Co-evolutionarily stable community structure in a patchy environment

Abstract

Co-evolutionarily stable community structures are studied when predators search for several species of prey distributed in a number of patches. Prey may migrate between the patches. In a co-evolutionarily stable community (CSC) with two trophic levels, (1) if a predator species feeds on a prey species in some patch, then the predator should in fact exploit the prey in any patch that contains the prey and are utilized by the predator. If a CSC is at a steady state, then (2) the number of predator species which utilize a particular patch does not exceed the number of prey species in the patch. A simpler model, in which only one species of prey lives in each patch, ignoring prey migration and handling times of predators, is analyzed to investigate dynamical properties. At a steady state in a CSC of this system, (3) all prey species can co-exist. Moreover, (4) no two predator species can attack the same prey species, and (5) the steady state in a CSC is always globally stable. Also, in a case when a prey-predator system does not have a stable steady state, the dynamic structure of a co-evolutionarily stable community is discussed. In a non-equilibrium case, (6) the number of predator species which simultaneously utilize the common two patches does not exceed the sum of the numbers of prey species in the two patches.