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Entire solutions of time periodic bistable Lotka–Volterra competition-diffusion systems in \({\mathbb {R}}^N\)

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Abstract

This paper is concerned with entire solutions of a two species time periodic bistable Lotka–Volterra competition-diffusion systems in \({\mathbb {R}}^N\). Here an entire solution refers to a solution that is defined for all time and in the whole space. It is known that “annihilating-front” type entire solutions have been obtained recently in \({\mathbb {R}}\). In the present paper, we first prove that there is a new type of entire solution which behaves as three time periodic moving planar traveling fronts as time goes to \(-\infty \) and as a time periodic V-shaped traveling front as time goes to \(+\infty \) in \({\mathbb {R}}^N\). Furthermore, we show that the propagating speed of such entire solutions coincides with the unique speed of the time periodic planar front, regardless of the shape of the level sets of the fronts.

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Acknowledgements

We are very grateful to the anonymous referee and the editors for their valuable comments and suggestions that helped to improve the manuscript. The first author’s work was partially supported by NSF of China (11971128, 11401134) and by postdoctoral scientific research developmental fund of Heilongjiang Province (LBH-Q17061) and by the Heilongjiang Provincial Natural Science Foundation of China (LH2020A003). The second author’s work was partially supported by NSF of China (11771110). The third author’s work was partially supported by NSF of China (12071193, 11731005).

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Authors and Affiliations

  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang, People’s Republic of China

    Wei-Jie Sheng & Mingxin Wang

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Zhi-Cheng Wang

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  1. Wei-Jie Sheng
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Correspondence to Wei-Jie Sheng.

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Sheng, WJ., Wang, M. & Wang, ZC. Entire solutions of time periodic bistable Lotka–Volterra competition-diffusion systems in \({\mathbb {R}}^N\). Calc. Var. 60, 37 (2021). https://doi.org/10.1007/s00526-020-01887-2

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