Dynamic Analysis Of Several Classes Of Fractional Predator-prey Models

Predator-prey model is one of the most important models in biological math-ematics.It has been concerned by ecologists and mathematicians for a long time.Fractional calculus is the extension of integral calculus in order,and has been widely used in the fields of physics,chemistry and biology.The predator-prey model based on fractional calculus has more practical significance.At present,it has become a hot research topic to study the dynamic behavior of fractional predator-prey model.In this paper,Hopf bifurcation theory and numerical simulation method are used to study the stability and bifurcation of the fractional predator-prey model.The main work is summarized as follows.Firstly,the Hopf bifurcation of a multi-fractional predator-prey model is ana-lyzed.By using the properties of Caputo fractional derivative,we proved that the stability of multi fractional order system is equivalent to that of a high-dimensional system of the same fractional order.The sufficient conditions for Hopf bifurcation occured are obtained by consideringas the bifurcation parameter in the case of₁=2₂=2(0<<1)and?as the bifurcation parameter in the case of2₁=₂=2(0<<1).The results are verified by numerical simulation.Secondly,we aim at controlling bifurcation of a three-species fractional predator-prey model through an extended feedback controller.By choosing the time delay as the bifurcation parameter,some sufficient conditions for the existence of bifurcation induced by delay are obtained.In order to control the Hopf bifurcation of the sys-tem,an extended feedback controller on the zooplankton population is designed.The results show that both fractional exponential and feedback parameters can effectively control the dynamic characteristics of the system.The extended feedback controller is enough to delay the occurrence of Hopf bifurcation and we can obtain ideal dy-namic behavior by changing parameters.Numerical simulations are used to verify the correctness of the theoretical analysis.Finally,the Hopf bifurcation of a class of delayed fractional predator-prey model with economic is discussed.By using the Hopf bifurcation theory,some sufficient conditions for the existence of Hopf bifurcation induced by delay are obtained.The results show that in the case of zero economic profit,the biological equilibrium point of the system is asymptotically stable.Under positive economic profit condition,the system produces limit cycles at the positive equilibrium point as the delay increases through a certain threshold.Finally,the correctness of the theory is verified by nu-merical simulation,and the effects of time delay,economic and fractional exponential on the stability of the system are further discussed.