Post on 16-Dec-2015
Quantitative Biology: populations
graham.medley@warwick.ac.uk
Lecture 1. Basic Concepts & Simplest Models
• Definitions• Basic population dynamics
– immigration-death• discrete & continuous
– birth-death• discrete
– logistic equation• discrete & continuous
• Multiple species: competition and predation
Definitions
• Population– a “closed” group of individuals of same spp.– immigration and emigration rates zero
• Metapopulation– a collection of populations for which the migration
rates between them is defined• Community
– a closed group of co-existing species
Fundamental Equation
• Populations change due to– immigration, emigration
• additive rates; usually assumed independent of population size
– birth, death• multiplicative rates; usually dependent on
population size
tttttt EdNIbNNN 1
Immigration-Death (Discrete)
• Time “jumps” or steps– N is not defined between steps
• Immigration & death rates constant• Death rate is a proportion
– the proportion surviving is (1- )– limits: 0 1
ˆ1ˆˆˆ1 tttt NNNN
Immigration-Death (Continuous)
Ndt
dN
• Re-expressed in continuous time– N defined for all times
• Death rate is a per capita rate– the proportion surviving a period of time, T, is
exp(-T)– limits: 0
Immigration-Death Solution
0
5
10
15
20
25
0 1 2 3 4 5 6
Time
N
Total Original Immigrants
I-D Equilibrium• When dN/dt = 0
– population rate of change is zero– immigration rate = (population) death rate
0
10
20
30
40
50
0 10 20 30 40 50
Population
Rat
es
Immigration Death Equilibrium
Characteristic Timescales
• Life expectancy, L, determines the timescale over which a population changes (especially recovery from perturbations)
• L is reciprocal of death rate (in continuous models)
• In immigration-death model increasing death rate (decreasing life expectancy) speeds progress (decreases time) to equilibrium
Immigration-death model with different L
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5 6
Time
N
L = 2.00 L = 1.00 L = 0.50
Simplest Discrete Birth-Death Model
• R is the reproductive rate– the (average) number of offspring left in the
next generation by each individual• Gives a difference equation
– check with fundamental equation• Population grows indefinitely if R>1
tt NRN 1
Birth-Death Continuous
• r is the difference between birth and death rates– R = er ; r = ln(R)
• If r > 0, exponential growth, if r < 1 exponential decay
rtt eNNNrN 0
Density Dependence: necessity
• To survive, in ideal conditions, birth rates must be bigger than death rates– ALL populations grow exponentially in ideal
circumstances• Not all biological populations are growing
exponentially– ALL populations are constrained (birth death)– Density dependence vs. external fluctuations
• Stable equilibria suggest that density dependence is a fundamental property of populations
Factors & Processes• Density Independence Factors
– act on population processes independently of population density• Limiting Factors
– act to determine population size; maybe density dependent or independent• Regulatory Factors
– act to bring populations towards an equilibrium. The factor acts on a wide range of starting densities and brings them to a much narrower range of final densities.
