Spatiotemporal Dynamics And Control Of Two Kinds Of Fractional Reaction-diffusion Systems

Hopf bifurcation theory is not only a classical theory for studying oscillatory period solutions of differential equations,but also has an important significance in the study of self-excited oscillations in engineering.In recent years,with the increasing research on nonlinear dynamics,the inscription of dynamical models is not only limited to the ordinary differential models,but also fractional order calculus and reaction diffusion terms are often used to describe more advanced and accurate dynamical models.Numerous studies have shown that time delay is crucial for nonlinear dynamics.For example,the latent time delay of infectious disease model and the fear time delay of ecological competition model are good examples of how time delay affects the dynamical behavior of models.Currently,the study of Hopf bifurcation with time delay has become a popular topic in nonlinear dynamics research,including stability,bifurcation control,synchronization,etc.Based on the medical and ecological backgrounds and previous studies on the dynamics of infectious disease models and predator-prey models,this paper designs a class of fractional-order SEIR model with latent time delay and diffusive ecological competition model with fear time delay.The stability of these models,Turing instability,Hopf bifurcation,and bifurcation control strategies are investigated by using bifurcation theory.The specific work is as follows:(i)A fractional-order SEIR epidemic model with latent delay is proposed.The latent delay of the model is selected as the bifurcation parameter,the stability of the model and Hopf bifurcation are analyzed,and it is concluded that the fractional order has a delaying effect on the bifurcation delay.(ii)A diffusion predator-prey model with fear delay is developed.The effects of diffusion coefficients and fear delay on the stability of the model is investigated,the direction of Hopf bifurcation is calculated.It is found that fear factor can reduce the population density.(iii)A hybrid control strategy based on the diffusive predator-prey model with fear delay is designed.Through this control strategy,the controlled model is stabilized at the preset equilibrium point,and the dynamics of the controlled model is improved.