) = 11+x

or (1 − exp(−x))/x), one can show (see the exercises) thatwithout environmental fluctuations in the parameters, the species with the larger value of aibi/(1− si)competitively excludes the other species.

To understand how environmental fluctuations may alter this outcome, we follow Chesson [1988]and consider the degenerate (but very informative) case where the species are competitively equivalentin the absence of these fluctuations. Namely, a1 = a2 = a, b1 = b2 = b, s1 = s2 = s for all time. In thiscase, the deterministic dynamics are

ni,t+1 = ni,t (baR(a(n1,t + n2,t)) + s) with i = 1, 2.

As n1,t+1/n2,t+1 = n1,t/n2,t for all t, all radial lines in the positive orthant are invariant. Consequently,provided baR(0) + s > 1, there exists a line of equilibria connecting the two axes. Regarding theseneutral dynamics, Chesson [1988] wrote

Classically, when faced with a deterministic model of this sort ecologists have con-cluded that only one species can persist when the likely effects a stochastic environ-ment are taken into account. The reason for this conclusion is the argument thatenvironmental perturbations will cause a random walk to take place in which eventu-ally all but one species becomes extinct. Our conclusion supports this conclusion inonly a narrow range of circumstances.

To see how Chesson arrived at the final sentence, lets consider injecting stochasticity in the extractionintensities. Specifically, the ai,t are independent and identically distributed in time. Moreover, to makethings tractable and to add no systematic advantage of one species of the other, we follow Chessonand assume that a1,t and a2,t are exchangeable: P[(a1,t, a2,t) ∈ A] = P[(a2,t, a1,t) ∈ A] for any Borel setA ⊂ R2.

We can writer2(n1) = E[log (ba2,tR(a1,tnj) + s)] = E[h(a1,t, a2,t)]

whereh(a1, a2) = E[log (ba2R(a1n1) + s)]

By the fundamental theorem of calculus,∫ a2

a1

∫ a2

a1

∂2h

∂a1∂a2(x, y) dx dy =

∫ a2

a1

∂h

∂a2(a2, y)− ∂h

∂a2(a1, y)dy

= h(a2, a2)− h(a2, a1)− [h(a1, a2)− h(a1, a1)]


As r1(n1) = r2(n2) = 0, we have E[h(a2,t, a2,t)] = E[h(a1,t, a1,t)] = 0. Hence,

E[h(a2,t, a2,t)− h(a2,t, a1,t)− [h(a1,t, a2,t)− h(a1,t, a1,t)]] = −2r2(n1) = −2r1(n2)

and

r2(n1) = −1

2E

[∫ a2,t

a1,t

∫ a2

a1

∂2h

∂a1∂a2(x, y) dx dy

]Computing this mixed partial yields

∂2

∂a1∂a2log (ba2R(a1n1) + s) =

∂

∂a1

bR(a1n1)

ba2R(a1n1) + s

=bR′(a1n1)n1s

(ba2R(a1n1) + s)2

which is negative provided that s > 0. Hence, provided there are fluctuations and they are not perfectlycorrelated (i.e. a1,t 6= a2,t with positive probability), we get r2(n1) = r1(n2) > 0 and the species coexist.Hence, as pointed out by Chesson, contrary to naive expectations, adding fluctuations in the ai,t doeslead to a random walk toward extinction. Rather it facilitates coexistence.

We can also perform the same analysis with respect to fluctuations in the bs or the ss (try it foryourselves!). However, in each of these cases, the mixed partials of the corresponding h function equalzero. Hence, fluctuations in these parameter have no effect on the ri (namely, they remain zero).

This highlights that the coexistence mechanism for this model requires three things: (i) there aretimes at which each species has a higher per-capita growth rate than the other (i.e. ai,t vary but don’tperfectly co-vary), (ii) growth realized during good years can be “stored” through bad years (i.e. s > 0),and (iii) during “good” years for each species, its increase in their per-capita growth rate is higher inthose years which are worse for their competitor, than years which are also good for their competitor(i.e. the mixed partial is negative). Hence, there is more nuance to this effect than as first envisionedby Hutchinson in the quote at the start of this chapter. Chesson [1988] calls this trifecta “the storageeffect.”

