Study On The Characteristics Of Several Eco-dynamic Systems With Time Delay And Spatial Effect

Pattern dynamics is an important part in the field of nonlinear science.As an applied science,it involves mathematics,biology,ecology and so on.At present,the research objects of pattern dynamics are mainly reaction diffusion systems,nonlinear optical systems,Rayleigh-Benard systems and so on.A Pattern is a regularity in temporal space or spatial space in the world.The formation of Turing pattern is a kind of pattern formation mechanism which is caused by local instability.In the study of pattern dynamics,there is another kind of pattern,spiral wave pattern,which originated from the global instability of the system.This paper will use the linear analysis theory,Routh-Hurwitz criterion and multi-scale analysis method to study three kinds of preaator-prey systems with reaction 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 predator-prey system with cross diffusion and modified Leslie-Gower terms.Firstly,the Turing space is obtained by linearization analysis.Secondly,the ampli-tude equation of the system is derived by using the multi-scale analysis method,the selection results of Turing patterns are given.Finally,Matlab software is used to simulate the system.We can obtain different types of Turing patterns,such as spot,stripe and coexistence patterns.In the second part,we study the spatial patterns of a delayed Leslie-Gower predator-prey system with Holling-Ⅳ functional response.By using the stability theory and bifurcation theory,the local stability of the positive equilibrium point of the system and the condition of Hopf bifurcation are obtained.Then the effects of time delay,diffusion and intrinsic growth rate on the system are investigated by nu-merical simulation.It is found that the time delay affects the dynamic behaviour of the system.In the third part,we study the spatial dynamics in a delayed diffusive predator-prey system with Leslie-Gower terms and linear harvesting effect.By using the stabil-ity theory and bifurcation theory,the stability conditions of the positive equilibrium point of the system and the condition of Hopf bifurcation are obtained.