Dynamical and numerical analyses of a generalized food-chain model

This paper focuses on the qualitative and quantitative analysis of a generalized three-species food-chain model. The conditions for the existence and stability of the boundary and positive equilibria of the model are established. By constructing an appropriate "bifurcation function", it is shown that the model undergoes a Hopf bifurcation from the positive equilibrium for certain parameter values. The root of this bifurcation function gives the bifurcation value. A robust non-standard numerical method is constructed and used to obtain the solution of the model. Numerical simulations using this method show that the Hopf bifurcation is supercritical with a stable limit cycle in the positive octant.