What happens when r1(n2) < 0 or r2(n1) < 0? Under two additional technical assumptions 2,Chesson and Ellner [1989] proved

Theorem 4.3. (Exclusion) If ri(nj) < 0 for some i = 1, 2 with j 6= i, then P[limt→∞ ni,t = 0|n0]can be made arbitrarily close to 1 whenever nj,0 > 0 and ni,0 is sufficiently small.

If r2(n1) > 0, r1(n2) < 0, and (0, ε] × [ε,∞) is accessible from (0,∞)2 for all ε > 0 sufficientlysmall, then limt→∞ n1,t = 0 with probability one whenever n2,0 > 0.

If r2(n1) < 0, r1(n2) > 0, (0, ε] × [ε,∞) is accessible from (0,∞)2 for all ε > 0 sufficiently small,and an additional technical assumption (see A6 in [Chesson and Ellner, 1989]3), then

P[

limt→∞

ni,t = 0 for some i]

= 1

whenever ni,0 > 0 for i = 1, 2.

Example 4.2. (Reproductive interference for competing species) This example is based on work inprogress with Sharon Strauss Let λi be the maximal number of offspring produced by a mated individualof species i, αi the strength of intraspecific competition, and βj the strength of interspecific competition.In the presence of conspecific and heterospecific individuals, the expected number of offspring producedby a mated individual of species i is

λi1 + αini + βjnj

with j 6= i

2see A2’ and A3’ in Chesson and Ellner [1989]3I believe that the existence of a V for stochastic boundedness can be used to replace this assumption


0 100 300 500

020

040

060

080

0

time

dens

ities

0 100 300 500

010

0030

00time

dens

ities

Figure 4. Dynamics of coexistence and exclusion for the stochastic exploitative com-petition model. No storage (s = 0) on the left, lots of storage (s = 0.8) on the right.

We assume the fraction of mated individuals increases with conspecific density and decreases withheterospecific density:

(4.4)ni

ni + bjnj

where bj measures the strength of reproductive interference of species j on species i. One interpretationof this fraction is that the populations are panmictic and probability of mating with a heterospecific orconspecific is proportional to their population densities. Alternatively, equation (4.4) corresponds tothe fraction of ovules or eggs that get fertilized (at the individual or population scale) by conspecificpollen or sperm.

We assume that environmental fluctuations cause fluctuations in the low-density reproductive rates.Specifically, λi,t are given by log-normally distributed random variables with log means µi, log variancesσ2i , and log correlation ρ. Putting together these components together yields the following model

n1,t+1 = n1,tn1,t

n1,t + b2n2,t︸︷︷︸% mated

λ1,t1 + α1n1,t + β2n2,t︸︷︷︸

# potential offspring

(4.5)

n2,t+1 = n2,tn2,t

n2,t + b1n1,t

λ2,t1 + α2n2,t + β1n1,t

(4.6)

If µi > 0, the the single species systems have a unique positive stationary distribution by results atthe end of Chapter 3 (both theorems apply!). Furthermore, the dynamics are stochastically boundedusing V (n1, n2) = n1 + n2. Provided that ρ < 1, the sets [ε,∞)× (0, ε) are accessible for all ε > 0. Asri(nj) = −∞ for i = 1, 2 and j 6= i, the previous theorem implies that whenever ρ < 1, one species is

3. MANY SPECIES 58

0 20 40 60 80 100

010

020

0

years

min

. den

sity

ρ = 0.5

0 20 40 60 80 100

020

6010

0

years

min

. den

sity

ρ = − 0.5

0.0 0.2 0.4 0.6 0.8 1.0

−0.

50.

00.

