Qualitative Analysis,Parameter Estimation And Simulation Of Two Discrete Predation Models

In recent years,differential equations have been widely applied in various fields,especially in the field of biomathematics.In this thesis,we mainly study the bifurcation problems of two kinds of discrete predator-prey models and estimate the parameters in a stochastic differential equations with Brownian motion.In Chapter 1,introduces the background,significance of the research and the maj or work.In Chapter 2,we recall the basic concepts for discrete dynamical system,the local bifurcation theory,and the method to estimate the parameters in a stochastic differential equation.In Chapter 3,some complicate dynamical behaviors are formulated for a discrete predator-prey model with group defense and nonlinear harvesting in prey.After considering the existence and stability for all of its nonnegative fixed points,it is main for us to present those conditions for the occurrences of transcritical bifurcation,saddlenode bifurcation and Neimark-Sacker bifurcation,respectively.Numerical simulations not only verify the theoretical results for saddle-node bifurcation and Neimark-Sacker bifurcation,but also display more interesting dynamical properties of the model.In Chapter 4,we revisit a discrete predator-prey model with Allee effect and Holling type-I functional response.The most key is for us to find the bifurcation diffrenece:a flip bifurcation occurring at the fixed point E3 in the known results can not happen in our results.The reason leading to this kind of difference is the different discrete method.In order to demonstrate this,we first simplify corresponding continuous predator-prey model,then apply a different discretization method to this new continuous model to derive a new discrete model.Next,we detailedly consider the dynamics of this new discrete model.By using a key lemma,the existence and local stability of nonnegative fixed points E0,E1,E2 and E3 are completely studied.By employing the Center Manifold Theorem and bifurcation theory,the conditions for the occurrences of Neimark-Sacker bifurcation and transcritical bifurcation are obtained.Our results complete the corresponding ones in a known literature.Numerical simulations are also given to verify the existence of Neimark-Sacker bifurcation.In Chapter 5,we consider the continuous predator-prey model with group defense and nonlinear harvesting,introduce the stochastic perturbation term,and construct the stochastic differential equation model.Euler-maruyama method is used to discretized the stochastic differential equation,and pseudo maximum likelihood estimation method is used to estimate the parameters of the newly constructed discrete biological mathematical model.