Spatio-temporal Dynamic Analysis Of A Kind Of Mutualism System

The population dynamics and community structure in ecology strongly depend on the inter-specific relationship,among which predation,competition and mutualism are the three basic types.Extensive studies have been carried out about predation and compe-tition,while not too many people have dedicated themselves to the study of mutualism.It is not until recently that people have started to study the topic by constructing a model and great efforts still need to be done in order to illustrate its ecological effects.This paper aims to explore the dynamics and ecological role of mutualism in a changing habitat by using the mathematical model constructed as a result of the combination of the overcompensation with the predator-prey relationship.This paper constructs a predation-mutualism model by taking into account the ani-mal's overcompensation for plants arising from its overexploitation.Based on this model,it draws the following conclusions by using such different modeling methods as mathemat-ical analysis and numerical simulation.(1)According to the ordinary differential model,in the absence of overcompensation,if the predation pressure is below the critical value,the two species co-exist in a stable system which converges at a positive equilibrium point;if the predation pressure goes above the critical value,there is a limit cycle oscillation in the system with a Hopf bifu rcation the critical value;if the predation pressure drops below a certain value,there is a positive equilibrium point in the system.However,as the predation pressure goes up,the amplitude of the limit cycle increases so sharply that the violent system oscillation in the wake might lead to the extinction of the prey and even the collapse of the whole system.(2)According to the ordinary differential model,when the overcompensation coefficient ? is greater than 0,in the case of the same parameter,the addition of overcompensation makes it possible for the two species that cannot coexist in the original parameter range live together,thus expanding their coexistence area;as for the system that was stable,the addition of overcompensation sfurther stabilizes the system;as to the systems which used to coexist in oscillation,the addition of a certain degree of overcompensation makes them coexist in a stable manner;as for the systems that could not co-exist,overcompensation can prevent them from collapse.(3)In the case of the time delay model,the addition of time delay does not change the stability of the system.(4)In the case of the reaction diffusion model,the conditions of Hopf and Turing bifurcation are revealed and a conclusion is drawn that overcompensation can reverse the stability of the system.(5)According to the meta-population model under the uniform field hypothesis,when habitat decreases,there is a greater chance that species N can survive after some adjustments.However,if the birth rate of species P exceeds a certain value,no matter what changes might occur in the birth rate of species N,N will extinct before the collapse of the system due to the following excessive predation pressure.(6)In the case of the pair approximation model,the high connectivity of the habitat will contribute to the coexistence of the two species;but on the other hand,in real life,the deterioration and discontinuation of the habitat reduces its connectivity,which will hinder the survival of species P.Based on detailed researches,this paper concludes that overcompensation helps to stabilize the ecosystem despite that the positive feedback of mutualism in many other mechanisms can destabilize the system.This gives a testament to the complexity of the eco-system.This paper has added some fresh ideas to the existing conclusions of mutualism and enriched the theory of mutualism system.