Kollár Richard, Novak Sebastian
Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia.
Institute of Science and Technology Austria, Am Campus 1, 3400, Klosterneuburg, Austria.
Bull Math Biol. 2017 Mar;79(3):525-559. doi: 10.1007/s11538-016-0244-3. Epub 2016 Dec 22.
Variation in genotypes may be responsible for differences in dispersal rates, directional biases, and growth rates of individuals. These traits may favor certain genotypes and enhance their spatiotemporal spreading into areas occupied by the less advantageous genotypes. We study how these factors influence the speed of spreading in the case of two competing genotypes under the assumption that spatial variation of the total population is small compared to the spatial variation of the frequencies of the genotypes in the population. In that case, the dynamics of the frequency of one of the genotypes is approximately described by a generalized Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation. This generalized F-KPP equation with (nonlinear) frequency-dependent diffusion and advection terms admits traveling wave solutions that characterize the invasion of the dominant genotype. Our existence results generalize the classical theory for traveling waves for the F-KPP with constant coefficients. Moreover, in the particular case of the quadratic (monostable) nonlinear growth-decay rate in the generalized F-KPP we study in detail the influence of the variance in diffusion and mean displacement rates of the two genotypes on the minimal wave propagation speed.
基因型的差异可能导致个体在扩散速率、方向偏差和生长速率方面存在不同。这些特性可能有利于某些基因型，并增强它们在时空上向优势较弱的基因型所占据区域的扩散。我们研究在两种竞争基因型的情况下，这些因素如何影响扩散速度，假设总种群的空间变化与种群中基因型频率的空间变化相比很小。在这种情况下，其中一种基因型频率的动态变化大约由一个广义的Fisher-Kolmogorov-Petrovskii-Piskunov（F-KPP）方程来描述。这个带有（非线性）频率依赖扩散项和平流项的广义F-KPP方程允许行波解的存在，这些解刻画了优势基因型的入侵。我们关于行波解存在性的结果推广了具有常系数的F-KPP方程的经典行波理论。此外，在我们所研究的广义F-KPP方程中二次（单稳）非线性生长-衰减率的特殊情况下，我们详细研究了两种基因型的扩散方差和平均位移速率对最小波传播速度的影响。