• Density Dependence Factors– act on population processes according to the density of the population– only density dependent factors can be regulatory
• Factors act through processes to produce effects (eg: drought-starvation-mortality)
Density Dependent Factors
• Mechanisms– competition for resources (intra- and interspecific)– predators & parasites (disease)
• Optimum evolutionary choices for individuals (e.g. group living, territoriality) may regulate population
Logistic - Observations
• Populations are roughly constant– K - “carrying capacity”– determined by species / environment combination– density dependent factors
• Populations grow exponentially when unconstrained– r - intrinsic rate of population change– i.e. before density dependent factors begin to operate
• r and K are independent
Logistic Equation - Empirical
• Empirical observations combined• Fits many, many data
treNNK
KtN
NK
rrN
K
NrN
N
NKrNN
0
0
2
1
)(
1
Logistic Equation - Mechanistic
• Linear decrease in per capita birth rate• Linear increase in per capita death rate N10
N10
11
0000
21100
1010
;
Kr
NN
NNNN
dbN
Stability of Logistic
• Linear birth and death rates (as functions of N) give a single equilibrium pointN = K
• Equilibrium is globally, stable
Logistic Equation - Dynamics
0
5
10
15
20
25
0 1 2 3 4 5 6
Time
N
0
10
20
30
40
50
60
0 1 2 3 4 5 6
Time
N
0
5
10
15
20
25
0 1 2 3 4 5 6
Time
N
Logistic Equation Properties 0
5
10
15
20
25
0 1 2 3 4 5 6
Time
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6
Time
Rat
e o
f C
han
ge
of
N
0
0.5
1
1.5
0 1 2 3 4 5 6
Time
per
cap
ita
Rat
e o
f C
han
ge
of
N
0
1
2
3
4
5
6
7
8
0 5 10 15 20 25
N
Rat
e o
f C
han
ge
of
N
0
0.5
1
1.5
0 5 10 15 20 25
N
per
cap
ita
Rat
e o
f C
han
ge
of
N
Logistic Equation (Discrete)
• Explicit equilibrium, K• Derivation is by considering the relative growth rate
from its maximum (1/R) to its minimum (1)• The growth rate (R) decreases as population size
increases
t
t
t
tt aN
RN
KNR
RNN
1)1(
11
1
10
100
1000
10000
0 10 20 30 40 50 60
Time
N Birth-Death
0
5
10
15
20
25
0 10 20 30 40 50 60
Time
N
Summary• Timescales
• the “system” (population) timescale is determined by the life expectancy of the individuals within the population
• Density dependence– Birth and death are universal for biological
populations– The direct implication is that populations are
regulated
Multi-population Dynamics • Two Species
– Competition (-/-)• intraspecific• interspecific
– Predation (+/-)• patchiness• prey population limitation• multiple equilibria
• Multi-species
Intraspecific Competition
• Availability of a resource is limited • Has a reciprocal effect (i.e. all individuals affected)• Reduces recruitment / fitness• Consequently produces density dependence• Important in generation of skewed distribution of
individual quality– Different individuals react differently to competition =
creates heterogeneity
• Inverse dd (co-operation)– Allee Effect
Interspecific Competition• Competition for shared resource
– results in exclusion or coexistence• which depends on degree of overlap for resource and degree
of intraspecific competition
• Aggregation & spatial effects– disturbance
• kills better competitor leaving gaps for better colonisers (r- & K- species)
– aggregation enhances coexistence• “empty” patches allow the worse competitor some space
Interspp Competition Dynamics
• Lotka-Volterra model– Structure– Statics
• What are the equilibria
– Dynamics• What happens over time
– Phase planes• isoclines
Lotka-Volterra Equations
• Based on logistic equations– One for each species– 21 represents the effect of an individual of species 2 on species 1
– i.e. if 21 = 0.5 then sp. 2 are ½ as competitive, i.e. at individual level interspecific competition is greater than intraspecific competition
1
2211111
1
K
NNKNr
dt
dN
Analysis
• Equilibrium points are given when the differential equation is zero– A single point (trivial equilibrium) and isocline– The line along which N1 doesn’t change
2211122111
1
1
2211111
0
0
0
NKNNNK
N
K
NNKNr
Phase Planes
• Variables plot against each other• Isoclines• Direction of change (zero on isocline)
– For spp. 1 these are horizontal toward isocline– For spp. 2 these are vertical toward isocline
• Combine two isoclines and directions on single figure…
Outcomes
K2 > K1/12
Spp 2 is more competitive at high densities
K2 < K1/12
Spp 2 is less competitive at high densities
K1 > K2/21
Spp 1 is more competitive at high densities
Exclusion(initial condition
dependent)Spp 1 wins
K1 < K2/21
Spp 1 is less competitive at high densities
Spp 2 wins Co-existence
Dynamics
• Exclusion or co-existence is not dependent on r– but dynamic approach to equilibrium is
0
20
40
60
80
100
120
0 20 40 60 80 100 120
Time
N
r=0.1 r=0.28 r=0.46
r=0.64 r=0.82 SB:r=0.1
Predation
• Consumers– inc. parasites, herbivores, “true predators”
• predator numbers influenced by prey density which is influenced by predator numbers– circular causality: limit cycles in simple models
• time delay – in respect of predator population’s ability to grow, r
• over-compensation– predators effect on prey is drastic
Predation Dynamics• Limit cycles rarely seen• heterogeneity in predation
– patchiness of prey densities• reduced density in prey population
– effect ameliorated by reduction in competition (i.e. compensation)
• increased density in prey population– effect ameliorated by increase in competition
(i.e. compensation)
Refuges• Prey aggregated into patches• Predators aggregate in prey-dense patches• Effect on prey population
– prey in less dense patches are most commonly in a partial refuge
– they are less likely to be predated
• Effect is to stabilise dynamics
Summary
• Individuals interact with each other– and compete
• Each individual is affected by the population(s) and each population(s) is affect by the individual– Population dynamics are reciprocal– and reciprocal across level
• Co-existence is sometimes hard to reproduce in models– How rare is it?
