Stability And Bifurcation Of Two Kinds Of Fractional Reaction-diffusion Biological Systems

Based on the diversity of interactions between ecosystems and species,research on ecosystem modeling and related dynamics has been widely carried out.With the deepening of nonlinear theory,the description of ecosystem models is not limited to ordinary differential models.Fractional order calculus and reaction-diffusion terms are more introduced into ecosystem models to characterize the regular or irregular migration process of species in space,and the interaction relationship between populations.In addition,time delay has become a factor that cannot be ignored in many ecosystems,and it has an important impact on the dynamic characteristics of the system,mainly including maturity time delay,capture time delay,and pregnancy time delay.Compared with non time-delay systems,time-delay ecosystems can more accurately reflect the dynamic evolution of populations and have more important biological significance.It is worth noting that the occurrence of time delay can disrupt the stability of the system and generate Hopf bifurcation.Bifurcation,as a cutting-edge topic in nonlinear science,has a profound application background and is extremely challenging.This article also builds on the current research status of fractional order systems and reaction diffusion systems,and uses nonlinear analysis methods such as partial functional differential equation theory,Hopf bifurcation theory,and Turing instability to study the stability and Hopf bifurcation of two types of nonlinear ecosystems.The specific work is as follows:1.A fractional order ecological epidemic model with predator population pregnancy delay was proposed,and the delay in the model was selected as the bifurcation parameter.The stability of the system and Hopf bifurcation were analyzed and discussed.Research has found that as the fractional order increases within the range,the bifurcation threshold decreases and the system bifurcation occurs earlier.Simultaneously comparing the stability domains of integer order systems and fractional order systems under the same parameters,it was found that the stability of fractional order systems is higher than that of integer order systems.Explored the influence of feedback gain parameters on system stability in time-delay feedback control,and found that as the feedback gain decreases,the stability of the system improves.2.A reaction diffusion ecological competition model was established under food constraints,and the effects of reaction diffusion coefficients on the stability of ecological competition systems containing Holling Type I and Holling Type II functional response functions were discussed.The study found that the introduction of the reaction diffusion term does not change the stability of the Holling Type I ecological competition system at the equilibrium point,but by adjusting the self diffusion coefficient,it can induce Turing instability in the Holling Type II ecological competition system and obtain corresponding Turing patterns.3.In the Holling Type I reaction diffusion ecological competition model proposed in Job 2,the pregnancy delay within the predator population was introduced,and the delay in the model was selected as the bifurcation parameter to analyze the impact of delay on system stability.Research has shown that when the hysteresis parameter is less than the bifurcation threshold,the system is locally asymptotically stable at the equilibrium point.When the hysteresis parameter is greater than the bifurcation threshold,the system generates a Hopf bifurcation.At the same time,the direction of the Hopf branch and the stability of the bifurcation periodic solution were discussed.In addition,the impact of capture term variables on system stability was explored,and it was found that increasing capture intensity within a range can have a beneficial effect on the stability of ecological competitive systems.Finally,a control strategy was applied in the uncontrolled system to explore the control effect of the hybrid controller on system stability.Research has shown that control parameters within a range can maintain a constant locally asymptotically stable state or increase the stability domain of the system,delaying the occurrence of bifurcation.