Dynamics of Three Species Food Chain Model with Holling type II Functional Response

TL;DR

This paper analyzes a three-species food chain model with Holling type II response, providing stability criteria, extinction conditions, and numerical insights into complex dynamics like chaos and bi-stability.

Contribution

It introduces parameters  and  to reformulate stability and existence conditions, offering a comprehensive classification of the model's dynamics.

Findings

Local stability and equilibria characterized by  and 

Three global extinction results analytically proved

Numerical simulations reveal chaos and bi-stability phenomena

Abstract

In this work, we revisit the classical Holling type II three species food chain model with a different viewpoint. Two critical parameters $\lambda_1$ and $\lambda_2$ dependent on other six parameters are defined. We show that local stabilities and existence of all equilibria can be reformulated by $\lambda_1$ and $\lambda_2$, and the complete classifications of parameters and its corresponding dynamics are given. Three global extinction results are showed by comparison principle analytically. Moreover, some interesting numerical simulations, including the chaotic phenomena, and the bi-stability phenomena, including equilibrium-cycle bi-stability and cycle-cycle bi-stability, are presented numerically. Finally, a brief discussion and biological implications are given.