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

Document Type : Research Paper


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

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



‎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.