The Study Of Some Classes Of Predator-prey Systems With Multi-species

In this paper, we mainly study the dynamics of some classes of predator-prey systems with multi-species. The article includes four chapters.The preface is in chapter 1. we introduce the research background of this article, the main task and some important preliminaries.In Chapter 2. we study the dynamics of a predator-prey system which is consisted of t-wo competitive prey populations and one predator population, which feeds on both the prey species. Firstly, we investigate the positivity and boundedness of the solutions and the existence of all the possible equilibrium points of this system. Further by using qualitative method of ordinary differential equation, we discuss the local asymptotically stability(LAS) of equilibri-a and the persistence of the system. Constructing Lyapunov function, we obtain the global asymptotic stability of the positive equilibrium. Finally, a numerical analysis is given to show the effectiveness of the main results.In Chapter 3. we propose and investigate a predator-prey model with harvesting and reserve area in a polluted environment. It is assumed that the habitat is made up of reserved area, where the prey lived safely, and unreserved area, where the predator attack its sole food the prey. The predator consumes the prey according to the Holling type II functional response. We investigate the positivity and boundedness of the solutions of this system, as well as the existence of the equilibria of this system. By analyzing the characteristic equations, the local asymptotically stability of feasible equilibria of the system is discussed, as well as globally stability with the help of Lyapunov function. We also discuss the bionomic equilibrium is said to be achieved when the total revenue obtained by selling the harvested biomass equals the total cost utilized in harvesting it. Then an optimal harvesting policy is also discussed using the Pantryagin'.s Maximum Principle. In the end. numerical simulations are carried out to explain the mathematical conclusions.In Chapter 4. we propose and investigate a predator-prey model with additional food to top-predator. We investigate the positivity and boundedness of the solutions of this system, as well as the existence of the equilibria of this system. By analyzing the characteristic equations. the local asymptotically stability of feasible equilibria of the system is discussed. In order to make the model more realistic. We also discuss a delay system corresponding to this system.By analyzing the characteristic equation associated with the model, its linear stability is inves-tigated. Choosing delay terms as bifurcation parameters the existence of Hopf bifurcations is demonstrated. Stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced. Furthermore, some of the bifurca-tion properties including direction, stability and period are given. Numerical simulations were carried out to illustrate the main theoretical results.