• Heterogeneity (e.g. patches) tends to enhance co-existence
Lecture 2: Structuring Populations
• Age– Leslie matrices
• Metapopulations– Probability distributions
• Metapopulations– Levin’s model
Types of Structuring• Individuals in a population are not identical
– heterogeneity in different traits• trait constant (throughout life)
– DNA (with exceptions? e.g. somatic evolution)– gender (with exceptions)
• trait variable– stage of development, age, infection status,
pregnancy, weight, position in dominance hierarchy, etc
Rate of Change of Structure
• If trait constant for an individual throughout life, then it varies in the population on time scale of L– e.g. evolutionary time scale; sex ratios
• If trait variable for an individual, then varies on its own time scale– infection status varies on a time-scale of
duration of infectiousness– fat content varies according to energy balance
Modelling Stages (Discrete)
• Discrete time model for non-reversible development– at each time step a proportion in each stage
• die (a proportion s survives)• move to next stage (a proportion m)
– a number are born, B– complication: s-m
taxtyyty
xxtxtxxtxxtx
NmNsN
BmsNBNmNsN
,,1,
,,,1,
• easiest to chose a time step (which might be e.g. temperature dependent) or stage structure (if not forced by biology) for which all individuals move up
tyytz
txxty
tx
NsN
NsN
BN
,1,
,1,
1,
Leslie Matrix
• This difference equation can be written in matrix notation
tt MNN 1
zy
x
zyx
tz
ty
tx
t
ss
s
bbb
N
N
N
0
00;
,
,
,
MN
Properties of Matrix Model• No density dependence or limitation
– as discrete birth-death process, the population grows or declines exponentially
• The equivalent value to R is the “dominant eigenvalue” of M– associated “eigenvector” is the stable age distribution
• If the population grows, there is a stable age distribution– after transients have died away
• Density dependence can be introduced– but messy
Leslie Matrix Example
• This matrix has a dominant eigenvalue of 2 and a stable age structure [ 24 4 1 ]
• i.e. when the population is at this stable age structure it doubles every time step
0210
0031
1290
M
Spatial Structure
• Many resources are required for life– e.g. plants are thought to have 20-30 resources
• light, heat, inorganic molecules (inc. H2O) etc.
• Habitats are defined in multi-dimensional space– “niche” is area of suitability in multidimensional space– Areas of differing suitability
• Disturbance– No habitat will exist forever– Frequency, duration and lethality– Dispersal is a universal phenomena
Metapopulations
• A collection of connected single populations– whether a single population with heterogeneous
resources or metapopulation depends on dispersal• if dispersal is low, then metapopulation• degree of genetic mixing• human populations from metapopulation to single
population?
