Spatio-temporal Dynamic Evolution And Control Of Populations In Ecological Competition Systems

In nonlinear dynamical systems,there exist complex dynamical behaviors such as bifurcation,chaos and oscillation,among which bifurcation is an important index to study the periodic solutions and system characteristics.Hopf bifurcation,as the most common phenomenon in bifurcation research,not only has great research significance in theoretical research and control research,but also has been widely used in medicine,finance,computer science,biology and other practical disciplines,with a wide range of research prospects and practical significance.As is known to all,time delay is ubiquitous in dynamic systems such as ecological competition.The system is not only affected by various factors at the current moment,but also closely related to the state of the system at the past moment.Because of the existence of time delay,the dynamic system will lose stability,even bifurcation and chaos.In addition,all kinds of dynamic systems,including ecological competition system,will not only produce adverse dynamic behavior in time,but also be affected by space.In nature,factors such as temperature,climate,latitude and longitude,and seasonal changes will lead to uneven distribution of spatial density.High density materials tend to flow to low density areas,and their diffusion coefficient can change the dynamic behavior of the system to some extent,resulting in Turing instability.Therefore,it is of great theoretical significance and practical value to study the ecological competition system under the multiple influences of time and space.This paper is dedicated to complex dynamic problems of the research on ecological competition system,based on the predecessors’ work,this paper further explores the infectious diseases under the influence of fractional order prey-the dynamics of predator-prey system model,with Holling type Ⅱ prey-predator model Turing instability and under the influence of time delay and diffusion of Marine planktonic bifurcation analysis and control of the system.The specific work is as follows:(1)Considering the effects of gestation period of predator and incubation period of infectious disease on the system,a fractional ecosystem model with double delays was proposed to study the dynamic evolution process of predator and prey under the influence of infectious disease.Firstly,the existence of equilibrium point is proved,then the stability and Hopf bifurcation conditions of fractional delay ecosystem are given,and two bifurcation criteria caused by different delays are determined.The results show that Hopf bifurcation occurs when the hysteresis is greater than the bifurcation point.(2)In this paper,the Turing instability of a predator-prey model with diffusion and Neumann boundary conditions is investigated.The sufficient and necessary conditions for the stability of the system without diffusion are obtained by eigenvalue theory to reveal the influence of diffusion effect on the system.In addition,the Turing instability condition is established,and the spatial diffusion coefficient and the Turing instability of the system are closely related to the strip pattern.(3)A proportional differential(PD)feedback controller is proposed for Marine plankton systems with time delay and diffusion.The stability and Hopf bifurcation conditions of the system are obtained by analyzing the distribution of characteristic roots of the system without control and diffusion and taking time delay as bifurcation parameter.The Turing bifurcation critical value of a controlled Marine plankton system with proportional differential(PD)feedback controller is given.It is found that adjusting the parameters of the controller can optimize the dynamic system and obtain the desired dynamic behavior.In this paper,it is shown that the fractional order,time delay,diffusion coefficient and controller parameters are the important parameters that affect the dynamic behavior of the system.By controlling the variation of these parameters,the different conditions of system stability and multiple bifurcation points are derived,and the appropriate system parameters are selected to achieve the ideal dynamic behavior.The results obtained in this paper can not only extend the theoretical research and application of nonlinear systems,but also provide guidance for practical engineering applications.