Non-Standard Finite Difference Methods For Two Types Of Biological Models With Spatial Diffusion

Theory of differential equations is widely used in natural sciences and engineering technology, which is one of the important means for researching model. Based on learning differential equations, it is aware that the majority differential equations are difficult to find out analytical solutions, then, a large number of numerical methods have been proposed for solving approximate solutions of differential equations under specific initial and boundary condition. In recent decades, partial differential equations are widely applied to research various issues in biology, physics, chemistry, mathematics ecology, and other subjects, the numerical solution of partial differential equations are obtained by constructing appropriate numerical methods, and using computer to make simulation, which is a common method for researching partial differential equations.There are a lot of research results for solving partial differential equations, it is still a very effective method for partial differential equations using numerical methods to obtain approximate solutions. After the discretization of continuous system, its dynamic properties will occur in some changes, for a specific differential equations, then, how to select an appropriate difference schemes for preserving unconditionally the dynamic properties of continuous system are an important research topic. This article is the application of non-standard finite difference method to construct difference schemes incorporating numerical solution of partial differential equations and study the behaviors of discrete dynamical systems; Verifying if the discrete system unconditionally remained the main dynamical behavior of the original continuous system solutions independence of the time step and space step, for example, the nonnegative, the stable of equilibrium solution. Based on the selected difference schemes, structuring the discrete Lyapunov function incorporating time and space,Applying Lyapunov stability theory, it is obtained the equilibrium solution is stable for any time step and space step.This paper constructs numerical schemes for solving two types of a continuous biological modes with spatial diffusion. One is the discrete rabies model about fox as transmission carrier with diffusion, it is verified the nonnegative of numerical solution and the global asymptotical stable of the disease-free equilibrium for the rabies model.the other is the discrete Lotka-Volterra predator prey model having a constant number of prey refuge with spatial diffusion, firstly, we prove that the numerical solution of predator prey model is nonnegative; Then, we find that when the prey refuge is constant number and reaches a certain conditions, the only positive equilibrium solution of the discrete-time system is globally asymptotically stable; Finally verifying the value of the prey refuge strengthen, the only positive equilibrium solution of the discrete-time system is unstable, the boundary equilibrium solution occurs and is locallyasymptotically stable.