On behaviors of solutions in non-linear reaction-diffusion systems

在本研究中，我们得到了一些非线性反应扩散系统的解的行为。(1)研究了具有扩散项的Lotka-Volterra三竞争种系统，揭示了种群的范围扩张行为。三个竞争物种的种群可以入侵并扩大它们的活动范围。种群密度在一段时间间隔内几乎是恒定的。在那之后，人口密度在混乱中变化。我们还得到了范围扩展速度的近似表达式。(2)我们考虑一个反应扩散系统，它描述了种群密度在移动和静止状态下的动态变化。结果表明，对于具有Alee效应的生长函数，其范围扩展的速度小于logistic生长函数，并且存在扩展停止的情况。在运动状态和静止状态的转换速率相等的情况下，得到了距离扩展速度的估计表达式。(3)考虑周期性破碎化环境中种群的范围扩展，用偏微分方程来描述。对“二维行周期波”进行了观测，得到了二维行周期波速度的隐式表达式，以及二维行周期波在起始点处的一些种群前缘分布规律。

In this research, we have the following results of the behaviors of solutions in some non-linear reaction-diffusion systems. (1) We investigate a Lotka-Volterra three-competing-species system with diffusion terms and reveal the range expansion behaviors of populations. The populations of three competing species can invade and expand their ranges. The densities of populations are nearly constant during some time interval. After that, the population densities vary in chaos. We also get an approximate expression for the speed of range expansion. (2) We consider a reaction-diffusion system, which describes a dynamics of the population densities with mobile and stationary states. We result that for the growth function with Alee effects, the speed of range expansion is smaller then that for the logistic growth function and also there is a case where the expansion ceases. We also get some estimated expressions for the speed of range expansion in case of equal transition rate between mobile state and stationary one. (3) We consider the range expansion of a population in periodically fragmented environments, which is described by a partial differential equation. We observe some waves called by "traveling periodic wave in tow-dimension (TPW)" and get an implicit formula for the speed of TPW and some patterns of the front edges of population localized at origin at the beginning.