Dynamic Behaviors Of The Ratio-dependent Predator-prey System

Population dynamics is the important branch of studying mathematical biology. Considering the characteristics of the species and the intra-relations with the other species during its evolution, we establish the mathematical models. These models can describe quantitatively the species evoluation. Further, we apply the mathematical theoreies, methods, and computer software to analysis the dynamicla behaviors of pop-ulations, which shows and predicts the evoluation laws of population, and supply the theoretical references when the policy of protecting the species and maintaining the balance of biological system will be enacted. Predator-prey is a common intra-relation coexisting the same bilogical system. It is reasonable that we assume that the predator-prey response funtion is dependent on the ratio of the predator to the prey. In this paper, we study the dynamical behaviors for three classes of the predator-prey systems with ratio-dependent functional response.In chapter2, we assume that the Gipin-Ayala evoluation equations is satisfied by the prey, and the predator-prey response funtion is the ratio-dependent functional response. We establish the predator-prey system discribed by the ordinary differential equations, and obtain the existence and stability of the equilibrium with the qualitative theory and stability of ordinary differential equations. Further, the extinction and permanence is investigated.A fact is the population evolution processes undergo relatively long periods of smooth variations followed by a instantaneous rapid change, which may be caused by human activity, epidemic disease, and season change. Therefore, impulsive differen- tial equations are the more suitable for simulating the evolutionary processes of the species. In chapter, we estabilish the predator-prey system with ratio-dependent func-tional response and the constant stocking of the predator at the fixed moments. With the Floquet theory and comparison theorem on the impulsive differential equations, the existence and stability of the boundary periodic solution are considered. Moreover, we discuss the extinction and permanence for such a system.In chapter4, we assume that Logistic equation is satisfied by the predator, which implies that the predator is density dependence. And we assume that the predator is harvesting by the constant ratio of its level at the fixed moments. We establish a predator-prey model with ratio-dependent functional response and the ratio-dependent harvesting of the predator impulsively. We discuss the extinction and permanence of system, and illustrate our results by the numerical simulation.