The Analysis Of The Solution Of The Reaction-diffusion System With Strong Allee Effect

In this thesis,we study a predator-prey model with strong Allee effect under homogeneous second boundary conditions:Considering the relationship between the living condition and growth trend of species in real ecological environment and the space of species,this thesis studies the reaction diffusion system by introducing diffusion terms on the basis of reference[10].The properties of solutions of the predator-prey model are discussed by means of the theories and methods of partial differential equations?modern analysis and nonlinear analysis in this thesis.The main contents in the thesis are as follows:In Chapter 1,the research background and current situation of the relationship between species and species in ecosystem are introduced,and the reaction diffusion system to be studied in this thesis is also introduced.In Chapter 2,we study the existence of non-constant positive solutions for equilibrium state problems of the reaction-diffusion system.The main contents are as follows:the stability of the positive equilibrium point of the system is proved by the linearization method;The priori estimate of the solution of the equilibrium state problem is given by the maximum principle and the Harnack inequality;Finally,the nonexistence and existence of non-constant positive solutions are obtained by means of energy integration method and topological degree theory,respectively.The results show that if the predator's vaiishing threshold is changed within a certain range,the two species can coexist under the condition that the prey's migration rate remains constant and the predator's vanishing threshold is changed within a certain range.In Chapter 3,the Hopf bifurcation of the system is studied in one-dimensional domain(0,l?).Firstly,the global stability of the internal equilibrium solution is proved by constructing Lyapunov function.Secondly,the global stability of the equilibrium solution and the possibility of existence of the Hopf bifurcation solution are verified by numerical simulation.Finally,by using the central manifold theorem and the regularity theory,the existence of the bifurcation periodic solution and the conditions for judging the bifurcation direction and stability of the bif:urcation periodic solution axe obtained.The results show that the stable periodic behavior of the two species will occur by reasonably controlling the migration rate of the prey and the vanishing threshold of the predator.