Abstract 

Three species food web models are fundamental building blocks of large scale ecosystems. To clarify the local or global and short-term or long-term behavior of ecosystems, it is essential to understand the interacting dynamics of three species food web models. A monotone ecosystem whose interactions among n-species are all cooperative or competitive (n = 2) have been well studied in the past three decades thanks to the theory of monotone dynamical systems. However, for a non-monotone system whose interactions are blended at least with one consumption (i.e. herbivory, predation or parasitism), most known results are constrained on two species cases since the classical Poincaré-Bendixson Theorem can be applied. Hence recently attentions have been attracted to the dynamics of a non-monotone ecosystem with at least three species.

In this talk, we will first survey some recent progress on the non-diffusive predator-prey food web models. Then, with diffusion, the existence of traveling wave solutions by upper lower solutions/shooting method are investigated. Finally, some open problem of these topics are proposed.