The Stability Analysis Of (Almost) Periodic Lotka-Vloterra Systems With Grazing And Diffusions

In the Lotka-Volterra ecological systems, the differential equations governing the population densities possess multiple steady-state (or equilibrium) solutions. An ecologically interesting and mathematically challenging problem is to determine whether and when the time-dependent solution converges to a steady-state solution, and to which one. Since the environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment are considered as important selective forces on systems in a fluctuating environment. However, in fact, it is more realistic to consider almost periodic system than periodic system. For the periodic Lotka-Volterra systems, many skills and techniques have been developed. Comparably, there are few methods to analyze the almost periodic systems since the interaction among multi-species for the almost periodic systems is more complex. In the meantime, the grazing is also an important phenomenon of species. Motivated by the above discussions, the objective of this paper is to study the almost periodic solution for several kinds of Lotka-Volterra ecological systems with grazing rates and diffusions. The main results obtained in this dissertation are as follows:In Chapter1, the background and significance of the subject is first introduced, then the research status of the Lotka-Volterra ecological systems is elaborated, finally, the content and organizational structure of this paper is introduced.In Chapter2, firstly, almost periodic solution of a three-species competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we give sufficient conditions to ensure the existence and globally asymptotically stable for the strictly positive space homogenous almost periodic solution, and then the results obtained are extended n-species competition models.Chapter3is concerned with the almost periodic solution of a diffusive two-competing-predator and one-prey model with grazing rates. By using the method of upper and lower solutions and the exponential dichotomy of linear system as well as Banach contract fixed point theorem, some new results are obtained.In Chapter4, by using the software package MATLAB7.0, some numerical simulations are shown to illustrate our theoretical analysis.