Stability Analysis Of Two Predator-Prey Models With Stage Structure

Many scholars have explored the population dynamics system,stage structure,functional response,time-delay response,and so on.Based on the existing research,this paper mainly studies two classes of predator-prey models with stage structure,time delay,and Holling type-Ⅱ functional response function.One is a predator-prey model with mature time delay,and the other is a predator-prey model with nutritional transformation time delay.This paper mainly uses the qualitative and stable methods of ordinary differential equations to study.Firstly,the theory of functional differential equations is used to verify the correctness of the solution.Secondly,the Hurwitz criterion is used to discuss the stability of the positive equilibrium point of these two kinds of models without and with time delay,and the sufficient conditions for the local asymptotic stability of these two kinds of models are obtained,and the sufficient conditions for the existence of Hopf bifurcation of these two kinds of predator-prey models are obtained according to the bifurcation conditions of the time delay system.Then,by using the continuation theorem in coincidence degree theory and Mawhin continuation theorem,the sufficient conditions for the existence of periodic solutions and positive periodic solutions for two kinds of predator-prey systems with stage structure,time delay and functional response function are obtained.Finally,the two models are numerically simulated by Matlab software,and the numerical simulation diagram is drawn.Finally,the two kinds of models are numerically simulated by using Matlab software,and the numerical simulation diagram is drawn.By selecting appropriate parameters and different time-delay values,the changes and laws of population number in different states are deduced through numerical simulation,and the components and curves of the solution of positive equilibrium point in local asymptotic stability and instability are obtained respectively.The results show that the stability of the system will change with the change of bifurcation parameters,and Hopf bifurcation will occur under certain conditions,and the system can produce periodic solutions when certain conditions are met.