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

Document Type: Research Paper

Authors

1 Esfarayen university

2 Department of Mathematics University of Neyshabur

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 transcritical 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 occured‎.

‎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 althought‎
‎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