Nonlinear Diffusion In The Ecosystem

It was 1920 when American mathematician A.J.Lotka and Italian mathematician V.Volterra first use the theory of nonlinear differential equations make a qualitative description of species dynamical systems. In 1930, Fishe cite diffusion into Genetics to describe the diffusive phenomena of species between spaces. In the year 1950, Skenaln cite diffusion into dynamical system of species, then Takeuchi. Y. considered the normal diffusive single species autonomy system inthe year 1989 and 1990. After Mr. Cui Yin-gan defined the "Dispersal" problem, more and more scholars join into the discussion of this thesis. Most of these researches are based on linear diffusion while papers mentioned non-linear diffusion are not familiar. Thus, this thesis discussed some three dimensional non-linear diffusive ecosystems which educed some salutary conclusions.This thesis is make up by four parts: the Introduction, the Theory, the Modeling and the Conclusion.First, in the Introduction part(see Chapter 1), the research history and actuality of "Dispersal Ecosystem" and "Diffusive System" were introduced. What is more, present the whole figure of the thesis in the order of this article.Then in the Theory part (see Chapter 2), extend a method of reducing dimension which combined the Theory of Center Manifold and the method mentioned in Zhang Xingan and Chen Lansun's paper. In all, the method of dimensional reducing of three systems, "a Kind of Multidimensional Polynomial System with Identity Matrix Linear Part" system (2.2), "a Kind of Multidimensional Quadratic Polynomial System with Diagonal Matrix Linear Part" system (2.8), "a Kind of Multidimensional High Order Polynomial System with Diagonal Matrix Linear Part" system (2.16), were discussed. An interesting phenomena that some systems can be reduced to one dimension was found and proofed (see Corollary 2.1 and Corollary 2.2).After above, the Modeling part is concluding Chapter 3, Chapter 4, Chapter 5 and Chapter 6, which discussed six different systems. Including a kind of non-linear diffusive Lotka-Volterra competition system, two different non-linear diffusive Lotka-Volterra prey-predator system, a kind of non-linear diffusive Lotka-Volterra prey-predator system with Holling-II reactive function, a kind of non-linear prey-predator system with Leslie number reactive function, and a kind of non-linear prey-predator system with Leslie number reactive function, Holling-II reactive function and time delay. The main discussion is about the global stability of each system and Hpof bifurcation about some of them.At last, the Conclusion part is Chapter 7. In this part, the ecological meaning of the six systems are concluded, and the comparison of global stability and Hopf bifurcation between diffusive system and the corresponding non-diffusive system are given, which more clearly expatiate the characteristic of diffusion system.