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性质的结果。(1)研究了一类具有扩散项的Lotka-Volterra三种群竞争系统，揭示了种群的值域扩展行为.三个相互竞争的物种的种群可以入侵并扩大它们的范围。在一定的时间间隔内，种群的密度几乎是恒定的。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)考虑一个反应扩散系统，它描述了一个具有移动的和稳态的种群密度动力学。我们的结果是，对于具有Alee效应的增长函数，范围扩展的速度小于逻辑斯谛增长函数，也有一个情况下，扩展停止。在移动的态与静止态之间的转移率相等的情况下，得到了距离扩展速度的估计表达式。(3)我们考虑周期性破碎环境中种群的活动范围扩展问题，它由一个偏微分方程描述。我们观察到了一种称为“二维周期行波"的波，得到了二维周期行波速度的一个隐式公式和初始定域在原点的布居前沿的一些模式。

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.