The Algorithms For Predator-prey Model On Surfaces

The predator-prey model is one of the most basic models for studying between populations and between populations and the environment in ecology,it is not only a powerful mathematical tool for describing the dynamics of biological systems,but also helps humans to understand ecological problems.The predator-prey models on surfaces can be directly used in simulating biological population distributions on biofilms or solid surfaces.The difficulty of solving this model is that the solution area is a surface,which lacks a fast and efficient algorithm.Therefore,this paper studies two kinds of methods for a type of predator-prey models on surfaces:1.we present a lumped mass finite element method for solving the predator-prey model with self-diffusion term on surfaces.The first-order backward Euler format is used in time dispersion.In order to better deal with strong nonlinear terms,we perform decoupling processing.The semi-discrete and fully-discrete formats and their stability analysis are given.The main purpose of this method is to overcome the difficulty of the positivity preservation of the solutions.Besides,numerical simulations are considered to illustrate the feasibility of the numerical method by convergence tests.The periodic traveling waves and chaos caused by the invasion of predators are simulated on three different implicit surfaces.2.For the predator-prey model with cross-diffusion term on surfaces,the radial basis function-difference method with the backward Euler format is used.In the discrete scheme,decoupling is used to deal with nonlinear terms.At the same time,we give the stability analysis of the algorithm.Finally,numerical experiments simulate the conversion between Turing patterns due to changes in the diffusion coefficient d11.