International Journal of Nonlinear Analysis and Applications

Investigating the dynamics of Lotka$-$Volterra model with disease in the prey and predator species

Document Type : Research Paper

Authors

1 Esfarayen University of Technology‎, ‎Esfarayen‎, ‎North Khorasan‎, ‎Iran

2 Department of Mathematics‎, ‎University of Neyshabur‎, ‎Adib BLVD‎, ‎Neyshabur‎, ‎Iran‎

Abstract

‎In this paper, a  predator$-$prey model  with logistic growth rate in the prey population was proposed.  It included an SIS infection in the prey and predator population.  The stability of the positive equilibrium point, the existence of Hopf and transcortical  bifurcation with parameter $a$ were investigated, where $a$ was regarded as  predation rate. It was found that when the parameter $a$ passed through a critical value,  stability changed and Hopf bifurcation occurred.  Biologically, the population  is positive and bounded. In the present article,  it was also shown that the model was bounded and that it had the positive solution.  Moreover, the current researchers came to the conclusion that although  the disease was present in the system, none of the species would be extinct. In other words, the system was persistent. Important thresholds, $R_{0}, R_{1}$ and $R_{2}$, were identified in the study. This theoretical study indicated that under certain conditions of $R_{0}, R_{1}$ and $R_{2}$,  the disease remained in the system or disappeared.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 633-648
  • Receive Date: 11 December 2017
  • Revise Date: 15 November 2018
  • Accept Date: 03 March 2019