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Minimal-Speed Selection to a Lotka–Volterra Competition System with Local Versus Nonlocal Diffusions and Cubic Nonlinearity

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Abstract

In this paper, the minimal-speed selection mechanism to a Lotka–Volterra type of model with local versus nonlocal diffusions and cubic nonlinearity is considered. By using comparison principle and upper-lower solution method, general criteria for both linear and nonlinear selection of the minimal wave speed are derived. Based on these results, several explicit conditions for determining the linear selection of the minimal wave speed are obtained via the constructions of various upper solutions. These conditions can help us to confirm that the species will spread with the speed of a linearised system and take an insight into thresholds for different model parameters. Moreover, it is found that the model with cubic nonlinearity causes a very different mechanism compared to the one caused by quadratic nonlinearity. For example, the linear selection can be realized when the diffusion strength is in an infinite interval for the model with cubic nonlinearity. Finally, it should be pointed out that concrete conditions for nonlinear selection are not obtained due to the failure of construction of a lower solution, and are left for further work.

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Authors and Affiliations

  1. School of Mathematics and Physics, University of South China, Hengyang, 421001, China

    Jiali Zhan, Jiding Liao & Hongyong Wang

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  1. Jiali Zhan
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  2. Jiding Liao
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  3. Hongyong Wang
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Contributions

Jiali Zhan and Hongyong Wang wrote the main manuscript, Jiding Liao and Hongyong Wang were responsible for conceptualization and methodology. All authors reviewed the manuscript.

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Correspondence to Jiding Liao or Hongyong Wang.

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This work is partially supported by the Scientific Research Fund of Hunan Provincial Education Department Grant (23A0342), Natural Science Foundation of Hunan Province Grant (2021JJ30019).

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Zhan, J., Liao, J. & Wang, H. Minimal-Speed Selection to a Lotka–Volterra Competition System with Local Versus Nonlocal Diffusions and Cubic Nonlinearity. Qual. Theory Dyn. Syst. 23, 152 (2024). https://doi.org/10.1007/s12346-024-01006-y

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