Oscillatory Coexistence of Species in a Food Chain Model With General Holling Interactions

Abstract

Proper choice of interaction function is one of the most important parts for modelling a food chain. The interaction functions of Holling type-I, type-II or type-III are not always appropriate for modelling a food chain. Therefore, the study of food chain models with general Holling interactions is necessary for constructing realistic food chain models. In this paper, we propose a tri-trophic food chain model with general Holling interactions. We derive extinction criterion for species and global stability conditions. Bifurcation analysis of the model is done with respect to general Holling parameters. Conditions for Hopf bifurcation are discussed theoretically and existence of Hopf bifurcation is shown numerically. We utilize the MATCONT package to show the existence of Hopf point, limit point and branch point as the two important general Holling parameters are varying. Our analysis demonstrates that the dynamics of predator–prey model strongly depends on interaction function characterized by Holling parameters. We have shown stable coexistence as well as oscillatory coexistence of species in the model for specific range of values of Holling parameters.