Two Predation Models With Allee Effect And Stage Structure And Optimal Capture

Predation model is a kind of ecological mathematical model favored by biologists and mathematicians,the predation model has been paid much attention by many researchers.By carefully observing the living conditions of various populations,a large number of ecologists have found that the growth process of many populations is related to age,time delay,and Allee effect.In order to make the predation model more practical,the Allee effect,Age structure and time delay are added into the predation model in this paper.In addition,considering optimal capture in predation models not only maintains species diversity,but also promotes social progress and development.Therefore,to make the model have better research value,this paper studies two kinds of predation models with Allee effect and Age structure and the optimal capture.In chapter 1,this paper describes the research background and significance of the predation model with Allee effect and Age structure and the optimal predation problem,and introduces the research status at home and abroad,and finally gives the main research content of this paper.In chapter 2,this paper introduces the theory of differential equation and the optimal capture problem.In chapter 3,this paper establishs and studies a predation model with additive Allee effect and Age structure of prey growth rate,The local stability of the positive equilibrium is proved by Routh Hurwitz theorem,and the global stability is proved by constructing Lyapunov function.The optimal capture effort is obtained by using Pontryagin maximum principle.Finally,the influence of Allee effect on the stability of the system is simulated.In chapter 4,a predation model with Allee effect and Age structure was established,and a time delay was added to the model,analyze the absence and existence of predators respectively.Firstly,we study the case that the predator does not exist,and discuss the existence of the positive equilibrium point and local stability of the model.Then,we study the case that the predator does exist,and the positive equilibrium point of the system is calculated,and the local stability of the model with and without time delay is analyzed respectively.Then,the existence and expression of the optimal capture effort are proved,Finally,the effects of time delay and juvenile feeding rate on the stability of the system and the existence of the optimal harvesting effort were numerically simulated.In chapter 5,this paper summarizes the main research content and research results,and puts forward the innovation and shortcomings,which is convenient for others to study further.Finally,acknowledgements and references are given in this paper.