一类三种群反应扩散模型的定性分析及最优控制

发布时间：2018-04-12 15:40

本文选题：存在性 + Turing不稳定性　；参考：《陕西科技大学》2017年硕士论文

【摘要】：在反映客观世界运动过程量与量之间的关系中,大量存在满足微分方程关系式的数学模型,且微分方程作为研究生态系统所需的重要工具,不仅可以描述单个种群内部之间的作用关系,也可以描述多个种群之间的捕食、竞争和互惠等相互作用关系。本文主要在Neumann边界条件下,对一类具有竞争-竞争-捕食关系的三种群反应扩散模型作了以下定性分析。首先,给出该模型常数解的存在性和稳定性。通过求解代数方程得到该模型平凡解、弱半平凡解、强半平凡解的存在性条件,特别地,得出了在一定条件下,该模型正常数解的唯一存在性。利用Lyapunov第一方法,判断出五个常数解在ODE系统和PDE系统下都不稳定;在一定条件下,强半平凡解5E在ODE系统下局部渐近稳定,在PDE系统下不稳定,即由于种群扩散导致系统失稳而形成了新的空间模式,产生了Turing不稳定性;一定条件下的强半平凡解6E在ODE系统和PDE系统下都局部渐近稳定,在其相反条件下,都不稳定。对唯一的正常数解,一方面,给出了其局部渐近稳定性;另一方面,运用Lyapunov第二方法构造V函数,通过判断其全导数的正负得出了唯一存在的正常数解在ODE系统下的全局稳定性。其次,建立该模型对应的非常数正平衡态解的存在性。利用最大值原理和Harnack不等式给出正解的先验估计,并利用Poincaré相关不等式给出非常数正平衡态解的不存在性条件,同时,运用Leray-Schauder度理论通过计算不动点指数证明了非常数正平衡态解的存在性。再次,给出时滞系统Hopf分支的存在性。由于生物种群的发展不完全依赖于当前的状态,还依赖于此前的某一时刻或者某一时间段的状态,所以对该模型的ODE系统加入时滞项进行讨论。利用比较定理给出时滞系统解的有界性和一致持久性。以时滞为参数,给出发自两个强半平凡解的Hopf分支的临界点0?.结果表明时滞会对平衡点的稳定性产生影响,当时滞参数超过某一临界值时,平衡点的稳定性会发生变化,且在该临界值处发生分支现象。最后,给出加入捕获项后的最优控制策略。在该模型对应的ODE系统中加入单种群及两种群捕获项,与经济理论相结合,利用Pontryagin极大值原理得出模型的最优控制策略。结果表明,当贴现率无限大时,收益趋于零;相反,当贴现率为零时,收益达到最大。生物数学模型为研究生物现象提供了便利,模型的相关性质即可解释和预测一些种群行为,从而达到趋利避害的目的。同时,结合生物数学理论和经济知识理论,在种群发展过程中实施人为干预,使生态资源得到合理的利用和开发,以期在保证生态系统和谐发展的情况下,最大程度上实现经济效益,这些对生物种群和人类的发展都有着不可替代的作用。
[Abstract]:In the relationship between the quantity and the quantity of the objective world motion process, there are many mathematical models which satisfy the relation of differential equation, and the differential equation is an important tool for studying ecosystem.It can describe not only the interaction within a single population, but also the interaction between multiple populations, such as predation, competition and reciprocity.In this paper, under the Neumann boundary condition, the following qualitative analysis is made for a class of three species reaction-diffusion model with competition-competition-predator-prey relationship.Firstly, the existence and stability of the constant solution of the model are given.By solving the algebraic equation, the existence conditions of the model's trivial solution, weak semi-trivial solution and strong semi-trivial solution are obtained. In particular, under certain conditions, the unique existence of the normal number solution of the model is obtained.By using Lyapunov's first method, it is found that five constant solutions are unstable in both ODE system and PDE system, and under certain conditions, the strong semi-trivial solution 5e is locally asymptotically stable in ODE system and unstable in PDE system.In other words, a new spatial model is formed because of the instability of the system caused by population diffusion, and the strong semi-trivial solution 6e is locally asymptotically stable under the ODE system and the PDE system under certain conditions, and is unstable under the opposite conditions.For the unique normal number solution, on the one hand, the local asymptotic stability is given, on the other hand, the Lyapunov second method is used to construct the V function.By judging the positive and negative of its total derivative, the global stability of the unique existence of the normal number solution in the ODE system is obtained.Secondly, the existence of positive equilibrium solutions corresponding to the model is established.A priori estimate of the positive solution is given by using the maximum principle and Harnack inequality, and the nonexistence condition of the positive equilibrium solution of a nonconstant number is given by using the Poincar 茅 correlation inequality. At the same time,The existence of positive equilibrium solutions of nonconstant numbers is proved by using the Leray-Schauder degree theory by calculating the fixed point exponents.Thirdly, the existence of Hopf bifurcation for time-delay systems is given.Because the development of biological population is not completely dependent on the current state, but also depends on the state of a certain time or a certain time before, the ODE system of this model is discussed in terms of adding time delay term.The boundedness and uniform persistence of solutions for time-delay systems are obtained by using comparison theorem.Taking time delay as a parameter, the critical point of Hopf bifurcation starting from two strongly semi-trivial solutions is given.The results show that the time delay will affect the stability of the equilibrium point. When the hysteretic parameter exceeds a certain critical value, the stability of the equilibrium point will change and the bifurcation will occur at the critical value.Finally, the optimal control strategy is given after the capture term is added.The single population and two species capture items are added to the corresponding ODE system of the model. Combined with the economic theory, the optimal control strategy of the model is obtained by using the Pontryagin maximum principle.The results show that when the discount rate is infinite, the income tends to be zero, on the contrary, when the discount rate is 00:00, the income reaches the maximum.The biological mathematical model provides convenience for the study of biological phenomena, and the related properties of the model can explain and predict some population behaviors, so as to achieve the purpose of seeking advantages and avoiding harm.At the same time, combined with the theory of biological mathematics and economic knowledge, artificial intervention is carried out in the process of population development, so that ecological resources can be rationally utilized and exploited in order to ensure the harmonious development of the ecosystem.To maximize economic benefits, these play an irreplaceable role in the development of biological populations and human beings.
【学位授予单位】：陕西科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175;O231

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