Wang Mao-Xiang, Lai Pik-Yin
Department of Physics, Graduate Institute of Biophysics, and Center for Complex Systems, National Central University, Chungli, Taiwan 320, Republic of China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Nov;86(5 Pt 1):051908. doi: 10.1103/PhysRevE.86.051908. Epub 2012 Nov 7.
We consider the competitive population dynamics of two species described by the Lotka-Volterra model in the presence of spatial diffusion. The model is described by the diffusion coefficient (d(α)) and proliferation rate (r(α)) of the species α (α = 1,2 is the species label). Propagating wave front solutions in one dimension are investigated analytically and by numerical solutions. It is found that the wave profiles and wave speeds are determined by the speed parameters, v(α) ≡ 2 sqrt [d(α)r(α)], of the two species, and the phase diagrams for various inter- and intracompetitive scenarios are determined. The steady wave front speeds are obtained analytically via nonlinear dynamics analysis and verified by numerical solutions. The effect of the intermediate stationary state is investigated and propagating wave profiles beyond the simple Fisher wave fronts are revealed. The wave front speed of a species can display abrupt increase as its speed parameter is increased. In particular for the case in which both species are aggressive, our results show that the speed parameter is the deciding factor that determines the ultimate surviving species, in contrast to the case without diffusion in which the final surviving species is decided by its initial population advantage. Possible relations to the biological relevance of modeling cancer development and wound healing are also discussed.
我们考虑在存在空间扩散的情况下，由Lotka-Volterra模型描述的两个物种的竞争种群动态。该模型由物种α（α = 1,2为物种标签）的扩散系数（d(α)）和增殖率（r(α)）描述。通过解析方法和数值解研究了一维中的传播波前解。发现波剖面和波速由两个物种的速度参数v(α)≡2√[d(α)r(α)]决定，并确定了各种种间和种内竞争情况下的相图。通过非线性动力学分析解析地获得了稳定波前速度，并通过数值解进行了验证。研究了中间稳态的影响，并揭示了超出简单Fisher波前的传播波剖面。一个物种的波前速度随着其速度参数的增加可能会突然增加。特别是对于两个物种都具有侵略性的情况，我们的结果表明，速度参数是决定最终存活物种的决定性因素，这与没有扩散的情况相反，在没有扩散的情况下，最终存活物种由其初始种群优势决定。还讨论了与模拟癌症发展和伤口愈合的生物学相关性的可能关系。
