众所周知, 扩散和时滞在事物的演化过程中往往是不可避免的. 因此时滞反应扩散方程和非局部时滞反应扩散方程引起了许多学者的关注, 在对具有 (非局部) 时滞的反应扩散方程的研究中, 最受关注的一个领域就是其行波解并已有许多重要结果. 但是对于系统来讲, 这方面的结果还很少, 而且这些结果不能揭示行波解的许多重要性质. 本文将从Lotka-Volterra 系统出发研究 (非局部) 时滞系统的行波解. 由于种群共存在实际中的重要意义, 因此本文研究的行波解皆与模型的共存平衡态有关. 主要内容分为四章阐述.
本文首先运用 Schauder 不动点定理结合上下解方法考虑了一类时滞反应扩散系统的行波解存在性, 并将结果应到具有时滞的竞争型 Lotka-Volterra 系统建立了连接平凡平衡态和共存平衡态之间的行波解. 特别还发现对应于种间竞争的时滞对于行波解的存在性影响不明显, 而对应于种内竞争的时滞对于行波解存在性有影响. 此外, 研究表明该系统的正行波解在零点附近的增长性和指数函数相当.其次考虑了具有时滞的 Lotka-Volterra 合作系统连接平凡平衡态和共存平衡态行波解的存在性, 渐近稳定性和最小波速. 首先通过构造适当的上下解, 证明了该系统行波解的存在性, 并且该行波解在平凡平衡态附近几乎是指数增长的. 然后将基于上下解方法和比较原理的挤压技术发展到该系统, 从而证明了行波解是渐近稳定的. 结果表明, 即使对于具有时滞的单稳系统其行波解依然可以决定其相应初值问题的长时间行为. 特别还给出了系统行波解的不存在性以及最小波速, 并指出了最小波速就是两种群系统中传播较快的物种的渐近传播速度.关于具有非局部时滞的竞争型 Lotka-Volterra 系统的双稳波. 通过引入一个无时滞但含有更多变元的辅助系统, 研究了该系统行波解的存在性, 稳定性以及波速的唯一性.结果表明, 对于具有无限时滞的双稳系统, 其双稳波关于大时滞依然具有持久性, 而且其行波解可以决定其相应初值问题的长时间行为, 因此从理论上说明了具有非局部时滞双稳系统行波解的重要性. 这为解释种群动力学中的生物入侵和空间分离现象提供了理论依据. 此外还给出了相应系统单稳波的存在性.本文最后研究了具有非局部时滞的 Lotka-Volterra 合作系统波前解. 通过采用前面类似的思想及完全不同的技巧, 研究了该系统行波解的存在性, 不存在性以及稳定性等,并探讨了非局部性的影响. 主要的研究技巧包括上下解, 谱分析以及挤压技术.
It seems that the time delay and diffusion are inevitable in many evolutionary pro-cesses, thus the delayed reaction diffusion equations and reaction diffusion equations with
nonlocal delay attracted much attentions, one important topic is its travelling wave solu-tions. But the result concerning with the system is scarce, and many important properties can not be displayed by the existed results. This paper is focused on the travelling wave solution of Lotka-Volterra systems with (nonlocal) delays, and the main interesting is the travelling wave solutions relating the coexistence equilibrium due to the importance of coexistence of species in ecology. The main result is divided into four chapters.
Firstly, the existence of the travelling wave solutions of delayed reaction diffusion systems are considered by Schauder’s fixed point theorem and upper-lower solutions.
The result is also applied to consider the two-species diffusion-competition models with delays and establish the existence of travelling wave solution connecting 0 with coexistence equilibrium. Especially, the result implies that the delays appeared in the interspecific competition terms do not affect the existence of traveling waves while the delays appeared in the intraspecific competition terms do. Moreover, its positive travelling wave solution grows like an exponential function near 0.Then, the existence, asymptotic stability and minimal wave speed of travelling wave fronts of cooperative Lotka-Volterra system with delay is studied. By constructing proper upper-lower solutions, the existence of travelling wave fronts connecting 0 with coexistence equilibrium was established and the result indicates that the travelling wave fronts grows exponentially near 0. Then the asymptotic stability of the travelling wave fronts is established by developing the so-called squeezing technique based on upper-lower solutions technique and comparison principle, which implies that even for delayed systems,its travelling wave front also can determine the long time behavior of the corresponding initial value problem. Moreover, the minimal wave speed and nonexistence of travelling wave fronts are established and is equivalent to the faster spreading speed of two species.Subsequently, we consider the bistable waves of a diffusive Lotka-Volterra type model with nonlocal delays for two competitive species. By introducing an undelayed reaction diffusion system with more variables, the existence, asymptotic stability and the uniqueness of wave speed are established for the system with nonlocal delay. It is interesting that the bistable wave is persistent even for large delay and the stability of travelling wave fronts implies that the bistable wave of nonlocal delayed system also can determine the long term behavior of the corresponding initial value problem which indicates the importance of travelling wave fronts of reaction diffusion system with nonlocal delay. Especially,the result concerning with the bistable waves can be generalized in mathematics and implies the spatial isolation or species invasion in ecology. The corresponding monostable case is also be studied.In the last, the monostable travelling wave fronts of Lotka-Volterra cooperative diffusion systems with nonlocal infinite delay is studied by an idea similar to that in Chapter4, but the technique is different to that. The existence, nonexistence and stability of the travelling wave fronts of the system is established by the upper-lower solutions,spectral theory and squeezing technique. The stability result of travelling wave fronts also extends the importance of travelling wave fronts of monostable reaction diffusion systems with nonlocal infinite delay.