Travelling wave solutions of diffusive Lotka-Volterra equations
Travelling wave solutions of diffusive Lotka-Volterra equations
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DOI：
10.1007/bf00276112
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发表时间：
1983-05
影响因子：
1.9
通讯作者：
S. Dunbar
中科院分区：
文献类型：
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作者：
S. Dunbar
研究了一类基于Lotka-Volterra捕食模型的反应扩散方程组行波解的存在性.为了简单起见，我们只考虑一维空间。波是过渡阵面型的，类似于Fisher和Kolmogorov等人讨论的标量反应扩散方程的行波解。这里讨论的波不一定是单调的。对于任何速度，都有一个过渡波前型行波解。对于这里讨论的系统之一，存在将波分成两种类型的区别速度c*，速度c < c* 的波是一种类型，速度c <c* 的波是另一种类型。我们提出的数值证据表明，对于这个系统的波的速度c* 是稳定的，并且c* 是在某种意义上的渐近传播速度。对于另一个系统，所有速度的波在某种意义上都是稳定的。存在性的证明使用一个射击参数和一个李雅普诺夫函数。我们还讨论了这些波的存在的一些可能的生物学意义。
We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speedcthere is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c*dividing the waves into two types, waves of speed c < c*being one type, waves of speed c ⩾ c*being of the other type. We present numerical evidence that for this system the wave of speed c*is stable, and that c*is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves.