Hopf Bifurcation and Stability Analysis of Delayed Lotka–Volterra Predator–Prey Model Having Disease for Both Existing Species

Abstract

Abstract The study of biological problems using mathematical models not only has had significant advances but also it has attracted the attention of many scientists. Mathematical modeling of diseases enables one to predict when the disease occurs, and therefore it leads to the successful control of the diseases before it gets epidemics. This paper constructs a biological model in the mathematical aspect. In this paper, a delay predator–prey model is proposed with logistic growth in the prey population. It is assumed that this model includes an SIS infection in both prey and predator species. After dealing with disease in the prey population, we analyze the disease predator population. Moreover, we prove the existence of Hopf bifurcation for this system by analyzing characteristic equations. Then important threshold quantities are identified. Our theoretical study indicates that threshold quantities \(R_1\) and \(R_2\) are important when a transmissible disease runs among the prey population and the predators are disease, respectively.KeywordsHopf bifurcationAsymptotically stabilityPrey–predatorLotka–VolterraDelayAMS Subject Classification34C2334L2092D40