Permanence and coexistence in a diffusive complex ratio-dependent food chain

In this article, we study a complex ratio-dependent food chain model with diffusion in bounded habitat and various boundary conditions. As an extension of simple food chain models studied in earlier literature (Ko and Ahn, Math Biosci Eng 4:1–11, 2007; Kuang and Beretta, J Math Biol 36:389–406, 1998), this model has a more complicated (and general) dynamical structure where the super-predator consumes both the prey and predator. For Dirichlet, Robin, and Neumann boundary conditions, we obtain straightforward criterion for the existence of a positive global attractor which ensures the permanence effect in the ecological system and the presence of a positive steady-state solution. Sufficient conditions on the interaction rates are further given for the uniqueness and global stability of the coexistence state. Several examples give specific parameter sets which satisfy the permanence criteria, or violate the criteria with multiple coexistence states. Numerical simulations of the diffusive food-chain models are also provided to demonstrate and compare the asymptotic behavior of the time-dependent density functions. © 2014, Springer-Verlag Berlin Heidelberg.