A Class Of Delay And Ratio-dependent Holling Type Iv Type And Leslie Predator - Prey Model And Its Control

The dynamic behavior between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology. In previous studies, Considerable progress has been made about predator- prey theory, but there are still many problems to be studied further.In this paper, a delayed ratio-dependent predator-prey system of holling-â…£and Leslie is considered,In addition to taking into account the ratio functional response function, the control and Anti-control theory is first applied in such biological system. Through the control of the different coefficient values, the stability of the equilibrium point can be changed from stable to unstable or from unstable to stable.Our paper is separated into five chapters. The first chapter is the introduction and preparation knowledge; Chapter 2 presents the results on boundedness of solutions. While there has no control, the behavior of the system is discussed in Chapter 3. No defines in the point (0,0) of the system brings much difficulties in the research on the kinetic nature near (0, 0). By applying the series theories and methods in the literature on the singular point of nonlinear equations, we solve this problem satisfactorily and give the all possibility of the orbits of the system converging (0,0) as tâ†'+∞(or tâ†'-∞) in the First Quadrant. Then we discuss the stabilities of other fixed points, and prove the conditions appeared Hopf bifurcation by the Center manifold theorem and normal form. While there has control, the behavior of the system is discussed in Chapter 4. The stability and branch direction of Hopf bifurcation are discussed in this chapter by Hassard methods. In chapter 5, some numerical simulations are performed to illustrate the analytical results found.