Lotka-Volterra反应扩散竞争系统的动力学研究

The competitive Lotka-Volterra system with diffusion is an important ecological model, which is used to illustrate the competitive relationships between different species. Although there have been a large number of conclusions in the research of all aspects of the system, some of the key problems have still not been solved. Therefore, in this project, based on the previous works, we further focus on the relevant dynamic problems of the competitive Lotka-Volterra system with diffusion. This project is composed of three parts. Firstly, under the unbalanced competition, for the competitive Lotka-Volterra system with diffusion, via Wazewski theorem and the phase plane analysis, we will discuss the minimal speed of traveling front solutions, which connects two different equilibria of the system on each axis. And furthermore, we will investigate the sufficient conditions for the nonlinear determinacy of this minimal speed and its specific expression. Thus our results will advance the research the conjecture of Murray et al. Secondly, under the weak competition, with the aid of the weighted energy method and some estimates, we will explore the global exponential stability of traveling front solutions of this system connecting different equiliria. This conclusion will complete the relative research of stability of traveling front solutions. Thirdly, under different kinds of competition, by using the super-sub solution method, the comparison theorem and the threshold solution method, we will investigate the existence and uniqueness of entire solutions of this system. In fact, these results will supplement the results obtained by Morita et al. and enrich the relative aspects of entire solutions.

Lotka-Volterra反应扩散竞争系统是描述种群间相互竞争的一类重要生态模型.对该系统各方面的研究虽已取得了许多的成果,但一些关键问题仍未得到解决.本项目在前人已有的研究基础之上,将进一步深入研究Lotka-Volterra反应扩散竞争系统的相关动力学问题,具体包含如下三个问题.首先,在不均衡竞争情形下,结合Wazewski定理和相平面分析方法,研究该系统连接两个轴上平衡点的波前解的最小波速问题,探究最小波速由非线性决定的充分条件,估计最小波速具体表达式,从而推进Murray猜想和Hosono猜想的研究.其次,在弱竞争情形下,利用能量加权方法和一些估计,研究该系统连接不同平衡点的波前解的全局指数稳定性,完善波前解稳定性的相关结论.最后,在不同竞争情形下,通过上下解方法,比较原理及临界解等方法,研究该系统整体解的存在性和唯一性,补充Morita等人的研究结果,丰富了整体解的研究内容.

传播问题的研究是扩散方程与系统研究领域的热点,本项目围绕扩散方程和系统行波解和整体解的存在性与稳定性、最小波速的选择机制等展开研究. .本项目将加权能量方法应用到带有退化反应项的Fisher型方程和Lotka-Volterra反应扩散竞争系统非临界波前解的稳定性、移动环境下非局部扩散方程强迫波的稳定性、非单调的时滞Nicholson方程的狄利克雷初边值问题解的指数收敛性的研究,改进了已有的结果;利用上下解方法等方法证明了带有退化反应项的时滞Fisher型方程波前解的存在性,推广了时滞微分方程波前解存在性的范围;通过改进上下解的构造技巧证明了不同类型扩散系统行波解的存在性,建立了对应的最小波速的选择机制,推进了相关研究;将单调动力系统方法、消失粘性法、打靶法等方法应用到更为复杂系统,证明这些系统行波解的存在性,扩展了研究内容;在存在性的基础上,利用上下解方法证明了双稳反应扩散方程整体解的稳定性,丰富了已有的研究成果..在研究成果、人才培养和学术交流等方面,均实现了预期目标,在Proceedings of the Royal Society of Edinburgh. Section A. Mathematics,Journal of Mathematical Physics等期刊已发表学术论文10篇,1名项目组青年教师晋升副教授,培养硕士研究生13名(毕业2名,转为硕博连读1名,在读10名),其中,2名硕士继续攻读博士学位,1名硕士获国家奖学金,1名硕士获山西大学校级优秀学位论文并推荐参评山西省优秀学位论文..本项目在科学研究、人才培养等方面都取得了较好的成效.