• Depends on tempo-spatial habitat distribution & dispersal
Levins Model
• Ignore “local” (within patch) dynamics– single populations are either at N=0 or N=K
population size– equilibrium points of logistic equation, ignore
dynamics between these points (i.e. r)
KNorNN
K
NrNN
00
1
• Let p be the proportion of patches occupied (i.e. where N=K)– (1-p) is proportion of empty patches– a is rate of extinction (per patch)– m is per patch rate of establishment in empty
patch and depends on proportion of patches filled (dispersal)
appmpdt
dp
rateextinctionrateoncolonizatidt
dp
)1(
Model Results
• Equilibrium only for m > a– i.e. metapopulation can only exist if local
establishment is greater than local extinction• Dynamics similar to logistic equation
mapappmp 1)1(
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
p
ap mp(1-p)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
p
a m(1-p)
Extinction Rates• Stochastic probability of extinction
– disturbance (not related to population size)– demographic (related to population size - the
smaller the population the greater the risk of extinction)
• Appears to decrease with increasing p• Dispersal occurs all the time, tending to
increase small populations
Model with Decreasing Extinction
• Include dependence of local extinction on total patch occupancy
• Exponential assumption gives implicit equilibrium result with two possibilities
01lnln
)1(
)1(
10
0
0
1
1
papa
m
eapm
epapmpppa
pa
0
0.02
0.04
0.06
0.08
0 0.2 0.4 0.6 0.8 1
p
tota
l rat
es
a0 p e{-a1 p} mp(1-p)
0
0.1
0.2
0.3
0.4
0 0.2 0.4 0.6 0.8 1
p
per
pat
ch r
ates
a0 e{-a1 p} m(1-p)
0
0.02
0.04
0.06
0.08
0 0.2 0.4 0.6 0.8 1
p
tota
l rat
es
a0 p e{-a1 p} mp(1-p)
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1
p
per
pat
ch r
ates
a0 e{-a1 p} m(1-p)
dt
dp
p*p
Decreasing probability of extinction with increasing proportion of patchesoccupied is an example of positive density-dependence. The effect here isto create a threshold proportion of patches that need to be occupied toavoid metapopulation extinction.
If the metapopulation level effect is due to negative per patch rate ofchange of patch occupancy at low p, then this is called a metapopulation‘Allee effect’ (Amarasekare 1998. Allee effects in metapopulation dynamics.American Naturalist. 152). Just as the Levin’s model can be thought of asanalogous to the logistic model of population growth rate, if the per capitapopulation growth rate becomes negative at small population size, thiscreates a threshold population size, below which extinction results– a phenomenon known as the Allee effect (Stephens et al. 1999. What isthe Allee effect? Oikos. 87, 185-190).
Patches as “Networks”
• Simplest models have all patches equally connected
• But patches may be connected as networks, for example:
Networks in Matrix Format
• This can be written as a matrix of (direct) connections:
0000
1000
0100
0010
C
Structure of Connections
• Multiplying this matrix together once gives the patches connected by two steps etc:
0000
0000
0000
1000
;
0000
0000
1000
0100
32 CC
Summary
• Heterogeneity between individuals is what biology is about– Extends to heterogeneity between populations
• Nothing is the same and doesn’t stay the same– We haven’t touched on evolution
• Dispersal is universal and leads to metapopulations
Lecture 3. Small Populations
• Stochastic effects– demographic & environmental– demographic stochasticity in small populations– stochastic modelling
• e.g. death process; immigration-death process• Monte Carlo simulation
• Probabilities of extinction
Stochasticity• Deterministic models
– give expected (average) outcome (in most cases)
• Demographic– individuals come in single units– e.g. if deterministic model predicts 5.6 individuals, at
the limit of accuracy the number can only 5 or 6
• Environmental– environments (resource availability) fluctuates
“randomly”– chance events (c.f. disturbance)
Demographic Stochasticity• More predominant in small populations
– relative error is greater (c.f. plotting on log scale)
• Gender problems– if a population of 10 individuals produces 15 offspring,
there is a 15% chance of 5 or less females
0
0.05
0.1
0.15
0.2
0.25
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0.2
0.4
0.6
0.8
1
1.2
Pure Death Process
• Deterministic model– negative exponential decay: N(t)=N(0)exp(-t)
• Stochastic model– exp(-t) is the probability of an individual
surviving to t – if individual survival is independent, then the
numbers surviving to t have a binomial distribution with mean N(0)exp(-t)
Immigration-Death Process
• Solution is sum of two populations• Immigration process is a Poisson process
