title: Stochastic eco-evolutionary model of a prey-predator community.
abstract: We are interested in the impact of natural selection in a prey-predator community. We introduce an individual-based model of the community that takes into account both prey and predator phenotypes. Our aim is to understand the phenotypic coevolution of prey and predators. The community evolves as a multi-type birth and death process with mutations. We first consider the infinite particle approximation of the process without mutation. In this limit, the process can be approximated by a system of differential equations. We prove the existence of a unique globally asymptotically stable equilibrium under specific conditions on the interaction among prey individuals. When mutations are rare, the community evolves on the mutational scale according to a Markovian jump process. This process describes the successive equilibria of the prey-predator community and extends the polymorphic evolutionary sequence to a coevolutionary framework. We then assume that mutations have a small impact on phenotypes and consider the evolution of monomorphic prey and predator populations. The limit of small mutation steps leads to a system of two differential equations which is a version of the canonical equation of adaptive dynamics for the prey-predator coevolution. We illustrate these different limits with an example of prey-predator community that takes into account different prey defense mechanisms. We observe through simulations how these various prey strategies impact the community.
authors.affiliation: CMAP, École Polytechnique, CNRS UMR 7641, Route de Saclay, 91128, Palaiseau Cedex, France. manon.costa@cmap.polytechnique.fr.
authors.forename: Manon
authors.initials: M
authors.lastname: Costa
authors.affiliation: Institute of Ecology and Environmental Sciences-Paris (UPMC-CNRS-IRD-INRA-UPEC-Paris Diderot), Université Pierre et Marie Curie, UMR 7618, Paris, France.
authors.forename: Céline
authors.initials: C
authors.lastname: Hauzy
authors.affiliation: Institute of Ecology and Environmental Sciences-Paris (UPMC-CNRS-IRD-INRA-UPEC-Paris Diderot), Université Pierre et Marie Curie, UMR 7618, Paris, France.
authors.forename: Nicolas
authors.initials: N
authors.lastname: Loeuille
authors.affiliation: CMAP, École Polytechnique, CNRS UMR 7641, Route de Saclay, 91128, Palaiseau Cedex, France.
authors.forename: Sylvie
authors.initials: S
authors.lastname: Méléard
mesh_terms: D000222:Adaptation, Physiological; D000818:Animals; D005075:Biological Evolution; D019295:Computational Biology; D017753:Ecosystem; D008390:Markov Chains; D055641:Mathematical Concepts; D008954:Models, Biological; D009154:Mutation; D010641:Phenotype; D011157:Population Dynamics; D011235:Predatory Behavior; D012641:Selection, Genetic; D013269:Stochastic Processes
keywords: Adaptive dynamics; Long time behavior of dynamical systems; Lotka-Volterra equations; Multi-type birth and death process; Mutation selection process; Polymorphic evolution sequence; Predator-prey