ANALYSIS OF MULTI-SPECIES ECOLOGICALAND EVOLUTIONARY DYNAMICS

Ecole Normale Supérieure, ParisDecember 9-13, 2013

Adaptive Dynamics (AD) and its canonical equation (F. Dercole)Introduction to evolutionary dynamics with examples within and beyond biology. Modeling approaches to evolutionary dynamics. The AD approach through a representative example: the evolution driven by the competition for resources.The AD canonical equation.Further readingsAnalysis of Evolutionary Processes, Princeton Univ. Press, 2008, Chaps. 1-3 and Appx. B, CTechnovation (2008) 28:335-348 J. Theor. Biol. (1999) 197:149-162

6.

A naive introduction to innovation and competition processesInnovations and competition

Evolution

stationary non-stationary(Red Queen Dynamics)

multiple

Evolutionary attractors

Evolutionary branching Evolutionary extinction

Evolution in biology

Evolution outside biology

(see f.r. 1 Chap. 1)

Modeling approaches to innovation and competition processes

(see f.r. 1 Chap. 2)

Each individual is characterized by 0, 1, or more inheritable traits (phenotypes/strategies)

Adaptive Dynamics – Basic assumptions

Reproduction is clonal (asexual)thus offspring are either characterized by the trait of the parent or are mutants

Mutations in different traits of the same individual are independent

Mutations are rare on the ecological time scale

Mutations are small

The coexistence of populations is stationary

The (abiotic) environment is isolated, uniform, and invariant

Traits are quantitative characteristicsdescribed as continuous variables (symbol ), possibly through a scaling

See f.r. 1. See also the original contributionsMetz et al. (in Stochastic and Spatial Structures of Dynamical Systems, Elsevier 1999)Geritz et al. (Phys. Rev. Lett. 78, 2024-2027, 1997; Evol. Ecol. 12, 35-57, 1998)Dieckmann & Law (J. Math. Biol. 34, 579-612, 1996)

The AD working scheme

The AD canonical equation through a simple exampleQuestion: Does the competition for resources optimize a morphological phenotype,e.g. body size, or promote genetic diversity? (see f.r. 2 and 3)

Let’s start with a single resident population: the resident model is the logistic one!

The resident (ecological) equilibrium

The AD working scheme

The carrying capacity

The AD working scheme

The AD canonical equation through a simple exampleQuestion: Does the competition for resources optimize a morphological phenotype,e.g. body size, or promote genetic diversity? (see f.r. 2 and 3)

Let’s start with a single resident population: the resident model is the logistic one!

The competition function

The resident-mutant model

The resident (ecological) equilibrium

The competition function

The carrying capacity

symmetric competition asymmetric competition

The model parameters :

The AD canonical equation through a simple exampleQuestion: Does the competition for resources optimize a morphological phenotype,e.g. body size, or promote genetic diversity? (see f.r. 2 and 3)

Let’s start with a single resident population: the resident model is the logistic one!

The competition function

The resident-mutant model

The resident (ecological) equilibrium

The AD working scheme

The mutant invasion fitness

The pairwise invasibility plot: the sign of the fitness

It is the initial per-capita rate of growth of the mutant population

moreover, invasion implies substitution(see f.r. 1 Appx. B)

The selection derivative :

We expect to to have the same sign of

Technically, it is the eigenvalue determining invasion

and have opposite sign

if then

where is the probability of a mutation at birth, is the standard deviation of mutations,and in the limit of extremely rare and small mutations

The AD canonical equation

(see f.r. 1 Chap. 3 and Appx. C)

The AD working scheme

where is the probability of a mutation at birth, is the standard deviation of mutations,and in the limit of extremely rare and small mutations

The AD canonical equation

The evolutionary equilibrium

such that . . It results

Stability via linearization

eigenvalue

is stable for all parameter settings, so there are no bifurcations

(see f.r. 1 Chap. 3 and Appx. C)

At , , so that invasion does not necessarily implysubstitution. Can residents and mutants coexist and undergoevolutionary branching?

?

And what if we have a large mutation (or, most likely, the introduction of an alien species)?The resident-mutant model (or a suitable resident model) gives the resulting ecological attractor(an equilibrium?) for which we can derive the corresponding canonical equation