Hopf Bifurcation And Control Analysis Of Two Kinds Of Fractional-order Prey-predator-scavenger Models With Double Delay

Fractional calculus is more suitable for describing practical problems in physics,biology,engineering,and many other fields because of the“non-locality”and“memorability”.There is a memory effect on the behavior of various populations in the ecological model,and the evolution of the ecosystem not only depends on the current state of various populations but also is affected by the previous state.Therefore,it is more precise to describe the development and change of the biological model by introducing the fractional-order and delay terms.It facilitates researchers to better understand the various dynamic characteristics of the population evolution process and has practical meaning for predicting population development trends.In this thesis,the stability,Hopf bifurcation,and control of two kinds of fractionalorder prey-predator-scavenger models with double-delay are studied by using the stability theory of fractional-order differential equations.The main research contents of the thesis are as follows:1.The stability and Hopf bifurcation of a fractional-order prey-predator-scavenger system(FPSS)with hunting delay and competition delay is studied.Firstly,the existence and uniqueness of the solutions for the system are proved.Secondly,the characteristic equation of the linearized system for FPSS is obtained by using the methods of linearization and Laplace transform.Based on the characteristic equations,the stability of the system,the conditions for the Hopf bifurcation and the critical value of delay are discussed for the cases of zero delay,single delay,and non-zero double delays,respectively.Finally,the reasonableness and correctness of the theoretical analysis are performed by the AdamsBashforth-Moulton predictor-corrector scheme with appropriate model parameters.2.The stability,Hopf bifurcation and control of a fractional-order prey-predatorscavenger system(GFPS)with gestation delay is studied.Firstly,the existence and uniqueness conditions of the solutions for GFPS are discussed.Secondly,local stability conditions of various equilibria of the non-delay GFPS are derived by using the stability theory of the fractional-order differential equations.Thirdly,choosing gestation delay as a bifurcation parameter,the stable region and conditions of emergence of Hopf bifurcation at the coexistence equilibrium of the GFPS are deduced.Fourthly,two control methods are introduced for GFPS.One is an enhanced state feedback controller,and the other is a hybrid control strategy consisting of a linear delay feedback controller and a fractionalorder proportional derivative controller.Furthermore,the stable region and the conditions of emergence of Hopf bifurcation at the coexistence equilibrium of the control system are analyzed.Finally,numerical simulations are implemented to validate the correctness of the theoretical analysis and the effectiveness of the various controller.Particularly,it is respectively verified that the enhanced state feedback controller and the hybrid control strategy have a better control effect than that of a single controller by selecting different sets of parameters.