5

environmental std. σ

cros

s−co

rrel

atio

n ρ

0.0 0.2 0.4 0.6 0.8 1.0

probability of species loss

Figure 5. Extinction in the face of environmental fluctuations. In the left panels, thedynamics of the minimum species density (i.e. the smaller value of N1,t and Nt,2) for twovalues of the cross-correlations. In the right panel, the probability of the loss of a specieswithin 50 years as a function of the magnitude of the fluctuation σ1 = σ2 = σ and thecross-correlation ρ.

asymptotically lost with probability one. In fact, species being lost goes to zero at a super-exponentialrate. Interestingly, one can prove that when ρ = 1, contingent coexistence in the deterministic versionof the model implies contingent coexistence in the stochastic model (proof in progress). Figure 5illustrates the effect of ρ on how quickly species are lost. Due to the super-exponential convergence tozero for one of the species, simulating the dynamics leads quickly to numerical zeros on these events. Ifwe interpret hitting the numerical zero as an extinction event, then we can compute how the probabilityof extinction (in finite time) depends on the various parameters of the model (Figure 5)

3. Many species

Partly to escape these complications, we count like the Australian Arunta Tribe, “one,two, many, ” and move on...directly to multispecies communities. – Robert May [1981]

For two species communities, the “mutual invasiability” paradigm does a good job of characterizingwhether species coexist or not.4 However, does this condition extend to communities with three ormore species? There are two natural candidates for extending the mutual invasiblity condition. Thefirst is that every missing species can invade a stationary distribution supporting a subset of species.While we will show this condition is sufficient for coexistence, it is generally too strong of a condition.

4Of course once allows for Allee effects or obligate mutualisms, this paradigm is not very useful. Many opportunitiesfor new results in this area but they are likely to be much more nuanced if one allows for contingent coexistence.

3. MANY SPECIES 59

For example, in a system consisting of a plant, herbivore, and predator, the herbivore might be able tohave a positive per-capita growth rate when introduced at low densities in the plant-only system, butthe predator (assuming its not an omnivore) would not. However, the predator could have a positiveper-capita growth rate when introduced at low densities in the plant-herbivore system (assuming thesetwo species coexist).

This simple gedanken experiment suggests another way to extend the “mutual invasibility” con-dition. Namely, for every stationery distribution supporting a subset of species, at least one of themissing species can invade i.e. have a positive per-capita growth rate at low densities. While thiscondition is necessary for the form of coexistence that we focus on, it is not sufficient in general. Forexample, May and Leonard [1975] introduced a deterministic model of three competing species engagedin a “rock-paper-scissor” game: species 2 displaces species 1 in pairwise competition, species 3 displacesspecies 2, and species 1 displaces species 3. In this system, the only subcommuntiies consist of a singlespecies and one of the missing species can always invade these subcommunities. However, May andLeonard [1975] showed numerically that trajectories of this system can cycle toward extinction (SeeFigure 7 for the stocahstic counterpart). It wasn’t until many years latter that mathematicians (see,Hofbauer and Sigmund 1998 for a delightful exposition on this theory in the context of Lotka-Volterrasystems) developed a theory to identify when these “rock-paper-scissor” systems are extinction proneor allow for coexistence. Here we introduce the stochastic analog of this deterministic theory.

For this theory, we study the dynamics of k interacting populations in a random environment. Letnit denote the density of the i-th population at time t and nt = (n1,t, . . . , nk,t) the vector of populationdensities at time t. As before, we let ξt represents the state of the environment (e.g. temperature,nutrient availability) at time t. The fitness fi(nt, ξt+1) of population i at time t depends on thepopulation state and environmental state at time t + 1. Under these assumptions, we arrive at thefollowing stochastic difference equation:

(4.7) ni,t+1 = ni,tfi(nt, ξt+1).

Following Schreiber et al. [2011], we make four assumptions:

A1: There exists a compact set S of Rk+ = {n ∈ Rk : xi ≥ 0} such that nt ∈ S for all t ≥ 0.

A2: {ξt}∞t=0 is a sequence of i.i.d random variables taking values in a separable metric space E,having distribution m.

A3: fi(n, ξ) are strictly positive functions, continuous in n and measurables in ξ.A4: For all i, supn∈S

∫(log fi(n, ξ))

2m(dξ) <∞Assumption A1 ensures that the populations remain bounded for all time. Currently there is activework being done to relax this assumption e.g. the work of Alex Henning on SDE counterparts to thistheory. Assumptions A2 and A3 imply that {Xt}∞t=0 is a Markov chain on S and that {Xt}∞t=0 isFeller 5 Assumption A4 is a technical assumption meet by many models.

The expected per-capita growth rate at state n of population i is

ri(n) = E[log fi(n, ξt)].

