多种群强耦合种群动态模型的稳定性分析
摘要
在数学生态学中,一个生态系统共存态的存在性及各种群的长时间行为是种群动态模型研究的主要内容.近年来,随着对种群动态模型研究的深入,同时考虑了扩散、自扩散和交错扩散作用的强耦合反应扩散方程组受到越来越多的关注.在这篇博士学位论文中,我们主要考虑种群动力学中的一类具有代表性的强耦合非线性反应扩散系统,通过对此类系统解的整体性态以及非常数稳态解存在性的研究,得到了一系列新的结果.整篇论文由五章组成.
第一章,介绍数学生态学及种群动态模型的背景、发展、研究进展及现状,给出本文所要讨论的主要问题和研究思想.
第二章,给出文中要用到的一些重要引理、命题等辅助知识.
第三章,讨论一类带非线性耗散的强耦合反应扩散系统的整体解.这类系统是同时考虑了扩散、自扩散和交错扩散作用的n种群SKT模型.以H.Amann建立的非负解的局部存在性为基础,采用PDE中标准的能量估计方法,结合恰当的Gagliardo-Nirenberg型插值不等式,我们在扩散矩阵和竞争矩阵正定的条件下得到该类系统解的整体存在性和一致有界性,并进一步由Lyapunov函数法得出非常数正平衡态的不存在性.
第四章,主要考虑三种群强耦合HP食物链模型.在零流边界条件下,运用Leray-Schauder度理论,得到这个强耦合系统的非常数正稳态解的存在性.同时,由种群动力学中的一些基本概念及方法,讨论该模型的常数正平衡点的稳定性.结果表明,对于强耦合情形,当第二个种群的扩散或自扩散作用较强时,系统不存在非常数正平衡态,而当第二个种群相对于第一个种群或第三个种群相对于第二个种群的交错扩散足够大时,系统至少存在一个非常数正稳态解.在弱耦合情形,当第三个种群的扩散充分大时会出现非常数正平衡态.因此,较强的扩散或交错扩散对系统生成稳态模式(stationary patterns)起着促进作用.
第五章,运用Rabinowitz局部和全局分歧定理,研究零边界条件下强耦合竞争-竞争-互惠模型的平衡态问题正解的大范围分歧.以第一个种群的内禀增长率作为分歧参数,根据Rabinowitz局部分歧定理得到了平衡态系统在第一个分量为0的半平凡平衡解支上的分歧解.进一步,由Rabinowitz正解的大范围分歧定理可知这个分歧解是全局存在的.因此当三个种群的内禀增长率适当大,并且某个相关的带非线性扩散项的特征值问题的第一特征值为0且为奇代数重数时,这三个种群至少有一个共存态.
In mathematical ecology, the coexistence and the long time behavior of various species in an ecosystem are the chief contents of population dynamical models. With the rapid development of researches, more and more attention is recently concentrated on the strongly coupled reaction-diffusion equations with diffusion, self-diffusions and cross-diffusions. In this doctoral dissertation, we mainly discuss a representational class of strongly coupled nonlinear reaction-diffusion systems arising in population dynamics. By investigating the global behavior and the existence of non-constant positive steady states, we get a series of new and meaningful results. This thesis consists of four chapters.
In Chapter 1, the background, historical development, advances in research and present situation and prospects of mathematical ecology and population dynamical models are introduced. The problems considered in this paper and the main ideas are presented.
In Chapter 2, the preliminaries used in the main body such as some important lemmas and propositions are listed.
In Chapter 3, we discuss the global solutions for a class of strongly coupled reaction-diffusion systems with nonlinear dissipative terms. Such system is n species SKT model including the effects of diffusions, self-diffusions as well as cross-diffusions. Based on the local existence of the nonnegative solutions from H. Amann, we consider the global existence and boundedness of the solutions by employing the standard technique of energy estimates and a few well-chosen Gagliardo-Nirenberg interpolation inequalities. If the diffusion matrix and competition matrix are all positive definite, the nonnegative solution for the system can exist globally and be uniformly bounded. Moreover, the non-existence of non-constant positive steady state is also obtained by constructing an appropriate Lyapunov function.
In Chapter 4, the strongly coupled HP food chain model with three interacting species is investigated. When zero population flux across the boundary, by using the Leray-Schauder degree theory, we obtain the existence of non-constant positive steady states for this model. In addition, the stability of constant positive equilibrium point is discussed by elementary concept and methods in population dynamics. The results indicate that for strongly coupled system, when the effect of diffusion or self-diffusion of the second species is strong, there is no non-constant positive steady state for the system, whereas there is at least one non-constant positive steady state if cross-diffusion of the second species due to the first species or the third species due to the second species is large enough, and there is at least one non-constant positive steady state for weakly coupled system when diffusion of the third species is strong enough. Therefore, the emergence of stationary patterns is due to strong diffusion or cross-diffusions.
In Chapter 5, we mainly discuss the golbal bifurcation of the steady state problem for the strongly coupled competitor-competitor-mutualist model under zero boundary conditions. The tools used in this chapter are the local and global bifurcation theorems of Rabinowitz. Regarding the intrinsic rate of natural increase of the first species as bifurcation parameter, we can obtain the positive solution branch of the steady state system which bifurcates from the semitrivial solution set the first component of which is zero by Rabinowitz's local bifurcation theorem. Further, Rabinowitz's global bifurcation theorem of positive solution shows that the local bifurcation branch is global. Therefore, the competitor-competitor-mutualist cross-diffusions system possess at least one coexistence if the intrinsic increasing rates of the three species fulfill proper conditions and the principal eigenvalue of an eigenvalue problem with nonlinear diffusion term is 0 and the algebra multiplicity of it is odd.
