Stochastic eco-evolutionary model of a prey-predator community. (English) Zbl 1335.60159
Summary: 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 a prey-predator community that takes into account different prey defense mechanisms. We observe through simulations how these various prey strategies impact the community.
MSC:
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
| 60J85 | Applications of branching processes |
| 60J75 | Jump processes (MSC2010) |
| 37N25 | Dynamical systems in biology |
| 92D25 | Population dynamics (general) |
| 92D15 | Problems related to evolution |
Keywords:
predator-prey community; multi-type birth-and-death process; Markovian jump process; Lotka-Volterra equations; dynamical systems; mutation selection process; polymorphic evolution sequence; adaptive dynamicsReferences:
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