– Poisson distribution for numbers immigrating in a given time
• Death process– binomial distribution for numbers surviving to
given time
Stochastic Population Processes
Process Popn. Distribution
Death Binomial
Immigration-Death Poisson
Birth-Death Negative Binomial
Imm.-Birth-Death Negative Binomial
“Logistic” “normal”
Monte Carlo Simulation
• More complex models require computer simulation to find solutions
• Collection of Poisson processes (i.e. assume independence between different processes)
• Process rates change with time
Monte Carlo Example• Immigration-death process• Process rates
– immigration: – death: N– total: T = + N
• Time to next event– from negative exponential distribution with
rate parameter T
T
psep Ts )ln(
;
Monte Carlo Iteration
• Calculate time to next event• Calculate which event has occurred
– /T is probability of immigration– N/T is probability of death
• Change population• Calculate T
Birth-Death Simulation
• Deterministic result is the mean of many stochastic results– not true for every model
• Individual simulations do not “look like” the mean– interpreting data
Random Walks
• Alternative view of stochastic models• Population size is performing a “random walk”
through time• Zero is an “adsorbing barrier” (extinction)
– Or 1 if dioecious• All populations (which have a death process)
have a non-zero probability of reaching zero
Time to Extinction• Death Process
– mean time to extinction is :• Birth-Death Process
– for b < d and N0 = 1 :
– depends on the absolute value of b & d (not just difference, r, as deterministic mean)
– faster reproducing spp. (big b) have longer times to extinction
0ln1
Nd
TE
d
b
bTE 1ln
1
Probability of Extinction
• Assured in pure death process• Birth-Death Process
– prob. of extinction by time t :
– N0 lines of descent have to become extinct
– prob. of ever extinction (b>d):
– ultimate extinction is increasingly unlikely as N0 increases
0
exp
exp)(
N
E rtdb
rtddtp
0
)(N
E b
dp
1 3 5 7 9
2.1
5.11E-14
1E-12
1E-10
1E-08
1E-06
1E-04
0.01
1
Prob. Ext.
N0
Birth Rate
Effect of Increasing Rates
• Increasing birth rates increases variability• Increasing death rates decreases variability• In extinction probabilities, increasing variation
increases extinction probabilities– Thus, for the same expected growth rate (r and R),
increasing birth rates (and increasing death rates) increases chance of eventually reaching absorbing barrier
Logistic Results
• Extinction is certain• Time to extinction
– with N0 = 1 :
– ln(TE) is a measure of stability of a population
KTrK
eT E
K
E ln;
0.05
0.15
0.25
0.35
0.45
10
50
90
110000
1E+101E+151E+20
1E+25
1E+30
1E+35
1E+40
1E+45
Te
r
K
Summary
• Dynamics and heterogeneity have a reciprocal effect on each other– Dynamics creates variability– Variability influences dynamics
• The mean is not always informative– The average human being…
• Studying variability (and the dynamics of variability) is usually more informative than studying the mean (and the dynamics of the mean)
Extinction Summary• Probability of extinction is less likely
– the greater the population size• each line of descent has to become extinct
– the carrying capacity is large• the random walk is further from the absorbing boundary
– the greater birth rates are compared to death rates• i.e. the larger the value of r
– the smaller the variation in population size• the smaller the birth rates, but see above
• Explains why the majority of populations of conservation concern tend to be large mammals in small habitats
Lecture 4. Modelling
Why prediction fails• Models are necessarily under-specified
– They have to be to be useful• The correspondence of the causal relationships they embody to actual
phenomena is never known to be perfect• The observable initial conditions are never perfectly observed• There are always unobservable initial conditions.
– What if a big asteroid hits? That might be predictable in the sense that the asteroid is already on its collision course with the Earth, but from a practical viewpoint it may be unobservable
• Parameters are estimates• Some processes are chaotic, such that arbitrarily small errors will
cumulate to arbitrarily large deviations from prediction
Principles• Exponential growth• Density Dependence
– positive• cooperation; aggregation; Allee effect
– negative• necessary but not sufficient for stability
• Circular Causality– pathways created by interaction with environment (inc. other species)– low frequency cycles / oscillations– time delays
• Limiting Factors– populations exist in complex webs of interaction, but only a few are
important at particular times / places