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Wave propagation for a cooperative model with nonlocal dispersal under worsening habitats

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Abstract

In this paper, we study a Lotka–Volterra cooperative system with nonlocal dispersal under worsening habitats. By constructing appropriate vector super-/subsolutions combined with the monotone iteration scheme, we obtain the existence of bounded and positive forced waves connecting zero equilibrium to the coexistence state of the limiting system corresponding to the most favorable resource with the wave speed at which the habitat is worsening. Compared with the existence result of traveling wave to the homogeneous system, we find that the wave phenomenon is more likely to occur in such shifting environment. Here, we remove the common assumption that the dispersal kernels are compactly supported and symmetric. Further, we investigate the tail behavior of the forced waves, especially for the rate of convergence to the extinction state, and derive the long-time dynamics of two species. Our result shows that weak interspecies cooperation and nonlocal diffusion pattern cannot prevent species from disappearing eventually under such a worsening habitat.

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Acknowledgements

We are grateful to two anonymous referees and the editor for their careful reading and valuable comments which led to an improvement in our original manuscript. The first author was partially supported by the Scientific Research Start-up Fee for High-level Talents (162301182740), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (CUGSX01) and the National Natural Science Foundation of China (11901543). The second author was partially supported by the National Natural Science Foundation of China (11731005, 11671180).

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Authors and Affiliations

  1. School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, P.R. China

    Jia-Bing Wang

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, P.R. China

    Wan-Tong Li

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  1. Jia-Bing Wang
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Correspondence to Wan-Tong Li.

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Wang, JB., Li, WT. Wave propagation for a cooperative model with nonlocal dispersal under worsening habitats. Z. Angew. Math. Phys. 71, 147 (2020). https://doi.org/10.1007/s00033-020-01374-w

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  • DOI: https://doi.org/10.1007/s00033-020-01374-w

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