Bifurcation analysis for the coexistence in a Gause-type four-species food web model with general functional responses

The dynamics of an ordinary differential equations (ODEs) system modelling the interaction of four species (one prey or resource population, two mesopredator populations, and one super-predator population) was analyzed. It was assumed that the functional responses for each interaction were general. We showed parameter conditions that ensured that the differential system underwent a supercritical Hopf bifurcation or a Bogdanov-Takens bifurcation, from which the coexistence of the four species was guaranteed. In addition, the results were illustrated by several applications, where the prey had a logistic growth rate. For the interaction of the mesopredators and prey, we considered classical Holling-type functional responses, and for the rest of the interactions, we proposed certain generalized functional responses similar to the well-known "Beddington-DeAngelis" or "Crowley-Martin" functional responses. At the end, some numerical simulations were given.