第一章,介绍数学生态学及种群动态模型的背景、发展、研究进展及现状,给出本文所要讨论的主要问题和研究思想.
第二章,给出文中要用到的一些重要引理、命题等辅助知识.
第三章,讨论一类带非线性耗散的强耦合反应扩散系统的整体解.这类系统是同时考虑了扩散、自扩散和交错扩散作用的n种群SKT模型.以H.Amann建立的非负解的局部存在性为基础,采用PDE中标准的能量估计方法,结合恰当的Gagliardo-Nirenberg型插值不等式,我们在扩散矩阵和竞争矩阵正定的条件下得到该类系统解的整体存在性和一致有界性,并进一步由Lyapunov函数法得出非常数正平衡态的不存在性.
第四章,主要考虑三种群强耦合HP食物链模型.在零流边界条件下,运用Leray-Schauder度理论,得到这个强耦合系统的非常数正稳态解的存在性.同时,由种群动力学中的一些基本概念及方法,讨论该模型的常数正平衡点的稳定性.结果表明,对于强耦合情形,当第二个种群的扩散或自扩散作用较强时,系统不存在非常数正平衡态,而当第二个种群相对于第一个种群或第三个种群相对于第二个种群的交错扩散足够大时,系统至少存在一个非常数正稳态解.在弱耦合情形,当第三个种群的扩散充分大时会出现非常数正平衡态.因此,较强的扩散或交错扩散对系统生成稳态模式(stationary patterns)起着促进作用.
第五章,运用Rabinowitz局部和全局分歧定理,研究零边界条件下强耦合竞争-竞争-互惠模型的平衡态问题正解的大范围分歧.以第一个种群的内禀增长率作为分歧参数,根据Rabinowitz局部分歧定理得到了平衡态系统在第一个分量为0的半平凡平衡解支上的分歧解.进一步,由Rabinowitz正解的大范围分歧定理可知这个分歧解是全局存在的.因此当三个种群的内禀增长率适当大,并且某个相关的带非线性扩散项的特征值问题的第一特征值为0且为奇代数重数时,这三个种群至少有一个共存态.
In mathematical ecology, the coexistence and the long time behavior of various species in an ecosystem are the chief contents of population dynamical models. With the rapid development of researches, more and more attention is recently concentrated on the strongly coupled reaction-diffusion equations with diffusion, self-diffusions and cross-diffusions. In this doctoral dissertation, we mainly discuss a representational class of strongly coupled nonlinear reaction-diffusion systems arising in population dynamics. By investigating the global behavior and the existence of non-constant positive steady states, we get a series of new and meaningful results. This thesis consists of four chapters.
In Chapter 1, the background, historical development, advances in research and present situation and prospects of mathematical ecology and population dynamical models are introduced. The problems considered in this paper and the main ideas are presented.
In Chapter 2, the preliminaries used in the main body such as some important lemmas and propositions are listed.
In Chapter 3, we discuss the global solutions for a class of strongly coupled reaction-diffusion systems with nonlinear dissipative terms. Such system is n species SKT model including the effects of diffusions, self-diffusions as well as cross-diffusions. Based on the local existence of the nonnegative solutions from H. Amann, we consider the global existence and boundedness of the solutions by employing the standard technique of energy estimates and a few well-chosen Gagliardo-Nirenberg interpolation inequalities. If the diffusion matrix and competition matrix are all positive definite, the nonnegative solution for the system can exist globally and be uniformly bounded. Moreover, the non-existence of non-constant positive steady state is also obtained by constructing an appropriate Lyapunov function.
In Chapter 4, the strongly coupled HP food chain model with three interacting species is investigated. When zero population flux across the boundary, by using the Leray-Schauder degree theory, we obtain the existence of non-constant positive steady states for this model. In addition, the stability of constant positive equilibrium point is discussed by elementary concept and methods in population dynamics. The results indicate that for strongly coupled system, when the effect of diffusion or self-diffusion of the second species is strong, there is no non-constant positive steady state for the system, whereas there is at least one non-constant positive steady state if cross-diffusion of the second species due to the first species or the third species due to the second species is large enough, and there is at least one non-constant positive steady state for weakly coupled system when diffusion of the third species is strong enough. Therefore, the emergence of stationary patterns is due to strong diffusion or cross-diffusions.
In Chapter 5, we mainly discuss the golbal bifurcation of the steady state problem for the strongly coupled competitor-competitor-mutualist model under zero boundary conditions. The tools used in this chapter are the local and global bifurcation theorems of Rabinowitz. Regarding the intrinsic rate of natural increase of the first species as bifurcation parameter, we can obtain the positive solution branch of the steady state system which bifurcates from the semitrivial solution set the first component of which is zero by Rabinowitz's local bifurcation theorem. Further, Rabinowitz's global bifurcation theorem of positive solution shows that the local bifurcation branch is global. Therefore, the competitor-competitor-mutualist cross-diffusions system possess at least one coexistence if the intrinsic increasing rates of the three species fulfill proper conditions and the principal eigenvalue of an eigenvalue problem with nonlinear diffusion term is 0 and the algebra multiplicity of it is odd.
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