The Study Of Reaction-diffusion Ecological Systems

The idea of modelling the dynamics of interacting populations with a system of nonlinear differential equations dates back at least to the pioneering work of Lotka and Volterra in the 1920s. The idea of using diffusion to model the spatial dispersal of the population genetics was introduced by Fisher in the 1930s and applied to population dynamics by Skellam and others in the early 1950s. Currently, reaction-diffusion systems are some of the most widely used models for population dynamics or genetics in situations where spatial dispersal plays a significant role. The problem of the existence of positive solutions to 2 x 2 reaction-diffusion systems has been well investigated. However, the problem of three variables or more than three variables is far from being solved. In this paper, several kinds of reaction-diffusion ecological systems with three or more than three equations are studyed and some valuable results are obtained. It consists of six parts. In part one, we quote a few familiar results, and introduce the notations and terminologies that will be useful in what follows. In part two, we study the Dirichlet problem for a reaction-diffusion system of three species: a predator, a mutualist-prey and a mutualist. There are many ways that a mutualist may affect a predator-prey interaction, but here we consider just one of these, we consider cases in which a mutualist modifies predation to the benefit of the prey, namely a mutualist deterring predation on a prey. Meanwhile, the mutualist also derives some benefit from the mutualist-prey, the presence of prey enhances the per capita growth rate of the mutualist. By means of the global bifurcation theory, starting with the semi-trivial solution which has only one zero-component, the sufficient conditions of existence for coexistence state and the corresponding parameter regions forthis system are established, and some local stability results for the coexistence states are obtained. In part three, By discussing the properties of linear cooperative system, the necessary and sufficient conditions for the existence of the positive solutions of an elliptic cooperative system in terms of the principal eigenvalue of the associated linear system are established, and some local stability results for the positive solutions are obtained. The problemton the weakly coupled cooperative elliptic system with two equations have beenstudied in many papers. Our model and some results can be seen extension for them. In part four, we discuss a reaction-diffusion system of a simple food web consisting of one predator and two prey populations in an un-stirred chemostat with a single nutrient input. Monod's model is employed for the dependence of the specific growth rates of the two prey populations on the concentration of the nutrient and a generalization of Monod's model for the dependence of the specific growth rates of the predator on the concentrations of both prey populations. The necessary and sufficient conditions for the existence of coexistence states are established. The main tools used here are Dancer fixed point index and the degree theory hi cones. In part five, By using degree theory in cones, combining with maximum principles, lower-upper solutions methods, a three-species ecological models with diffusion is studied and a method to solve the problem about the existence of positive solutions for the three species models which involve competition and predation is introduced. This method is to evaluate the fixed point index of certain operator on a slice of the positive cone. This slice contains all nonnegative solutions of the form, say, (u, v,0). Under appropriate conditions, the homotopy invariance can be applied on this slice, the calculation of fixed point indices can be proceeded, and so the existence of positive solution can be established. In part six, a reaction-diffusion system that models the situation in which two cooperative predators with a saturating interaction term for one species feed on a same prey is considered. Through suitable construction of upper...