As discussed before, when ri(n) > 0, the i-th population tends to increase when the current populationstate is n. When ri(n) < 0, the i-th population tends to decrease when the current population state isn. Of course, the other species rarely will be a fixed state n. Rather they will fluctuate around somestationary distribution. Recall that a random vector n (independent of ξt for all t) has a stationarydistribution for (4.7) if

P[nt+1 ∈ A|nt = n] = P[n ∈ A]

for every Borel set A ⊂ S. Furthremore, this stationary distribution is ergodic if n can not be writtenas a non-trivial convex combination of other random vectors with a stationary distribution. We define

5For this process, we can define the operator Ph(n) = E[h(nt+1)|nt = n] for any integrable function functionh : S → R. The Markov chain nt is Feller if Ph is continuous whenever h is continuous.

3. MANY SPECIES 60

the invasion rate of species i with respect to n as

ri(n) = E[log fi(n, ξt)]

The following theorem clarifies why ri(n) is called an invasion rate. A proof is given in [Schreiber et al.,2011, Appendix A].

Theorem 4.4. Let n have a distribution which is stationary and ergodic for (4.7). If n0 = n, then

limt→∞

1

t

t−1∑s=0

log fi(ns, ξs+1) = ri(n)

with probability one for each species i.

Before stating the main stochastic persistence theorem, we make a useful observation about theri(n) for species “supported” by n. Let n be as in the proposition. We say species i is supported ifP[ni > 0] > 0. Ergodicity implies that P[ni > 0] = 1 (otherwise one could decompose n into two piecesone of which has ni = 0 with probability one). What can we say about ri(n)? Choose m > 0 suchthat P[1 −m > ni > m] > 1/10. Let n0 = n. By Birkhoff’s ergodic theorem, ni,t enters the interval[m, 1/m] infinitely often i.e. there are random times t1 < t2 < t3 < . . . such that m ≤ ni,tk ≤ 1/m withprobability one for all k. Hence, with probability one

0 = limk→∞

1

tklog ni,tk

= limk→∞

1

tk

tk−1∑s=0

log fi(ns, ξs+1)

= ri(n)

where the first equality follows from ni,tk lying in the interval [m, 1/m] and tk → ∞ with probabilityone. Intuitively, a species supported by n (i.e. P[ni > 0] > 0) is on average neither growing (as itdynamics is bounded from above) or on average decreasing (as it isn’t tending toward extinction).Hence, its average per-capita growth rate is zero.

Now we state the main persistence theorem. For all δ ≥ 0, define Sδ = S ∩ (δ,∞)k i.e. communitystates corresponding to all species having densities strictly greater than δ. Then S \ S0 corresponds tocommunity states where at least one species is extinct.

Theorem 4.5. If there exists p ∈ [0, 1]k such that∑i

piri(n) > 0

for all ergodic and stationary n supported by S \ S0. Then for all ε > 0 there exists δ > 0 such that

lim inft→∞

#{1 ≤ s ≤ t : ns ∈ Sδ}t

≥ 1− ε almost surely

whenever n0 ∈ S0.

Outline of proof: Upcoming!! Currently, working on a proof of the following partial converse: ifthere exists p ∈ [0, 1]k such that ∑

i

piri(n) < 0

for all ergodic and stationary n supported by S \ S0, then (4.7) is extinction prone i.e. for all ε > 0there exists δ > 0 such that

P[ limt→∞

minini,t = 0|n0] ≥ 1− ε

whenever n0 ∈ S \ Sδ.

3. MANY SPECIES 61

Example 4.3. (Two species models) In the exercises, you are asked to verify that Theorem 4.5yields the mutual invasibility condition for competing species models from the previous section. Here,we examine a model of a predator-prey interactions where n1 is the prey density and n2 is the predatordensity. For the prey dynamics, we consider a classical stochastic Beverton-Holt model:

n1,t+1 = n1,tξt+1

1 + an1,t

where ξt+1 corresponds to the low density fitness of the prey, and a > 0 determines the strength ofintraspecific competition. The average, low density per-capita growth rate is r1(0) = E[log ξt] which weassume is positive. By results from Chapter 3, we get there is a unique positive stationary distributionfor this single species model. Let n1 be a random variable with this distribution which is independentof the ξt for all t.

