Bistability and Hopf bifurcation of a tritrophic system with Holling functional responses

Gamaliel Blé; Víctor Castellanos; Francisco Eduardo Castillo-Santos

doi:10.1016/j.heliyon.2021.e06212

Bistability and Hopf bifurcation of a tritrophic system with Holling functional responses

Heliyon. 2021 Feb 17;7(2):e06212. doi: 10.1016/j.heliyon.2021.e06212. eCollection 2021 Feb.

Authors

Gamaliel Blé¹, Víctor Castellanos¹, Francisco Eduardo Castillo-Santos¹

Affiliation

¹ UJAT, CONACYT-UJAT, División Académica de Ciencias Básicas, Km 1, Carretera Cunduacán-Jalpa de Méndez, c.p. 86690, Cunduacán, Tabasco, Mexico.

Abstract

In this paper we analyze the dynamics of a tritrophic food chain, with functional responses of Holling type III and II for the mesopredator and superpredator respectively and logistic growth rate for the prey. We show that there are parameter conditions for which the system has up to three equilibrium points. When three equilibrium points appear we show that each one of them exhibits a supercritical Hopf bifurcation. Moreover we can also show that there is a set of parameters for which the system exhibits a simultaneous Hopf bifurcation.

Keywords: Hopf bifurcation; Limit cycle; Tritrophic model.