Dynamic Stability Analysis Of Three Kinds Of Predator-prey System

In the ecosystem,the predator-prey system is a very classical complex system,and the population dynamics is very important to the stability of the predator-prey system.This paper considers the effects of three types of population dynamics on the predator-prey system:migration and Allee effects,odor disturbance and group defense,and toxin and high-order interactions(HOIs).By constructing the population dynamics model,the boundedness of the solution of the model is deduced,and the stability of the equilibrium point and the possibility of bifurcation are analyzed.In this paper,the results of theoretical analysis are verified by numerical simulation,and the dynamic stability of three types of prey-predator systems is studied.Firstly,a two-population system with artificially controlled migration and Allee effect was established.It is demonstrated that the solution of the system is non-negative and bounded,and then the types and stability of the equilibrium points are argued separately,and the reason of Hopf-bifurcation generated by the positive equilibrium point is argued by using bifurcation theory,and the correctness of the results is verified by numerical simulation.It is found that the system generates stable limit cycles,stable nodes and supercritical Hopf-bifurcation under the influence of the study parameters.Secondly,a three-species food chain system with odor disturbance and group defense was constructed.It is demonstrated that the solution of the population system is non-negative and finally bounded.Then,the types and stability of equilibrium points are illustrated theoretically,and the case of Hopf-bifurcation at boundary equilibrium point is discussed.It was found that group defense increased the density of prey population,while odor disturbance inhibited the density of prey population.In addition,some important dynamic behaviors such as stable node,Hopf-bifurcation,and saddle-node bifurcation are also produced.Then,a kind of three-species food chain system with toxin and HOIs is presented.The species and stability of the equilibrium points of the system were obtained,and it was found that both toxin and HOIs inhibited the growth of prey population.Interestingly,both toxins and HOIs induce ordered phase transitions from stable points to Hopf bifurcations,to limit cycles.In addition,the potential mechanism of toxin and HOIs inducing the system to produce non-monotonic phenomena is explored,and the two-parameter phase diagram is used to reflect the population system dynamics.Finally,this paper summarizes the content and introduces the innovation of the article.Then the paper briefly discusses the areas that need to be improved,and puts forward the specific work and prospects for future research.