Several Kinds Of Model Analysis With ALLee Effect And The Study Of The Optimal Capture Problems

In recent years, with the continuous change of the ecological environment, many species are affected by the Allee effect. The Allee effect will affect the stability of the population system, and even lead to the extinction of some species.Therefore, in order to the healthy de-velopment of the ecosystem, we should research to the Allee effect, so that we can maintain the ecological balance and make the resources get the reasonable development and utilization, thus promoting the sustainable development.This paper, we use the ordinary differential equations and optimal control theory to ana-lyze the following aspects in this paper, which is based on the attitude of active exploration.First, a generalized Logistic model with Allee effect is established.: In this chapter, when v=0,the existence and stability of positive equilibrium is analyzed firstly, and then it is extended to v≠ 0 that is called the generalized Logistic model with Allee effect, we discussed the stability of the positive equilibrium point, and considered the optimal control problem. Thus, it meets the needs of human beings and the development of the population.Second, taking into account the nature of the competition between the prey and the preda-tor, we establish a predator system by Allee effect model:According to the chapter, we consider the predator-prey system is influenced by Allee effect. Meanwhile,making the optimal capture of the prey and the predator simultaneously so as to control the ecological balance. In this part, we obtain the necessary and sufficient conditions for the positive equilibrium point by using the asymptotic stability of differential equation. The global stability of the system is proved by constructing the Lyapunov function and the economic model is established, then we study the ecological equilibrium point. Finally, the optimal harvesting of fresh bait is carried out.Third, ccording to the law of the development of the ecological system, we considered the predator model with Allee effect:In this chapter, we establish the model of the influence of the predator’s Allee effect, and analyze the performance of the five equilibrium points by using the method and theory of the last chapter. We get the sufficient condition for the asymptotic stability of the positive equilibrium point, and prove the global asymptotic stability.Then establishing the economic model that people need to study the ecological balance point,finally,to the development and utilization of resources, we start the optimal harvest of fresh bait.Finally, the overall framework and conclusions are summarized and prospected. Mean-while, we lay the foundation for further study on the paper.