If predators are randomly search for the prey with intensity b > 0, then the probability of a preyescapes predation is exp(−bn2,t). If each prey eaten leads to the production of c > 0 offspring (i.e. theconversion efficiency) and “adult” predators survive to the next year with probability 1 > s > 0, thenwe get the following model

n1,t+1 = n1,tξt+1

1 + an1,t

exp(−bn2,t)

n2,t+1 = cn1,t(1− exp(−bn2,t)) + sn2,t

At first glance this equation doesn’t look like it is of the right form i.e. there isn’t a factor of n2,t in thefirst term of the predator equation. However, if we define h(x) = (1− exp(−bx))/x (which is analytic!check for yourself), then the right hand side of the predator equation is of the form cn2,tn1,th(n2,t)+sn2,t

as desired.In the absence of the prey, the predator dynamics are given by exponential decay to extinction

i.e n2,t+1 = sn2,t with s < 1. Hence, there is no positive stationary distribution on the predatoraxis. Hence, the only ergodic, stationary distributions correspond to a Dirac measure at (0, 0) and thedistribution of (n1, 0). The persistence criterion states that we need weights p1, p2 such that∑

i

piri(0, 0) > 0 and∑i

piri(n1, 0) > 0

We have r1(0, 0) > 0 by assumption, r2(0, 0) = log s < 0, and r1(n1, 0) = 0 as species 1 is supported bythis stationary distribution. Hence, the conditions become

p1r1(0, 0) + p2 log s > 0 and p2r2(n1, 0) > 0

The second condition implies that we need r2(n1, 0) > 0. Assume this is true. Then setting p1 = 1 andchoosing p1 > 0 sufficiently small ensures that both conditions are meet. Hence, r2(n1, 0) > 0 sufficesfor stochastic persistence of both species. On the other hand, suppose that r2(n1, 0) < 0. Settingp2 = 1 and choosing p1 > 0 sufficiently small ensures the extinction prone condition is meet and thespecies do not stochastically persist.

What is r2(n1, 0)? As h(0) = b, we get

r2(n1, 0) = E[log(cn1b+ s)]

To get a sense of how this invasion rate depends on the mean and variance of n1, lets write n1 = n1+σXwhere n1 = E[n1], σ

2 = Var[n1] and σX = n1 − n1. If we assume that σ > 0 is sufficiently small (inthe next chapter we will see when this is true), we get the following second order approximation byTaylor’s theorem (try this yourself!):

r2(n1, 0) ≈ log(bcn1 + s)− 1

2

(bcσ

bcn1 + s

)2

3. MANY SPECIES 62

0.0 0.4 0.8

0.0

0.2

0.4

σ

redu

ctio

n in

r2

change in mean preychange in prey variance

Figure 6. Reduction in the predator invasion rate due to the change in the mean andvariance of the prey density.

Equivalently,

r2(n1, 0) ≈ logE[bcn1 + s]− 1

2CV[bcn1 + s]2

where CV denotes the coefficient of variation of a random variable.This expression implies that variation in the fitness of the prey might inhibit the persistence of

the predator in two ways. First, the fitness variability via Jensen’s inequality typically decreases themean prey density n1. This results in a reduction in the first term of the predator’s per-capita growthrate. Second, the fitness variability increases the variance in the prey’s stationary distribution. Thisleads to an additional reduction through the second term of the approximation. This second reductionstems for the concavity of the log function i.e. Jensen’s inequality once more! Figure 6 computes themagnitude of these two effects when increasing the variance experienced by the prey.

Example 4.4. Lottery model of evolutionary games The classical lottery model, as discussed at thebeginning of this chapter, assumes that individuals reproduce at rates independent of the frequenciesof the other populations. Here, we examine a lottery model that accounts for frequency dependentinteractions among different genotypes in an asexual population. Let ni,t denote the fraction of spaceoccupied by genotype i at time t. Individuals are randomly interacting and the effect of these interac-tions on their reproduction is determined by a payoff matrix, At, whose i–j-th entry corresponds to thepayoff to strategy i when playing strategy j. The number of offspring produced by strategy i is givenby∑

j Aji,t+1nj,t. After producing offspring individuals die and the frequencies in the next generationare given by the frequencies of the offspring. Namely,

(4.8) ni,t+1 =

∑j Aji,t+1nj,tni,t∑j`Aj`,t+1nj,tn`,t

The state space for this model is the k − 1 dimensional simplex: S = {x ∈ Rk+ :∑

i xi = 1}.

