發(fā)布時間：2018-04-12 15:40

  本文選題：存在性 + Turing不穩(wěn)定性　； 參考：《陜西科技大學(xué)》2017年碩士論文

【摘要】：在反映客觀世界運動過程量與量之間的關(guān)系中,大量存在滿足微分方程關(guān)系式的數(shù)學(xué)模型,且微分方程作為研究生態(tài)系統(tǒng)所需的重要工具,不僅可以描述單個種群內(nèi)部之間的作用關(guān)系,也可以描述多個種群之間的捕食、競爭和互惠等相互作用關(guān)系。本文主要在Neumann邊界條件下,對一類具有競爭-競爭-捕食關(guān)系的三種群反應(yīng)擴散模型作了以下定性分析。首先,給出該模型常數(shù)解的存在性和穩(wěn)定性。通過求解代數(shù)方程得到該模型平凡解、弱半平凡解、強半平凡解的存在性條件,特別地,得出了在一定條件下,該模型正常數(shù)解的唯一存在性。利用Lyapunov第一方法,判斷出五個常數(shù)解在ODE系統(tǒng)和PDE系統(tǒng)下都不穩(wěn)定;在一定條件下,強半平凡解5E在ODE系統(tǒng)下局部漸近穩(wěn)定,在PDE系統(tǒng)下不穩(wěn)定,即由于種群擴散導(dǎo)致系統(tǒng)失穩(wěn)而形成了新的空間模式,產(chǎn)生了Turing不穩(wěn)定性;一定條件下的強半平凡解6E在ODE系統(tǒng)和PDE系統(tǒng)下都局部漸近穩(wěn)定,在其相反條件下,都不穩(wěn)定。對唯一的正常數(shù)解,一方面,給出了其局部漸近穩(wěn)定性;另一方面,運用Lyapunov第二方法構(gòu)造V函數(shù),通過判斷其全導(dǎo)數(shù)的正負得出了唯一存在的正常數(shù)解在ODE系統(tǒng)下的全局穩(wěn)定性。其次,建立該模型對應(yīng)的非常數(shù)正平衡態(tài)解的存在性。利用最大值原理和Harnack不等式給出正解的先驗估計,并利用Poincaré相關(guān)不等式給出非常數(shù)正平衡態(tài)解的不存在性條件,同時,運用Leray-Schauder度理論通過計算不動點指數(shù)證明了非常數(shù)正平衡態(tài)解的存在性。再次,給出時滯系統(tǒng)Hopf分支的存在性。由于生物種群的發(fā)展不完全依賴于當前的狀態(tài),還依賴于此前的某一時刻或者某一時間段的狀態(tài),所以對該模型的ODE系統(tǒng)加入時滯項進行討論。利用比較定理給出時滯系統(tǒng)解的有界性和一致持久性。以時滯為參數(shù),給出發(fā)自兩個強半平凡解的Hopf分支的臨界點0?.結(jié)果表明時滯會對平衡點的穩(wěn)定性產(chǎn)生影響,當時滯參數(shù)超過某一臨界值時,平衡點的穩(wěn)定性會發(fā)生變化,且在該臨界值處發(fā)生分支現(xiàn)象。最后,給出加入捕獲項后的最優(yōu)控制策略。在該模型對應(yīng)的ODE系統(tǒng)中加入單種群及兩種群捕獲項,與經(jīng)濟理論相結(jié)合,利用Pontryagin極大值原理得出模型的最優(yōu)控制策略。結(jié)果表明,當貼現(xiàn)率無限大時,收益趨于零;相反,當貼現(xiàn)率為零時,收益達到最大。生物數(shù)學(xué)模型為研究生物現(xiàn)象提供了便利,模型的相關(guān)性質(zhì)即可解釋和預(yù)測一些種群行為,從而達到趨利避害的目的。同時,結(jié)合生物數(shù)學(xué)理論和經(jīng)濟知識理論,在種群發(fā)展過程中實施人為干預(yù),使生態(tài)資源得到合理的利用和開發(fā),以期在保證生態(tài)系統(tǒng)和諧發(fā)展的情況下,最大程度上實現(xiàn)經(jīng)濟效益,這些對生物種群和人類的發(fā)展都有著不可替代的作用。
[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.
【學(xué)位授予單位】：陜西科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175;O231

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本文編號：1740360