Analysis of a model of two parallel food chains

In this paper we study a mathematical model of two parallel food chains in a chemostat. Each food chain consists of a prey species $x$ and a predator species $y$. Two food chains are symmetric in the sense that the prey species are identical and so are the specialized predator species. We assume that both of the prey species in the parallel food chains share the same nutrient $R$. In this paper we show that as the input concentration $R^{(0)}$ of the nutrient varies, there are several possible outcomes: (1) all species go extinct; (2) only the two prey species survive; (3) all species coexist at equilibrium; (4) all species coexist in the form of oscillations. We analyze cases (1)-(3) rigorously; for case (4) we do extensive numerical studies to present all possible phenomena, which include limit cycles, heteroclinic cycles, and chaos.