Eur. Phys. J. Plus (2018) 133: 29
https://doi.org/10.1140/epjp/i2018-11864-8

Regular Article

Dynamical study of a chaotic predator-prey model with an omnivore

¹ Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, 11451, Riyadh, Saudi Arabia
² Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, 41522, Ismailia, Egypt
³ Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, 35516, Mansoura, Egypt

^* e-mail: aelsadany1@yahoo.com

Received: 2 July 2017
Accepted: 8 January 2018
Published online: 31 January 2018

Abstract

In this paper, the dynamics and bifurcations of a three-species predator-prey model with an omnivore are further investigated. The food web considered in this work comprises prey, predator and a third species, which consumes the carcasses of the predator along with predation of the original prey. The conditions for existence, uniqueness and continuous dependence on initial conditions for the solution of the model are derived. Analytical and numerical bifurcation studies reveal that the system undergoes transcritical and Hopf bifurcations around its equilibrium points. Further, the Hopf bifurcation curves in the parameters’ space along with codimension two bifurcations of equilibrium points and bifurcation of limit cycles that arise in the system are investigated. In particular, the occurrence of generalized Hopf, fold Hopf and Neimarck-Sacker bifurcations is unveiled and illustrates the rich dynamics of the model. Finally, bifurcation diagrams, phase portraits and Lyapunov exponents of the model are presented.