摘要
自然界中的许多生态现象都可以用数学模型来刻划.通过对数学模型的研究,我们可以对生态现象作出科学的解释和预测,从而为生态问题的解决提供合理的方案.早在十九世纪初,数学家们就利用常微分方程来描述生物种群的发展演变过程,这个时期所讨论的问题大都假定种群密度在空间分布均匀.如果密度分布不均匀,那么高密度位置的种群就会向低密度位置扩散,此时大量的生物数学模型可归纳为反应扩散方程.应用反应扩散方程来研究种群的动力学行为已成为偏微分方程领域研究的一个重要方向.鉴于种群的长时行为与反应扩散系统相应的平衡态问题密切相关,因此研究反应扩散系统正平衡态解的定性性质有着十分重要的理论和现实意义.
本文主要运用非线性分析和非线性偏微分方程,特别是反应扩散方程和对应椭圆型方程的理论和方法,深入研究了四类生物模型的动力学行为,包括正平衡态解的存在性、多重性、惟一性及稳定性.所涉及的数学理论包括上下解方法、比较原理、全局分歧理论、不动点指标理论和摄动理论等.主要内容有以下几个方面：
一、进一步研究了带Beddington-DeAngelis反应项的捕食-食饵模型正平衡态解的存在性和惟一性.首先利用分歧理论给出正解存在的充分条件,然后结合变分原理对主特征值进行估计,从而得到直接以参数形式出现的简单可计算的条件以保证正解的存在性和不存在性.同时,在一维情况下,我们给出了正解的存在惟一性.结果表明,当捕食者之间干涉作用较强时正解是惟一存在的.最后详细分析了体现捕食者之间相互干涉的参数k对正解渐近性的影响,进而在七充分大时得到了正解的惟一性.这与一维情况下的结果是一致的.
二、研究了一类具有扩散的捕食-食饵-互惠模型正平衡态解的多重性和惟一性.在种群的相互作用过程中,第三种群对捕食过程的干涉可以有很多方式.我们考虑的是第三种群阻碍捕食过程的情况,这种阻碍通过与食饵建立合作关系体现出来.首先利用不动点指标理论给出正平衡态解存在的两个充分条件,然后讨论了第三种群与食饵的合作关系对正解的影响.通过分歧理论和不动点指标理论的巧妙结合,我们在食饵对第三种群的影响较大时得到了正解的多重性结果,并且发现所有的正解只有两种类型,一种渐近稳定,一种不稳定；另一方面,通过细致分析解的渐近性态,我们在第三种群对捕食过程干涉较强(即γ充分大)时得到了正解的惟一性.进一步的研究发现,正解的惟一性不必然要求γ充分大,我们甚至在γ有界时得到了一个更一般的惟一性结果.
三、考虑了一类具有交叉扩散的捕食-食饵模型,其反应项为修正的Leslie-Gower和HollingⅡ型反应函数.首先利用分歧理论给出正解存在的两个充分条件,同时刻画了正解的共存区域.其次,通过分析共存区域的边界曲线对交叉扩散系数的依赖关系,我们发现当食饵对捕食者的扩散产生较大影响时,共存区域扩张,而当捕食者对食饵的扩散产生较大影响时,共存区域缩小.最后我们讨论了非线性扩散对正解的影响.结果表明,当非线性扩散较强时,所有正解只有两种可能的渐近行为.
四、研究了非均匀恒化器模型二重特征值处的分歧及稳定性.在两竞争物种的生长率均靠近其临界值时,得到了正解(u(s),v(s))的存在性,同时发现当s>0充分小时,该正解连接了分别发自两个半平凡解的分歧曲线.而且,通过复杂的计算,我们给出了该正解的渐近稳定性条件.
Many ecological phenomena in nature can be rationalized into mathematical models. By investigating these models, some ecological phenomena may be ex-plained and controlled scientifically, and some reasonable schemes may be provided for the solution of ecological problems.
As early as the beginning of the nineteenth century, mathematicians have made use of ordinary differential equations to describe the evolution of biological popula-tions. During that period, the issues discussed by people were under the assumption that the population density in space distribution is uniform. If the density distribu-tion is non-uniform, then high-density populations would diffuse to the position of low-density. At this point, a large number of bio-mathematical models can be sum-marized as reaction-diffusion equations. Using reaction-diffusion systems to study the dynamical behavior of biological populations has been an important research as-pect in the region of nonlinear partial differential equations. Since the population's long time behavior is closely related to the steady-state problem of reaction-diffusion system, studies on the qualitative properties of positive steady-state solutions have an important significance both in theory and in reality.
In this dissertation, mainly using the theories of nonlinear analysis and nonlin-ear partial differential equations, especially those of reaction-diffusion equations and the corresponding elliptic equations, we have systematically studied the dynamical behaviors of four biological models with reaction and diffusion, such as the coex-istence, multiplicity, uniqueness and stability of positive steady-states. The tools used here include super-sub solutions method, comparison principle, global bifurca-tion theory, fixed-point theory of topology and perturbation technique. The main contents and results are as follows:
(i) A diffusive predator-prey model with Beddington-DeAngelis functional re-sponse under homogeneous Dirichlet boundary conditions is studied once more. Making use of global bifurcation theory, we obtain a necessary and sufficient con-dition for the existence of positive solutions. Moreover, a range of parameters for the uniqueness of positive solution is described in one dimension. Furthermore, the effect of large k which represents the extent of mutual interference between preda-tors is extensively studied. By meticulously analyzing the asymptotic behaviors, we obtain a complete understanding of the existence, uniqueness and stability of positive solutions when k is sufficiently large.
(ii) We consider a reaction-diffusion system of three species:predator, prey and mutualist, and investigate the multiplicity and uniqueness of positive steady-states. Here we consider a case in which a mutualist modifies predation to the benefit of a prey, namely a mutualist deterring predation on a prey. By means of the index theory of fixed points, we obtain two sufficient conditions for the existence of positive solutions, and whenβis suitably large, we establish the multiplicity result and find all the positive solutions are of only two types, one is asymptotically stable and the other unstable. Moreover, by the classic regular and singular perturbation theory, we extensively study the effect ofγ, which represents the extent of a mutualist deterring predation on a prey, and gain a good understanding of the existence, uniqueness and stability of positive solutions whenγis sufficiently large. Our further results show that the uniqueness does not necessarily needγto be large at some moment, and we establish a more general result even whenγis bounded.
(iii) A cross-diffusion predator-prey model with modified Leslie-Gower and Holling-II functional responses is discussed. Making use of global bifurcation the-ory, we obtain two sufficient conditions for the existence of positive solutions and then describe the coexistence region. Moreover, by analyzing the cross-diffusion co-efficients dependence of two boundary curves, we find that the coexistence region spreads as the effect of prey on predator's diffusion increases, and narrows when the effect of predator on prey's diffusion is large. At last, we derive the effect of nonlin-ear diffusion on positive solutions. The results show that all the positive solutions have only two possible asymptotic behaviors.
(iv) A competition model in the unstirred chemostat is considered. The bifur-cation solution (u(s),v(s)) from a double eigenvalue is obtained. And we see that for sufficiently small s> 0, (u(s),v(s)) connects the bifurcating positive solution from the semitrivial solution (θa,0) with that from the other semitrivial solution (0,θb). Moreover, the asymptotic stability of (u(s),v(s)) is derived under certain conditions.
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