Study On The Dynamics And Control Of The Delayed Recation-Diffusion Population Models

The reaction-diffusion equation has attracted much attention due to its wide application in physics,chemistry and biology,and it has become one of the most popular research fields by using reactiondiffusion equation to study various real-world problems.The main aim is to discuss some delayed reaction-diffusion systems coming from biological population dynamics.By analyzing and summarizing the research status of population models,and according to partial functional differential theory and applying stability,Hopf bifurcation,Turing theory,we aim to determine the possible non-trivial role for the delay,diffusion,harvesting terms on the dynamics by employing eigenvalue analysis method,center manifolds,normal forms and Matlab simulations.The dynamic analysis and simulations provided some interesting phenomena and some meaningful results.1.We considered a delayed stage structured diffusive prey-predator model,in which predator is assumed to undergo exploitation.By using the theory of partial functional differential equations,the local stability of a interior equilibrium is established and the existence of Hopf bifurcations at the interior equilibrium is also discussed.By applying the normal form and the center manifold theory,an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived.Finally,the complex dynamics are obtained and numerical simulations substantiate the analytical results.The numercial simulations also show that delay and diffusion can lead system to chaotic behavior.2.A delayed reaction-diffusion predator-prey model with Michaelis-Menten Type Prey-Harvesting is considered.Firstly,we prove existence and uniqueness of the global solutions of the system by using the upper and lower solution method.Then,by analyzing the corresponding characteristic equations,the local stability of a positive steady state is established.The existence of Hopf bifurcations at the positive steady state is also discussed.Sufficient conditions are derived for the global stability of the positive steady state of the proposed problem by the Lyapunov functional.Numerical simulations are performed to illustrate the results.By numerical simulations,we also observe that as discrete delay increases more,the species may tend to extinct.The change of harvesting effort and nonlocal delay can transform an unstable system into a stable one.3.A delayed diffusive predator-prey system with interval biological parameters is constructed.Sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation.Furthermore,formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem.Then an optimal control problem has been considered.Research results shows that the discrete delays are responsible for the stability switch of the model system,and a Hopf bifurcation occurs as the delays increase through a certain threshold.The increasing delay may lead to the extinct of the prey or the predator.The diffusion we incorporate into the system can effect the convergence speed and the amplitude of the system.Over harvesting may be lead to the extinct of the species.This can provide theoretical basis for thoroughly studying about exploitation of ecological.4.A delayed diffusive predator-prey system with non-smooth continuous threshold harvesting is considered.First,the stability of the interior equilibrium and the existence of Hopf bifurcations,induced by diffusion and delay respectively,are investigated by analyzing the characteristic equations.Then discontinuous Hopf bifurcations are discussed.Numerical simulations reveal that the discrete delay is responsible for the stability switch of the model,and a Hopf bifurcation occurs as the delay increasing through a certain threshold,and the diffusion we incorporated into the model can affect the stability of system.As the diffusion coefficient decreasing through a certain threshold,inhomogeneous Hopf bifurcation occurs.Results also shows that harvesting may have a stabilizing effect on the ecosystem.This thesis is of great theoretical significance in guiding the research on dynamic analysis of non-smooth delayed reaction-diffusion model.5.We considered a diffusive plant invasion model with delay under the homogeneous Neumann boundary condition.The qualitative properties,including the existence and uniqueness of a nonnegative solution,persistence property,and local asymptotic stability of the constant steady states are obtained.We investigate Hopf bifurcation of this model as well as deriving some criteria by analyzing the associated characteristic equation and by taking delay as the bifurcation parameters.In some special cases,we investigate the system's discontinuous Hopf bifurcation.In addition,we also establish the existence and non-existence of nonconstant positive steady states of this model,indicating the effect of large diffusivity.Our simulations demonstrate that the numerically observed behaviors were in excellent agreement with the theoretically predicted results.Using numerical simulations,we provide some comparisons between our study and related work,and investigated the effects of delay and the diffusion term on dynamic behavior.Our numerical results showed that diffusion can make the system unstable and increasing delay may cause the plant extinction.