3. MANY SPECIES 63

Now consider the evolutionary game of rock, paper, scissors (see lecture slides for empirical exam-ples) where strategy 2 beats strategy 1, strategy 3 beats strategy 2, and strategy 1 beats strategy 3.Assume when species i wins, losses, or has a draw, it gets payoff of wi,t, `i,t, and di,t, respectively. Thenthe payoff matrix becomes

At =

d1,t `1,t w1,t

w2,t d2,t `2,t`3,t w3,t d3,t

Assume wi,t > dj,t > `k,t with probability one for all t and 1 ≤ i, j, k ≤ 3. Then one can show (seethe exercises!!) that for the pairwise dynamics, you get the winners out competing the losers. Hence,there are only three ergodic stationary distributions corresponding to the (1, 0, 0), (0, 1, 0) and (0, 0, 1)steady states.

Assume the wi,t all have the same distribution, di,t all have the same distribution, and `i,t have thesame distribution. Furthremore, assume there are no cross correlations. For any given strategy i, wecan compute the per-capita growth rate rw of the strategy that beats that strategy and the per-capitagrowth rate r` of the strategy that gets between by the strategy. These are given by

rw = E[logwtdt

] > 0 and r` = E[log`tdt

] < 0

where wt, dt, and `t have the common distributions and uncorrelated. Now for the strategies to coexist,we need to find weights pi > 0 such that

∑i piri(ej) > 0 for all j (Recall ej are standard basis element

of R3.) By symmetry, we should be able to choose all the pi to be equal, say 1. In which case, we needrw + r` > 0. Equivalently, stochastic persistence requires that

E[logwt] + E[log `t] > 2E[log dt].

Conversely, the reversed inequality implies that the strategies are extinction prone. Both forms ofdynamics are illustrated in Figure 7.

Example 4.5. (Stochastic Lotka-Volterra difference equations)

3. MANY SPECIES 64

0 20 40 60 80 100

0.0

0.4

0.8

time

freq

uenc

y

n1 n2

n3

0 20 40 60 80 100

0.2

0.4

0.6

time

freq

uenc

y

n1 n2

n3

Figure 7. UPCOMING

CHAPTER 5

Approximation methods

Finding closed form descriptions of the stationary distributions for the nonlinear difference equationsconsidered here is typically, at best, difficult or impossible. Hence, to get some analytical traction onthese stochastic models, we avail ourselves to a fundamental tactic of the applied mathematician: (i)identify models whose dynamics we understand and (ii) use approximation methods to understandthe effects of small perturbations on the dynamics. We will use this tactic to examine how feedbacksbetween deterministic and stochastic forces determine the mean and co-variance structure of stationarydistributions n and the per-capita growth rates ri(n) of species not supported by n.

We begin the illustrating the basic ideas for the stationary distribution of a single species modeland the implications for per-capita growth rate of another species being introduced at low densities.Then we extend these results to multivariate models. We conclude by providing some theorems thatallow one to analytically verify the persistence conditions from the previous chapter. The work beingpresented here is part of the in progress, thesis work of William Cuello. Hence, pleaselet us know if you have you seen similar results developed elsewhere in a mathematicallyrigorous fashion.

1. The scalar case

Many of the results in this chapter extend to arbitrary stochastic difference equations and areeasier to present in this level of generality. Hence, we begin by considering a general, scalar stochasticdifference equation of the form

(5.1) nt+1 = g(nt, ξt+1)

where ξt are i.i.d. scalars (for simplicity right now) and g is at least three times continuously differen-tiable with respect to both of its arguments.

