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Classification of the spreading behaviors of a two-species diffusion-competition system with free boundaries

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Abstract

In this paper, we revisit the spreading behavior of two invasive species modelled by a diffusion-competition system with two free boundaries in a radially symmetric setting, where the reaction terms depict a weak-strong competition scenario. Our previous work (Du and Wu in Cal Var PDE 57:52, 2018) proves that from certain initial states, the two species develop into a “chase-and-run coexistence” state, namely the front of the weak species v propagates at a fast speed and that of the strong species u propagates at a slow speed, with their population masses largely segregated. Subsequent numerical simulations in Khan et al. (J Math Biol 83:23, 2021) suggest that for all possible initial states, only four different types of long-time dynamical behaviours can be observed: (1) chase-and-run coexistence, (2) vanishing of u with v spreading successfully, (3) vanishing of v with u spreading successfully, and (4) vanishing of both species. In this paper, we rigorously prove that, as the initial states vary, there are exactly five types of long-time dynamical behaviors: apart from the four mentioned above, there exists a fifth case, where both species spread successfully and their spreading fronts are kept within a finite distance to each other all the time. We conjecture that this new case can happen only when a parameter takes an exceptional value, which is why it has eluded the numerical observations of Khan et al. (J Math Biol 83:23, 2021) .

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Acknowledgements

We appreciate the valuable comments and suggestions of the reviewer, which have helped us to improve the manuscript. YD was supported by the Australian Research Council. CHW was partially supported by the Ministry of Science and Technology of Taiwan under Grant MOST 109-2636-M-009-008 and MOST 110-2636-M-009-006, and he thanks University of New England for the hospitality during his visit.

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Authors and Affiliations

  1. School of Science and Technology, University of New England, Armidale, NSW, 2351, Australia

    Yihong Du

  2. Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu, 300, Taiwan

    Chang-Hong Wu

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  1. Yihong Du
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  2. Chang-Hong Wu
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Correspondence to Chang-Hong Wu.

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Communicated by P. H. Rabinowitz.

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Du, Y., Wu, CH. Classification of the spreading behaviors of a two-species diffusion-competition system with free boundaries. Calc. Var. 61, 54 (2022). https://doi.org/10.1007/s00526-021-02170-8

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