Study On The Effect Of Prey Refuge For Lotka-Volterra Predator-Prey System

This paper consists of three parts:Firstly, we study the influence of prey refuge on a Lotka-Volterra predator-prey model. The model assumes that the predator experiences density dependence. By constructing some suitable Lyapunov functions, some sufficient conditions are obtained which ensure the global asymptotic stability of the interior equilibrium and boundary equilibrium of the model. The trajectory distribution of system is given. The effect of prey refuge and density dependence of predator species on prey and predator population density is investigated when the interior equilibrium exists. Our results show that the density dependence of predator species plays an important role on equilibrium density of prey and predator.Secondly, we study the influence of mutual interference of predator species on the Lotka-Volterra predator-prey model incorporating a prey refuge. The dynamic behaviors of the system is not easy to investigate since the local stability of the equilibrium could not be judged by computing the Jacobian matrix of the system. To overcome this difficulty, we analyze the geometric topology of the trajectory near the equilibrium by judging the exceptional direction and normal sectors of the equilibrium. After that, by applying the Dulac’s criterion, we are able to show that the interior equilibrium is globally asymptotically stable. Some numerical simulations are carried out to illustrate the effects of mutual interference of predator species and prey refuge.Finally, we study the influence of prey refuge on the Lotka-Volterra predator-prey model with exponential growth on prey species. We consider two cases: the first one predator species rely merely on prey species for living while the other one predator species could take other species as food sources in addition to prey, respectively. In each case, we incorporate prey refuges which protect a constant fraction of the prey and protect a constant number of the prey, respectively. By constructing suitable Lyapunov functions, sufficient conditions which ensure the global stability of the positive equilibrium are obtained. Our findings show that in all those cases, increasing the refuge will increase both the density of prey and predator species.