We make two standing assumptions for this section. First, ξt = ξ + σZt where Zt has mean zero,variance 1, and compact support (i.e. there exists an interval [a, b] such that P[a ≤ Zt ≤ b] = 1). Weuse the parameter σ to control the magnitude of the noise. Second, we assume that n is a linearlystable equilibrium for the deterministic model

nt+1 = g(nt, ξ)

i.e the model corresponding to σ = 0. Namely, n = g(n, ξ) and∣∣ ∂g∂n

(n, ξ)∣∣ < 1.

Intuitively, if σ > 0 is sufficiently small, we should expect that nt fluctuates around n provided thatn0 is sufficiently close to n. Indeed, in the next section, we shall see that for sufficiently small σ > 0,there exists a random variable n such that such that nt converges in distribution and empirically to nwhenever n0 is sufficiently close n.

To understand the distribution of n when σ is sufficiently small, we will linearize the system around(n, ξ) using the first-order approximation

f(n, ξ) ≈ n+∂g

∂n(n, ξ)︸︷︷︸=:gn

(n− n) +∂g

∂ξ(n, ξ)︸︷︷︸=:gξ

(ξ − ξ).

65

1. THE SCALAR CASE 66

Our approximation xt for nt − n is given by the ARM(1) model

xt+1 = gnxt + gξσZt+1

Our assumption of linear stability implies that |gn| < 1. Hence, by the results from Chapter 2, we getthat xt converges to a distribution with mean 0 and variance σ2g2ξ/(1− g2n).

What does this linear approximation tell us about n? In the next section, we show that

E[n] = n+O(σ2)

Var[n] =σ2g2ξ

1− g2n+O(σ3)

Lets see how well these approximations work.

Example 5.1. (Beverton Holt model revisited) Consider the stochastic Beverton-Holt model:

nt+1 = ntξt+1

1 + nt

where ξt = ξ + σZt. Provided that ξ > 1, the deterministic model

nt+1 = ntξ

1 + nt

has a unique positive equilibrium n = ξ − 1 which is linearly stable as

gn =ξ

(1 + n)2=

1

ξ< 1.

We also have

gη =1

1 + n=

1

ξ.

Hence,

E[n] = ξ − 1 +O(σ2)

Var[n] =σ2

ξ2 − 1+O(σ3)

The approximation, as shown in Figure 1, of the variance is exceptionally good. However, for estimatingE[n], the error of order σ2 (all that we are guaranteed) is substantial.

Motivated by this example, we derive a second order correction term for E[n]. To this end, we usethe 2nd order approximation of g around (n, ξ):

g(n, ξ) = n+ gn(n− n) + gξ(ξ − ξ) +gnn2

(n− n)2 + gnξ(n− n)(ξ − ξ) +gξξ2

(ξ − ξ)2 +O(σ3)

where gnn, gξξ, gnξ are the second order derivatives of g evaluated at n = n, ξ = ξ. By stationarity, ifn0 = n, then

E[n] = E[g(n, ξt)]

= n+ gnE[n− n] + gξE[ξt − ξ] +gnn2

E[(n− n)2] + gnξE[(n− n)(ξt − ξ)] +gξξ2E[(ξt − ξ)2]

+O(σ3)

(1− gn)E[n− n] =gnn2

σ2g2ξ1− g2n

+gξξ2σ2 +O(σ3)

Hence, we get the higher order approximation

E[n] = n+σ2

2(1− gn)

(gnng

2ξ

1− g2n+ gξξ

)+O(σ3)

1. THE SCALAR CASE 67

0 1 2 3 4 5

05

1015

20

σ

dens

ity v

aria

nce

simulatedanalytic approx.

0 1 2 3 4 5

8.6

8.7

8.8

8.9

9.0

σm

ean

dens

ity

Figure 1. Simulated versus analytic approximations of the variance and means of n asa function of σ.

Implementing this approximation for the stochastic Beverton Holt model (check for yourself!) yieldsthe red curve in the right hand Figure 1.

We can use these analytical approximations of the expectation and variance of n to get estimatesof E[h(n)] for any smooth function h. Namely,

(5.2) E[h(n)] = h(n) +σ2

2

[h′(n)

(1− gn)

(gnng

2ξ

1− g2n+ gξξ

)+ h′′(n)

g2ξ1− g2n

]+O(σ3)

To illustrate the utility of this approximation, we can use to understand the stochastic persistence ofthe predator-prey model introduced in the previous chapter.

