Study On Spatial Pattern Dynamics Of Several Types Of Ecological System With Delay And Diffusion Effect

Spatial pattern dynamics is a relatively important part of the nonlinear field,which has attracted the attention of many scholars in recent years.It mainly studies the formation rule of the spatial pattern that,exists and has guiding significance among various systems.In the study of such problems,dynamic morphological analysis can be performed by constructing mathematical model with characteristics of population dynamics,which can be used to explain the spatial pattern formed by the interaction of populations.At the same time,combined with the results of numerical simulation,it can be shown that,the spatial distribution changes of populations to explain the problems of population persistence,extinction and evolution in space.This paper will use the linear analysis theory and bifurcation theory,Routh-Hurwitz criterion and multi-scale analysis method to study several kinds of predator-prey system with delay and diffusion.The following is the main content of the paper:In the first part,we study the formation and selection of Turing patterns for a class of ecological system with nonlinear harvesting effect.Firstly,the conditions of Turing instability induced by cross-diffusion term are given by using stability theory and the existence region of Turing patterns of the system is obtained by bifurcation theory.Secondly,the amplitude equations of the system and the selection results of Turing patterns are derived by using multi-scales analysis method.Finally,the pattern formation and selection results of the system are simulated by Matlab software.The results show that the system has rich Turing patterns,such as spot,stripe and the coexistence of the two types.In the second part,the spatial dynamics of a predator-prey system with delay and nonlinear prey harvesting effect is studied.The condition of Hopf bifurcation and Turing bifurcation has been obtained by stability theory and bifurcation theory.Numerical simulations show that the system has rich dynamic behavior,time delay and diffusion can not only affect the formation of Turing patterns,such as spot,stripe and the coexistence of the two types,but also affect the formation of spiral patterns.In the third part,we study the spatial dynamics in a delayed diffusive predator-prey system with Holling-III functional response and linear harvesting effect.Firstly,the local stability of positive equilibrium of the system and the condition of Hopf bifurcation are obtained by using the stability theory and the bifurcation theory.Be-sides,the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by the center manifold theorem and the normal form theory.Fi-nally,the correctness of the theoretical results is verified by a series of numerical simulations,which shows that the system has rich dynamic behavior.