Dynamics Of Some Prey-predator Models With Reaction And Diffusion

The reaction-diffusion prey-predator models have always been a hot topic in the field of biomathematics and has received much attention from many biologists and mathematicians.At present,an important direction of its researches is to integrate it with different ecological background,establish and analyze new biomathematical models,to understand more biological processes and mechanisms.This thesis studies the dynamics of several reaction-diffusion prey-predator models that combine different ecological background,including the Allee effect,spatially inhomogeneous environments and the infectious disease.The main work is as follows:Firstly,for a Leslie-Gower prey-predator model with strong Allee effect in prey,we discuss detailedly the existences of Hopf bifurcations and steady-state bifurcations.The direction of Hopf bifurcations and the stability of periodic solutions are obtained by using center manifold theory and normal form methods.This chapter not only improves and complements the dynamic analysis of this model by Mingxin Wang and Wenjie Ni,but also describes the influence of the Allee effect on the classical Leslie-Gower prey-predator model.The numerical simulation verifies some of the theoretical results.Secondly,a diffusive prey-predator model with strong Allee effect and a protection zone for the prey is considered.Using the method of upper and lower solutions and comparison principle,we show that the global existence and long-time behavior of positive solutions.With the help of an auxiliary equation and the comparison principle,it is proved that over-exploitation occurs in the system and solutions to avoid over-exploitation are proposed.At last,using the bifurcation theory and the linearization theory,we demonstrate that the existence and stability of steady-state bifurcations branching from constant semi-trivial solutions.Thirdly,a diffusive Beddington-De Angelis prey-predator model in the heterogeneous environment is investigated.By using the method of upper and lower solutions,the comparison principle and the regularity theory of parabolic equations,we obtain the conditions for the extinction of predator and the unboundedness of prey,and prove that the system is persistent when the size of crowding region is suitable.In addition,the unique positive steady-state solution is globally asymptotically stable when the growth rate of prey is appropriate.Finally,a diffusive prey-predator model with a transmissible disease in the prey population is discussed.The dissipation of the system is obtained by using the semigroup method.Through eigenvalue analysis and constructing the Lyapunov function,we prove that the local stability and globally asymptotic stability of the constant steady-state solution respectively.The a priori estimate of the positive steady-state solution is given by using the maximum principle and the Harnack inequality.The non-existence of nonconstant positive steady-state solution is obtained by the energy method.Taking advantage of the topological degree theory and the bifurcation theory,we show the existence of non-constant positive steady-state solution.