  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.
