Qualitative Analysis Of A Three-Species Reaction Diffusion Model And Optimal Control

Mathematical models can be frequently described by differential equations to associate with the real life.Not only can the interaction in single population be depicted but also the relationship between multiple populations including predation,competition and reciprocity as the differential equation is an essential tool for exploring ecosystem.This paper performs the qualitative analysis of a competitor-competitor-predator three-species reaction diffusion model under the Neumann boundary condition.Firstly,the existence and stability of constant solutions are discussed.After obtaining some existence conditions corresponding to trivial solution,weak semi-trivial solutions,strong semi-trivial solutions through solving algebraic equations,the unique positive constant solution is specially gained in some cases.The results display that five constant solutions are absolutely unstable judged by Lyapunov first method.Strong semi-trivial solution5 E is locally asymptotically stable in the ODE system and unstable in the PDE system because population diffusion leads to system instability and form a new spatial pattern,which is usually called Turing instability.Strong semi-trivial solution6 E is locally asymptotically stable in the ODE and PDE system under certain conditions while it is unstable in the opposite condition.For the unique positive constant solution,on the one hand,local asymptotic stability can be easily known,on the other hand,the global stability in the ODE system is decided by the total derivative of V function which is constructed by using the Lyapunov second method.Secondly,the existence of non-constant positive solutions are investigated.Priori estimates are given using the maximum principle and Harnack inequality.Non-existence conditions of the solutions are gained by related Poincaréinequalities.Leray-Schauder degree theory makes a contribution to the existence of the solutions owing to computing fixed point index.Thirdly,the existence of Hopf bifurcations are showed.Time delay is added into the ODE system since the development of the biological population is notwholly dependent on the current state,but also depends on the state of the previous time.Following the boundedness and permanence of solutions by using comparison theorems in the time delay system,the critical point0? of the bifurcations from two strong semi-trivial solutions can be found.It reveals that time delay can affect the stability of the equilibrium point.When the parameter exceeds a certain value,the stability of the equilibrium point will change and the bifurcation will occur at the critical point.Finally,conclusions about the optimal control strategy are demonstrated.The strategy of the model which contains the single capture item or double capture items and combined with economy is discussed via the Pontryagin maximum principle.The result indicates that when the discount rate tends to infinite,the income comes to zero while the discount rate goes to zero,the revenue achieves maximum.Mathematical biology models make the researching of biological phenomena easy,and relevant properties to models can evidently explain and predict popular behavior in order to foster strengths and circumvent weaknesses.Meanwhile,human intervention plays an important role in rational utilization and development of ecological resources,which can do good to the harmonious development of the ecosystem and maximize the economic benefits.