Research On The Stability And Almost Periodic Solution Of Some Biological Models

In this paper, we mainly investigate the stability and almost periodic solution of biological models. This paper consists of the following three parts:In Chapter2, we consider the stability of two types predator-prey systems with stage structure. First, we study a stage-structured predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes. By iterative technique and further precise analysis, sufficient conditions ensuring the global attractivity of the system are obtained, and a numerical simulation is presented to verify our main result. Next, we study a predator-prey system with stage structure and mutual interference, the existence of a unique interior equilibrium of the system is obtained. By analyzing the characteristic equations, we study the local stability of the interior equilibrium of the system. Using the iterative method, sufficient conditions ensuring the global attractivity of the system are obtained. We show that the mutual interference can enhance the stability of system.In Chapter3, we investigate an autonomous Lotka-Volterra competitive system with infinite delays and feedback controls. The existence and global stability of equilibriums are discussed using the Lyapunov functional method. We analyze the influence of feedback controls on the stability of equilibriums of system. When the two species competitive system is extinct, by choosing the suitable values of feedback control variables, we can make extinct species becomes permanent and globally stable, or still keeps the property of extinction. When the two species competitive system is globally stable, the dynamic behaviors of the system is independent of the coefficients of feedback controls, that is, the feedback controls only change the position of the unique positive equilibrium and retain the stable property.In Chapter4, we consider three types of discrete almost periodic biological models. First, we study a discrete n species Gilpin-Ayala type population model. By constructing suitable discrete Lyapunov functions, sufficient conditions are obtained to guarantee that r of the species in the system are permanent, while the remaining n-r species are driven to extinction. Using the theory of asymptotically almost periodic solution of discrete equation, we study the existence, unique and stability of the almost periodic solution of the system. Second, we consider an almost periodic discrete logistic equation with delay. By constructing suitable Lyapunov functional and almost periodic functional hull theory, we show the existence of a unique almost periodic solution which is globally attractive. Third, we consider a discrete mode of plankton allelopathy with delays. We prove that one of the components is driven to extinction while the other stabilizes at a unique strictly positive almost periodic solution of a corresponding logistic equation with delay.