Stability And Pattern Dynamics In A Predator-prey System With Hunting Cooperation

In the dynamics of interacting predator-prey populations,the functional response reflects the capture capacity of the predator population,and its different forms have important influence on the evolution and development of the population.In the mathematical modeling of predator-prey system,how to choose a reasonable functional response function to describe the development and evolution of different kinds of predator and prey populations in the objective world has become an important research topic in the fields of ecology and biological mathematics.Considering that carnivores such as lions,wolves and African wild dogs fight in groups when capturing prey,the more these predators are,the better they are for capturing.Recently,some scholars have proposed a kind of functional response function to describe this biological phenomenon.In this paper,the qualitative theory of differential equations is used to study the effects of such important biological factors as cross-diffusion and Allee effect on the stability,bifurcation,spatial pattern and other dynamics of a predator-prey system with hunting cooperation.The main work of this paper is divided into five chapters: In chapters 1,the biological background and research status of predator-prey system with hunting cooperation are briefly described,and the main work and structure of this paper are stated.In chapter 2,we study the existence and stability of positive equilibrium point and Hopf bifurcation of predator-prey system.It is shown that the ability of the hunting cooperation not only affects the existence of positive equilibrium point,but also affects its stability.In chapter 3,we investigate the effects of diffusion on the stability of positive equilibrium point,as well as instability and pattern induced by cross-diffusion.The contents of this chapter are based on the analysis of the model in Chapter2.Considering the population diffusion and spatial distribution,diffusion terms are introduced into the original model.According to the relationship between self-diffusion and cross-diffusion,we study the stability of diffusion system and Turing instability driven by cross-diffusion,and deduce the amplitude equation to judge the selection and stability of spatial pattern.We study the spatial pattern caused by cross-diffusion by using numerical method combined with the qualitative properties of amplitude equation,and find stable spot,strip and spot-strip pattern.In chapter 4,we study the stability,bifurcation and pattern of a kind of diffusion predator-prey system in which the growth of the prey population has Allee effect and the predator has hunting cooperation.The study shows that the hunting cooperation capability of the predator has important influence on the existence and stability of the equilibrium point of the system.However,its intrinsic growth rate only affects the stability and does not affect the existence of the positive equilibrium point.For the diffusion-driven spatial pattern phenomenon,it is found that the stronger Allee effect is,the more favorable it is for Turing instability.When Allee effect is constant,only when the hunting cooperation ability is greater than a certain critical value can the diffusion induce the pattern.In chapter 5,we summarize the main results of the whole paper and look forward to the further work.