Li Bingtuan, Weinberger Hans F, Lewis Mark A
Department of Mathematics, University of Louisville, Louisville, KY 40292, USA.
Math Biosci. 2005 Jul;196(1):82-98. doi: 10.1016/j.mbs.2005.03.008.
It is well known that in many scalar models for the spread of a fitter phenotype or species into the territory of a less fit one, the asymptotic spreading speed can be characterized as the lowest speed of a suitable family of traveling waves of the model. Despite a general belief that multi-species (vector) models have the same property, we are unaware of any proof to support this belief. The present work establishes this result for a class of multi-species model of a kind studied by Lui [Biological growth and spread modeled by systems of recursions. I: Mathematical theory, Math. Biosci. 93 (1989) 269] and generalized by the authors [Weinberger et al., Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol. 45 (2002) 183; Lewis et al., Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol. 45 (2002) 219]. Lui showed the existence of a single spreading speed c() for all species. For the systems in the two aforementioned studies by the authors, which include related continuous-time models such as reaction-diffusion systems, as well as some standard competition models, it sometimes happens that different species spread at different rates, so that there are a slowest speed c() and a fastest speed c(f)(). It is shown here that, for a large class of such multi-species systems, the slowest spreading speed c() is always characterized as the slowest speed of a class of traveling wave solutions.
众所周知，在许多用于描述更适应环境的表型或物种扩散到适应性较差的区域的标量模型中，渐近扩散速度可被表征为该模型一族合适行波的最低速度。尽管人们普遍认为多物种（向量）模型具有相同的性质，但我们并未发现有任何证据支持这一观点。本文针对Lui [《用递归系统建模生物生长与扩散。I：数学理论》，《数学生物学》93 (1989) 269] 所研究并由作者们 [Weinberger等人，《合作模型中扩散的线性猜想分析》，《数学生物学杂志》45 (2002) 183；Lewis等人，《两种群竞争模型的扩散速度与线性猜想》，《数学生物学杂志》45 (2002) 219] 推广的一类多物种模型建立了这一结果。Lui证明了所有物种都存在单一的扩散速度c()。对于作者上述两项研究中的系统，其中包括诸如反应扩散系统等相关的连续时间模型以及一些标准竞争模型，有时会出现不同物种以不同速率扩散的情况，从而存在一个最慢速度c()和一个最快速度c(f)()。本文表明，对于一大类这样的多物种系统，最慢扩散速度c()总是被表征为一类行波解的最慢速度。
