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Forced Traveling Waves in a Reaction-Diffusion Equation with Strong Allee Effect and Shifting Habitat

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Abstract

We study a reaction-diffusion equation that describes the growth of a population with a strong Allee effect in a bounded habitat which shifts at a speed \(c>0\). We demonstrate that the existence of forced positive traveling waves depends on habitat size L, and \(c^*\), the speed of traveling wave for the corresponding reaction-diffusion equation with the same growth function all over the entire unbounded spatial domain. It is shown that for \(c^*>c>0\) there exists a positive number \(L^*(c)\) such that for \(L>L^*(c)\) there are two positive traveling waves and for \(L<L^*(c)\) there is no positive traveling wave. It is also shown if \(c>c^*\) for any \(L>0\) there is no positive traveling wave. The dynamics of the equation are further explored through numerical simulations.

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Acknowledgements

B. Li was partially supported by the National Science Foundation under Grant DMS-1951482.

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Authors and Affiliations

  1. Department of Mathematics, University of Louisville, Louisville, KY, 40292, USA

    Bingtuan Li

  2. Department of Mathematics, SUNY Cortland, Cortland, NY, 13045, USA

    Garrett Otto

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  1. Bingtuan Li
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  2. Garrett Otto
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Correspondence to Garrett Otto.

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Li, B., Otto, G. Forced Traveling Waves in a Reaction-Diffusion Equation with Strong Allee Effect and Shifting Habitat. Bull Math Biol 85, 121 (2023). https://doi.org/10.1007/s11538-023-01221-9

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