Dynamics on a general stage structured n food chains

Abstract

There are many mathematical models to describe the dynamics of plankton community structure in literature, most of the models are based on the so-called preypredator model. The purpose of this thesis is to propose a general prey-predator model with stage structure and a constant maturation time delay with and without interaction between patches. First, we begin with the purpose of using delay differential equations in biological models, like those involving population dynamics, epidemiology, and physiology; and present a brief history of the delayed prey-predator models. We provide basic properties of delay differential equations, the Method of Steps to solve them, Chebotarev-Meimans Method and The D-Subdivision Method to determine the local stability. In Chapter (2), we propose a general model with n parallel food chains through the stage structured maturation time delay, which can cover most of the prey-predator models in the literature. We discuss some basic dynamical properties of the system with single or multiple patches and with general or some particular functional responses, including the existence of equilibrium points and their local and global stabilities. Then in Chapter (3), based on the model in (2.2), we include the consideration of migrations between all patches and present a more complex model for the multi-patch predator-prey interactions. We discuss the existence of equilibrium points and their local and global stabilities of the system with two patches and some properties of the general model. In Chapter (4), we give numerical simulations by choosing some different functions, parameters and time delay in several examples to illustrate the validity of the theoretical results given in Chapter (2) and (3). At last, in Chapter (5), we summarize the results in this thesis, and indicate some problems for future work.