article
Existence of traveling waves for the generalized F–KPP equation
published
yes
Richard
Kollár
author
Sebastian
Novak
author 461468AE-F248-11E8-B48F-1D18A9856A87
NiBa
department
Speed of Adaptation in Population Genetics and Evolutionary Computation
project
Limits to selection in biology and in evolutionary computation
project
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.
Springer2017
eng
Bulletin of Mathematical Biology10.1007/s11538-016-0244-3
793525-559
R. Kollár, S. Novak, Bulletin of Mathematical Biology 79 (2017) 525–559.
Kollár R, Novak S. 2017. Existence of traveling waves for the generalized F–KPP equation. Bulletin of Mathematical Biology. 79(3), 525–559.
R. Kollár and S. Novak, “Existence of traveling waves for the generalized F–KPP equation,” <i>Bulletin of Mathematical Biology</i>, vol. 79, no. 3. Springer, pp. 525–559, 2017.
Kollár, Richard, and Sebastian Novak. “Existence of Traveling Waves for the Generalized F–KPP Equation.” <i>Bulletin of Mathematical Biology</i>, vol. 79, no. 3, Springer, 2017, pp. 525–59, doi:<a href="https://doi.org/10.1007/s11538-016-0244-3">10.1007/s11538-016-0244-3</a>.
Kollár, R., & Novak, S. (2017). Existence of traveling waves for the generalized F–KPP equation. <i>Bulletin of Mathematical Biology</i>. Springer. <a href="https://doi.org/10.1007/s11538-016-0244-3">https://doi.org/10.1007/s11538-016-0244-3</a>
Kollár, Richard, and Sebastian Novak. “Existence of Traveling Waves for the Generalized F–KPP Equation.” <i>Bulletin of Mathematical Biology</i>. Springer, 2017. <a href="https://doi.org/10.1007/s11538-016-0244-3">https://doi.org/10.1007/s11538-016-0244-3</a>.
Kollár R, Novak S. Existence of traveling waves for the generalized F–KPP equation. <i>Bulletin of Mathematical Biology</i>. 2017;79(3):525-559. doi:<a href="https://doi.org/10.1007/s11538-016-0244-3">10.1007/s11538-016-0244-3</a>
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