Bifurcation of positive solutions for a three-species food chain model with diffusion

Abstract

In this paper, we consider a reaction–diffusion system describing a three-species Lotka–Volterra food chain model with homogeneous Dirichlet boundary conditions. By regarding the birth rate of prey r1 as a bifurcation parameter, the global bifurcation of positive steady-state solutions from the semi-trivial solution set is obtained via the bifurcation theory. The results show that if the birth rate of mid-level predator and top predator are located in the regions 0<r2<λ1a23u3r3 and r3>λ1, respectively. Then the three species can co-exist provided the birth rate of prey exceeds a critical value. Moreover, an explicit expression of coexistence steady-state solutions is constructed by applying the implicit function theorem. It is demonstrated that the explicit coexistence steady-state solutions is locally asymptotically stable.