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方程组解的行为的结果。(1)研究了具有扩散项的Lotka-Volterra三种群竞争系统，揭示了种群的值域扩张行为。三个相互竞争的物种的种群可以入侵并扩大它们的活动范围。种群密度在一定的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。我们还得到了射程扩展速度的近似表达式。(2)我们考虑了一个反应扩散系统，它描述了人口密度的动态变化。结果表明，对于具有Alee效应的增长函数，区间扩张的速度小于Logistic增长函数的扩张速度，并且存在扩张停止的情况。在动态态和静态态的转移率相等的情况下，我们还得到了距离扩展速度的一些估计表达式。(3)考虑了周期碎片化环境中种群的射程扩张问题，用一个偏微分方程来描述。我们观察到了一些被称为“二维行进周期波”的波动，得到了一个隐式的二维行进周期波速度公式和一些起源于原点的种群前沿的模式。

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.