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Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

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Abstract

In this paper, we consider the multidimensional stability of planar traveling waves for a class of Lotka–Volterra competition systems with time delay and nonlocal reaction term in n–dimensional space. It is proved that, all planar traveling waves with speed \(c>c^{*}\) are exponentially stable. We get accurate decay rate \(t^{-\frac{n}{2}} \textrm{e}^{-\epsilon _{\tau } \sigma t}\), where constant \(\sigma >0\) and \(\epsilon _{\tau }=\epsilon (\tau )\in (0,1)\) is a decreasing function for the time delay \(\tau >0\). It is indicated that time delay essentially reduces the decay rate. While, for the planar traveling waves with speed \(c=c^{*}\), we prove that they are algebraically stable with delay rate \(t^{-\frac{n}{2}}\). The proof is carried out by applying the comparison principle, weighted energy and Fourier transform, which plays a crucial role in transforming the competition system to a linear delayed differential system.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grants No. 12001502; Grants No. 11871007).

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Authors and Affiliations

  1. School of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, Shanxi, People’s Republic of China

    Yeqing Xue

  2. School of Science, China University of Geosciences, Beijing, 100083, People’s Republic of China

    Zhaohai Ma

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Zhihua Liu

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  1. Yeqing Xue
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  2. Zhaohai Ma
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Correspondence to Zhaohai Ma.

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Xue, Y., Ma, Z. & Liu, Z. Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term. Qual. Theory Dyn. Syst. 22, 122 (2023). https://doi.org/10.1007/s12346-023-00803-1

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