Spatio-temporal Dynamics Of Nonlinear Mathematical Biology Discrete Models

The spatio-temporal dynamics behavior for several classes of nonlinear mathematical biology discrete models are investigated in this paper. It is a synthesis of the author's research work when he is a Ph.D. candidate in Applied Mathematics.This paper can be disparted to seven chapters.In chapter 2, we construct three discrete modles of predator-prey nonlinear system, based on the Lotka-Volterra type, Holling-Tanner mixed type and Beddington-DeAngelis functional type of the continuous time systems, and investigate the permanence of the discrete analogue of the continuous systems. To verify the obtained conditions, some special numerical simulations are also included. Further, we propose a discrete predator-prey biodynamics system by biological stoichiometry, then we study the dynamics of this discrete model. We establish results on boundedness and global attractivity. Finally, several numerical simulations are given to support the complex phenomena of the biodynamics system.In chapter 3, we mainly concerns the spatio-temporal dynamics of the nonautonomous discrete biodynamics system with finite delays or infinite delay and feedback controls and a nonautonomous three-species predator-prey discrete time model with diffusion. First, A nonautonomous N-species discrete Lotka-Volterra competitive system of difference equations with delays and feedback controls is considered. New sufficient conditions are obtained for the permanence of this discrete system. The results indicate that one can choose suitable controls to make the species coexistence in the long run. It is very important to protect wildlife. Next, a discrete biodynamics system with infinite delay and feedback control is considered, sufficient conditions for the permanence of the system are obtained. Finally, a non-autonomous three-species predator-prey discrete time model with diffusion is studied, where the predators are confined to one patch and cannot disperse, the prey species can disperse between two-patches. It is proved the system is permanent under appropriate conditions. Furthermore, if the coefficients in the system are periodic, by employing the technique of Huo and Li [64], sufficient conditions which guarantee the existence and global stability of a positive periodic solution of the system are obtained. In this process, we give an example and simulation to illustrate the feasibility of our results.In chapter 4, we introduce and study a discrete multispecies general Gilpin-Ayala competition predator-prey model and a discrete multispecies competitive model. First, we propose a discrete multispecies general Gilpin-Ayala competition predator-prey model. By using new difference inequality and new technique, sufficient conditions are established for the permanence and the global stability. And a discrete nonautonomous multispecies growth competiyive system is investigated. By using the method of the fixed point theorems, a set of simple and easily verifiable conditions are given for the existence of convergent or divergent positive solutions.In chapter 5, we construct a delayed discrete time Lotka-Volterra type three species food-chain model by the continuous model in [160]. Sufficient conditions are established for the permanence and the feasibility of the obtained results are illustrated with an example. Next, by applying the comparison theorem of difference equation, we analyze the permanence of a kind of discrete n-species food-chain system with delays.In chapter 6, by some theorys of functional differential and difference equations, non-negative definite Lyapunov functions are employed to obtain sufficient conditions that guarantee boundedness of solutions of nonlinear functional discrete systems. However, the results are illustrated with several species biodynamics systems.In chapter 7, we first obtain some criteria for determining the asymptotic stability of the zero solution for some classes of delay-difference systems by using a discrete version of the second Lyapunov method. Furthermore, the results are applied to nonautonomous discrete-time dynamical networks with multiple delays, and some new results are obtained. Our results can be well suited for computational purposes. Finally, with the help of continuation theorem in conincidence degree theory and Lyapunov functional, we study existence of positive periodic solution of nonautonomous discrete-time dynamical networks system.