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ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS Ecole Normale Supérieure, Paris December 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 readings Analysis of Evolutionary Processes, Princeton Univ. Press, 2008, Chaps. 1-3 and Appx. B, C Technovation (2008) 28: J. Theor. Biol. (1999) 197:

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

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stationarynon-stationary (Red Queen Dynamics) multiple Evolutionary attractors Evolutionary branchingEvolutionary extinction

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Evolution in biology

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

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

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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 characteristics described as continuous variables (symbol ), possibly through a scaling 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, , 1997; Evol. Ecol. 12, 35-57, 1998) Dieckmann & Law (J. Math. Biol. 34, , 1996)

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The AD working scheme

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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) Lets start with a single resident population: the resident model is the logistic one! The resident (ecological) equilibrium

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The AD working scheme

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The carrying capacity

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The AD working scheme

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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) Lets start with a single resident population: the resident model is the logistic one! The competition function The resident-mutant model The resident (ecological) equilibrium

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The competition function The carrying capacity symmetric competitionasymmetric competition The model parameters :

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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) Lets start with a single resident population: the resident model is the logistic one! The competition function The resident-mutant model The resident (ecological) equilibrium

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The AD working scheme

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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

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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)

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The AD working scheme

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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)

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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

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