Example 5.2. (Predator-prey dynamics)Consider the predator-prey model

n1,t+1 = n1,tξt+1

1 + ntexp(−bnt)

n2,t+1 = cn1,tn2,th(nt) + sn2,t

where h(x) = (1 − exp(−bx))/x. Provided that E[log ξt] > 0, we had shown that there is a uniquepositive stationary distribution n1 for the prey and the per-capita growth rate of the predator at thisstationary distribution is

r2(n1, 0) = E[log(bcn+ s)]

2. THE MULTIVARIATE CASE 68

0 1 2 3 4 5

−0.

100.

000.

05

σ

pred

ator

per

−ca

pita

gro

wth

simulatedanalytic approx.

Figure 2. Simulated versus analytic approximations of r2(n1, 0).

Define f = bcn + s.Using equation (5.2) and our calculations from the previous example, we can getthe estimate

r2(n1, 0) = log f +σ2

2

[bc

f

1

1− gn

(gnng

2ξ

1− g2n+ gξξ

)−(bc

f

)2 g2ξ1− g2n

]+O(σ3)

= log f +σ2

2

[bc

f

ξ

ξ − 1

(−2

ξ4 − ξ2

)−(bc

f

)21

ξ2 − 1

]+O(σ3)

= log f − σ2

2(ξ2 − 1)

bc

f

[2

ξ2 − ξ+bc

f

]+O(σ3)

Hence, fluctuations in the prey’s fitness reduces the per-capita growth rate of the predator in twoways: the reduction in E[n] (the first term of the correction) and variance in n (the second term of thecorrection).

2. The Multivariate Case

This section is in progress... Basic idea is similar to the scalar case. As in the scalar case, weconsider the more general stochastic model

nt+1 = g(nt, ξt+1)

where we assume

Small noise: ξt = ξ + σZt lies in Rm for some m, E[Zt] = 0, Var[Zt] = 1, and Zt lies in a fixedcompact set with probability one.

Stable equilibrium: n is a linearly stable equilibrium for the deterministic model

nt+1 = g(nt, ξ)

2. THE MULTIVARIATE CASE 69

William has proven the following proposition

Proposition 5.1. If σ > 0 is sufficiently small, then there exists a unique stationary distributionn such that nt converges to n in distribution and empirically provided n0 is sufficiently close to n.

As before, we use the linear approximation

g(n, ξ) ≈ n+ (n− n)gn + (ξ − ξ)gξwhere gn is the k × k derivative matrix of g with respect to n and gξ is the k ×m derivative matrix ofg with respect to ξ; both evaluated at (n, ξ). The dynamics of nt − n are approximated by the AR(1)process:

xt+1 = xtgn + (ξt+1 − ξ)gξ.As the spectral radius of gn is less than one, this AR(1) process has a unique stationary distributionwith mean 0 and co-variance matrix Σ2 as described in the final section of Chapter 2.

One can prove the following properties of these approximations.

Proposition 5.2.E[n] = n+O(σ2) and Cov[n] = Σ2 +O(σ3)

To improve our approximation of E[n], we can do as before and use the second-order approximationof g:

g(n, ξ) ≈ n+ (n− n)gn + (ξ − ξ)gξ +1

2gξξ(ξ − ξ, ξ − ξ) +

1

2gnn(n− n, n− n) + gξn(ξ − ξ, n− n)

where gnn, gξξ, gnξ are the second-order derivatives of g which are 2-tensors and, consequently, act onpairs of vectors.

Index

accessible, 450,∞, 40

auto-regressive processes, 27

demographic stochasticity, 4density-dependence

negative, 35positive, 39

environmental stochasticity, 4ergodic, 17exchangeable, 56

irreducibility, 46iteroparous, 43

lottery model, 49Lyapunov exponent

dominant, 27

per-capita growth ratesensitivity, 21

first-order, 23second-order, 26

stochastic, 17Perron-Frobenius

constant environment, 14random environment, 17

primitiveconstant environment, 14stationary environment, 17

proper function, 43

regulated, 35reproductive values


sedentary, 43stable state distribution


stationary, 17stationary distribution

species support, 60stochastically bounded

from above, 43

70

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