ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS

1. ANALYSIS OF MULTI-SPECIES ECOLOGICALAND EVOLUTIONARY DYNAMICS 6. 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 readings Analysis of Evolutionary Processes, Princeton Univ. Press, 2008, Chaps. 1-3 and Appx. B,C Technovation (2008) 28:335-348 J. Theor. Biol. (1999) 197:149-162 EcoleNormaleSupérieure, Paris December 9-13, 2013

2. A naive introduction to innovation and competition processes Innovations and competition Evolution

3. Evolutionary attractors stationary non-stationary (Red Queen Dynamics) multiple Evolutionary branching Evolutionary extinction

4. Evolution in biology

5. Evolution outside biology (see f.r. 1 Chap. 1)

6. Modeling approaches to innovation and competition processes (see f.r. 1 Chap. 2)

7. Adaptive Dynamics – Basic assumptions Each individual is characterized by 0, 1, or more inheritable traits (phenotypes/strategies) Traits are quantitative characteristics described as continuous variables (symbol ), possibly through a scaling 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 See f.r. 1. See also the original contributions Metz 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)

8. The AD working scheme

9. The AD canonical equation through a simple example Question: 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

10. The AD working scheme

11. The carrying capacity

12. The AD working scheme

13. The AD canonical equation through a simple example Question: 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-mutantmodel The resident (ecological) equilibrium The competition function

14. The competition function The carrying capacity The model parameters: symmetric competition asymmetric competition

15. The AD canonical equation through a simple example Question: 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-mutantmodel The resident (ecological) equilibrium The competition function

16. The AD working scheme

17. The mutant invasion fitness It is the initial per-capita rate of growth of the mutant population Technically, it is the eigenvalue determining invasion The pairwise invasibility plot: the sign of the fitness if then and have opposite sign moreover, invasion implies substitution (see f.r. 1 Appx. B) The selection derivative: We expect to to have the same sign of

18. The AD canonical equation (see f.r. 1 Chap. 3 and Appx. C) 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

19. The AD working scheme

20. The AD canonical equation (see f.r. 1 Chap. 3 and Appx. C) 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 evolutionary equilibrium such that . . It results Stability via linearization eigenvalue is stable for all parameter settings, so there are no bifurcations

21. At , , so that invasion does not necessarily imply substitution. Can residents and mutants coexist and undergo evolutionary 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 ?