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Existence of Traveling Waves for the Generalized F–KPP Equation

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Abstract

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.

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Notes

  1. The quadratic f(p) is called monostable as the spatially homogeneous reduced dynamical system \(p_t = f(p)\) has one stable and one unstable equilibrium. On the other hand, a cubic f(p) is called bistable as in that case there are two stable equilibria.

  2. See Stokes (1976), van Saarloos (2003) and Dumotier et al. (2007) for more explanation of the term pushed and pulled wave in the context of front propagation in reaction–diffusion equations.

  3. Compare with the function h(p) given by (31) used in the proof of Lemma 1.

  4. Compare with (41) and (31). Also note that \(g(0)= g(1) =0\) and \(g(P) < 0\) for \(P\in (0,1)\) with respect to the flow (19)–(20) for all \(c \ge c^{*}\).

  5. Compare to (42).

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Acknowledgements

We thank Nick Barton, Katarína Bod’ová, and Srđan Sarikas for constructive feedback and support. Furthermore, we would like to express our deep gratitude to the anonymous referees (one of whom, Jimmy Garnier, agreed to reveal his identity) and the editor Max Souza, for very helpful and detailed comments and suggestions that significantly helped us to improve the manuscript. This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under Grant Agreement 618091 Speed of Adaptation in Population Genetics and Evolutionary Computation (SAGE) and the European Research Council (ERC) Grant No. 250152 (SN), from the Scientific Grant Agency of the Slovak Republic under the Grant 1/0459/13 and by the Slovak Research and Development Agency under the Contract No. APVV-14-0378 (RK). RK would also like to thank IST Austria for its hospitality during the work on this project.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia

    Richard Kollár

  2. Institute of Science and Technology Austria, Am Campus 1, 3400, Klosterneuburg, Austria

    Sebastian Novak

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  1. Richard Kollár
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Correspondence to Richard Kollár.

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Kollár, R., Novak, S. Existence of Traveling Waves for the Generalized F–KPP Equation. Bull Math Biol 79, 525–559 (2017). https://doi.org/10.1007/s11538-016-0244-3

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