Bifurcation analysis of a non-smooth prey–predator model by a differential linear complementarity system

The aim of this paper is to investigate a prey–predator system with threshold harvesting for both species, whereas most researchers are devoted to studying a threshold policy for one population to avoid intractable mathematical simulation occurred by their coupling. Due to the equivalence between differential inclusion and differential complementarity system, we reformulate this ecosystem as a differential linear complementarity system. Then specialized algorithms for the complementarity problem allow us to handle such non-smooth structure, thereby performing a numerical examination of the dynamics and bifurcations of our proposed system with success. As a result, we can observe that this system exhibits many peculiar bifurcation patterns that are inherent to a non-smooth dynamic system, including boundary node bifurcation, pseudo-saddle–node bifurcation, touching bifurcation, and sliding homoclinic bifurcation. Specifically, we observe a multiple crossing bifurcation that results from the superposition of a sliding homoclinic bifurcation and a boundary saddle bifurcation. They are originated from the coupling of their own independent non-smooth structures of the two species. This system also admits some conventional bifurcations like saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, which are identified previously in a prey–predator system without threshold policy. Both the theoretical and numerical results indicate that the non-smooth structure of the threshold harvesting policy increases equilibria, as well as the dynamical complications of the system.