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Entire solutions of Lotka-Volterra competition systems with nonlocal dispersal

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Abstract

This paper is mainly concerned with entire solutions of the following two-species Lotka-Volterra competition system with nonlocal (convolution) dispersals:

$\left\{{\matrix{{{u_t} = k * u - u + u(1 - u - av),} \hfill & {x \in \mathbb{R},\,\,t \in \mathbb{R},} \hfill \cr {{v_t} = d(k * v - v) + rv(1 - v - bu),} \hfill & {x \in \mathbb{R},\,\,t \in \mathbb{R}.} \hfill \cr}} \right.$
((0.1))

Here a ≠ 1, b ≠ 1, d, and r are positive constants. By studying the eigenvalue problem of (0.1) linearized at (ϕc(ξ), 0), we construct a pair of super- and sub-solutions for (0.1), and then establish the existence of entire solutions originating from (ϕc(ξ), 0) as t → −∞, where ϕc denotes the traveling wave solution of the nonlocal Fisher-KPP equation ut = k * u − u + u (1 − u). Moreover, we give a detailed description on the long-time behavior of such entire solutions as t → ∞. Compared to the known works on the Lotka-Volterra competition system with classical diffusions, this paper overcomes many difficulties due to the appearance of nonlocal dispersal operators.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Yuxia Hao & Wantong Li

  2. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

    Yuxia Hao

  3. School of Mathematics and Physics, Center for Mathematical Sciences, China University of Geosciences, Wuhan, 430074, China

    Jiabing Wang

  4. School of Mathematical Sciences, Capital Normal University, Bei**g, 100048, China

    Wenbing Xu

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  1. Yuxia Hao
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  2. Wantong Li
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Correspondence to Wantong Li.

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Additional information

Research of W.-T. Li was partially supported by the NSF of China (12271226), the NSF of Gansu Province of China (21JR7RA537) and the Fundamental Research Funds for the Central Universities (lzujbky-2022-sp07); research of J.-B. Wang was partially supported by the Basic and Applied Basic Research Foundation of Guangdong Province (2023A1515011757) and the National Natural Science Foundation of China (12271494), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (G1323523061) and research of W.-B. Xu was partially supported by the NSF of China (12201434).

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Hao, Y., Li, W., Wang, J. et al. Entire solutions of Lotka-Volterra competition systems with nonlocal dispersal. Acta Math Sci 43, 2347–2376 (2023). https://doi.org/10.1007/s10473-023-0602-9

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  • DOI: https://doi.org/10.1007/s10473-023-0602-9

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