Study On Almost Periodic Solution、Stability And Permanence Of Several Eco-models

The qualitative analysis of biological models is an important method to study the ecological species. At present, we do lots of work on ecological stability of biological mathematics, and we also get a lot of results. These work include cooperative population models, models considering the impact of environmental toxins on population and predator-prey models. But the environment patch, harvesting and delays have an impact on dynamic behavior. So, the related problems will be discussed and studied in this thesis. The thesis will be divided into the following parts:In the first part, we consider stability of almost periodic solution for nonautonomous system with time delays. Firstly, we get that the system is ultimately bounded. Then, by constructing a suitable Lyapunov function, we obtain that there exists a unique almost periodic solution and uniformly asymptotically stable in a certain conditions.In the second part, the persistence and extinction of a single species with diffusion and harvesting are considered. With the development of the economy, plants can not purify environmental toxins completely, so toxin’s damage to the living environment of populations are serious and it effects the persistence and extinction of the populations. In this part, the species can diffuse between the polluted patch and the protected patch. What’s more, harvesting is considered. By analyzing the right hand functions, the sufficient conditions for persistence and extinction of the species are obtained, furthermore, we found that the influences of harvesting and diffusion are important to the survival of the species. At last, several examples are given toillustrate the effectiveness of our results.In the third part, we study the dynamic system of cooperative models with harvesting. Dynamic behavior has been studied by many scholars, but this kind of system is not studied widely. So, we discuss this kind of system in this part. By calculating and analyzing the characteristic roots of the model and constructing suitable function, we prove that positive equilibrium of the model are globally asymptotically stable. Furthermore, almost periodic solution of the corresponding nonautonomous model is unique, which is uniformly asymptotically stable. Examples are given to illustrate the effectiveness of our results.Finally, the stability of prey-predator system with time delay is described in this part. By analyzing the eigenvalue roots of the system and different delays, the local stability of the equilibrium and the sufficient conditions for k switches are obtained. the globally asymptotical stability of positive equilibrium is derived. At last, numerical simulations are given to verify our main results.