Predator-Prey Model with Prey Harvesting, Holling Response Function of Type III and SIS Disease
Keywords:Prey, Infectious disease, Response function, Bifurcation, Global Stability.
AbstractThe populations of prey and predator interact with prey harvesting. When there is no predator, the logistic equation models the behavior of the preys. For interactions between preys and predators, we use the generalized Holling response function of type III. This function which models the consumption of preys by predators is such that the predation rate of predators increases when the preys are few and decreases when they reach their satiety. Our main goal is to analyze the influence of a SIS infectious disease in the community. The epidemiological SIS model with simple mass incidence is chosen, where only susceptibles and infectious are counted. We assume firstly that the disease spreads only among the prey population and secondly that it spreads only among the predator population. There are many bifurcations as: Hopf bifurcation, transcritical bifurcation and saddle-node bifurcation. The results indicate that either the disease dies out or persists and then, at least one population can disappear because of infection. For some particular choices of the parameters however, there exists endemic equilibria in which both populations survive. Numerical simulations on MATLAB and SCILAB are used to illustrate our results.
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