【作者】 王小利；

【导师】 王稳地；

【作者基本信息】 西南大学 ， 基础数学， 2016， 博士

【摘要】 本论文首先研究了下面的包含一种生物量(灌木,树)和一种资源(水)的模型的全局分支通过定性的方法,得到了系统的(前向／后向)跨临界分支,鞍-结点分支,(亚临界／超临界)Hopf分支,同宿轨分支,以及Bogdanov-Takens分支的存在性。通过数值计算,我们发现系统存在两个平衡点或者一个平衡点和一个极限环的双稳定现象,甚至存在两个平衡点和一个极限环的三稳定现象。此模型或许是同时具有平衡点后向跨临界分支,平衡点和极限环的鞍-结点分支,Hopf分支,极限环泡／心脏状分支,同宿轨分支和Bogdanov-Takens分支等动力学行为的最小的模型。模型中振荡植被模式的出现可以作为植被灾难性转变的一个预警信号。此模型也可以看作一个传染病模型。根据此研究方法和结果,也可以将此类传染病模型的参数空间根据平衡态稳定性的不同进行详细的情况分类,并且归类出此类传染病模型中的诸如“泡”,“心脏”,“莲蓬”及“辣椒”等不同形状的分支示意图。此研究结果为自然界中植被稳定态的优化提供了基本框架,也为水-植物系统的双稳定态的存在性提供了理论基础。接下来研究了下面的捕食者具有趋饵效应的捕食者-食饵模型的全局分支通过著名的Crandall-Rabinowitz分支定理和Shi-Wang的全局分支的理论,得到了从常数平衡解分支出非常数平衡解的存在性。同时,说明了在某些情况下非常数平衡解的稳定性。此研究结果表明,如果捕食者-食饵相互关系中食饵不具有群体防御的能力,则系统将会出现一个同质的稳定状态。但如果食饵具有了群体防御的能力,则系统将可能会出现异质的稳定状态。从生物学角度讲,捕食者的搜寻行为导致异质环境到同质环境的转变,而撤退行为则导致同质环境到异质环境的转变,从而也就会出现更丰富的动力学行为。然后研究了下面的考虑下渗反馈效应的水-植物模型的Turing分支通过Turing分支的研究方法,得到了系统Turing斑图模式解的存在性。通过有限差分的方法,给出了Turing斑图模式解的一维数值模拟。此研究结果意味着扩散可能导致斑图模式的形成,而降雨量和水植物之间的下渗反馈效应则影响植被进化的最终状态。如果考虑到植物根对水分的抽吸能力,研究结果说明由抽吸能力比较强的植物构成的植被区域比由抽吸能力比较弱的植被构成的植被区域更容易形成斑图模式。最后研究了下面的水-植物模型的Turing分支通过Turing分支的研究方法,得到了Turing失稳的参数空间。根据有限差分的方法,给出了Turing斑图模式解的二维数值模拟,进而得到了植物的竞争效应和根抽取引起的正反馈效应对系统Turing分支的影响,从而解释了自然界中植物斑图模式存在的多样性。此研究结果表明随着竞争强度的增加,系统产生斑图模式的可能性减小。但随着根的正反馈效应增强,系统产生斑图模式的可能性变大。因此,在根的正反馈效应和植物竞争负反馈效应的共同影响下,根的正反馈效应会促进植物斑图模式的形成,但植物之间的竞争则有相反的影响。更多还原

【Abstract】 In this thesis, we firstly study the global dynamics of an ordinary differential equation model for one biomass (shrubs or trees) and one resource (water) By carrying out qualitative analysis, we present a detailed partition of the pa-rameter space and find rich dynamic including a "bubble", or a "heart", or a "lotus", or a "pepper". Furthermore, we find that the system has bistability of two equilibria or one equilibrium and one limit cycle, or even tristability of two equilibria and one limit cycle. This is perhaps a minimal model to have the properties of backward transcritical bifurcation, saddle-node bifurcations for equilibria and limit cycles, Hopf bifurcations, limit cycle bubble, homoclinic bi-furcation, Bogdanov-Takens bifurcation, all in one simple model. In the model, oscillatory vegetation change may be an indicator of catastrophic shift. Our model apparently can also be considered as an epidemic model. This work pro-vides a fundamental framework for alternative stable states or stable cycles in native biomass density, also provides a theoretical foundation for the existence of bistability in natural water-biomass system.Then, we consider the global bifurcation of the following Lotka-Volterra predator-prey-taxis model By Crandall-Rabinowitz Bifurcation Theorem and Shi-Wang Bifurcation Theo-rem, we establish the result of the bifurcation of nonconstant solutions from the positive constant solution. Furthermore, we show that the nonconstant solutions are locally stable in some cases. Our study indicates that the predator-prey sys-tem undergoes a homogeneous stable state if the prey does not have group defense ability, but a heterogeneous stable state exists if the prey has the group defense ability. Biologically, the foraging behavior of predators may transform heteroge-neous environments into homogeneous environments, while the retreating behav-ior of predators may transform homogeneous environments into heterogeneous environments which induces richer dynamical behaviors.Next, we study the Turing bifurcation of the following water-biomass model which captures the "infiltration feedback" of biomass By using Turing’s idea, we obtain the existence of Turing pattern and its numer-ical simulations in 1-dimension space by finite difference methods. Our results imply that the diffusion may lead to the pattern formation, while the rainfall rate and the infiltration feedback between the water and plant may affect the final s-tate of vegetation evolution. If considering the root suction capability of water, our study shows that a vegetation cover comprised of biomass with stronger suc-tion ability is more likely to exhibit pattern formation than a vegetation cover comprised of biomass with weaker suction ability.Finally, we study the Turing bifurcation of the following water-biomass model By Turing’s idea, we present a detailed parameter space for Turing instability. By finite difference methods, we give the numerical simulations in 2-dimension space which in turn help to study the effects of the competition between plants and roots-induced positive feedback on Turing bifurcation and explain the diversity of dryland vegetation. Our results show that with the increase of the intensity of competition, the possibility of generating pattern formation is reduced, but as the positive feedback effect of root is increased, the possibility of generating pattern formation becomes larger. Thus, under the influence of the positive feedback effect and the negative feedback effect of plant competition, the positive feedback effect of root can promote the formation of pattern, but the competition among plants has the opposite effect.更多还原