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Wave speeds for the FKPP equation with enhancements of the reaction function

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Abstract

In classes of N-particle systems and lattice models, the speed of front propagation is approximated by that of the corresponding continuum model, and for many such systems, the rate of convergence to the continuum speed is known to be slow as N → ∞. This slow convergence has been captured by including a cutoff function on the reaction terms in the continuum models. For example, the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with a cutoff has fronts that travel at the speed \({c \sim c_{\rm FKPP} - \frac{\pi^2}{(\ln(N))^2}}\) , which agrees well with data from numerical simulations of the corresponding N-particle systems, where c FKPP is the linear spreading speed. In Panja and van Saarloos (Phys Rev E 66:015206, 2002), an example is presented in which a small enhancement of the reaction function causes the propagation speeds of fronts to be larger than c FKPP. Such front speeds are also observed in stochastic lattice models where the growth rates in the regime of few particles are modified. In this article, we analyze the dynamics of traveling fronts in the FKPP equation with the constant enhancement function employed by Panja and van Saarloos. We present formulas for the wave speeds, develop the criteria on the parameters for which the front speeds are larger than the linear spreading speed even in the limit in which the size of the cutoff domain vanishes, study the rate of approach as N → ∞, and identify the mechanisms in phase space by which the constant enhancement of the reaction function makes possible the larger than linear wave speeds. In addition, we extend these results to the FKPP equation with two other enhancement functions, which are also of interest for continuum level modeling of lattice models and many-particle systems in the regimes of small numbers of particles, namely a linear enhancement function and an enhancement that is uniform above the linearized reaction function. We also derive explicit formulas for the parameters in these problems. The mathematical techniques used herein are geometric singular perturbation theory, geometric desingularization, invariant manifold theory, and normal form theory, all from dynamical systems.

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Authors and Affiliations

  1. Universiteit Hasselt, Campus Diepenbeek, Agoralaan Gebouw D, 3590, Diepenbeek, Belgium

    Freddy Dumortier

  2. Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA, 02215, USA

    Tasso J. Kaper

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  1. Freddy Dumortier
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Correspondence to Tasso J. Kaper.

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Dumortier, F., Kaper, T.J. Wave speeds for the FKPP equation with enhancements of the reaction function. Z. Angew. Math. Phys. 66, 607–629 (2015). https://doi.org/10.1007/s00033-014-0422-9

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  • DOI: https://doi.org/10.1007/s00033-014-0422-9

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