The Two Types Of Discrete Functional Response Predator - Prey System

Since the pioneering work of Lorka and Volterr on predator-prey model, predator-prey models have received much attention from scientists. In these models, there are many, systemic and perfect research achievements for the models incorporating HollingI- IV functional response, and the research results of Leslie- Gower and Holling- Tanner models are also more many, moreover, more and more scholar who are attracted give their attention to these models. But besides, research on discrete models and discrete models with delays corresponding continuous systems are not enough for the moment.In this paper, we will consider two discrete predator-prey models with functional response, and discuss the dynamical behaviors of models, furthermore, some dynamical behaviors are simulated.In chapter 2, we firstly introduce some related the knowledge which will be applied, incorporating the type of bifurcation, the general condition of Jury, center manifold theorem and bifurcation theories. According to the continuous model, a discrete predator-prey model with Ivlev's functional response is established, the existence of the equilibria are analyzed, local asymptotic stability of the equilibria are argued by Jury's condition; sufficient conditions of the existence for fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation are also obtained by using center manifold theorem and bifurcation theory, and numerical simulations are used to illustrated dynamical behaviors of positive equilibrium.In chapter 3, concepts of stability, attractivity and some kind of persistence on the discrete dynamical system are introduced. According to the continuous system with delay, a discrete predator-prey model with delay is investigated, and permanence of the system is argued via citing previous results, and that the global attractivity of positive solutions of the system are testified by constructing a Lyapunov function.