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Long time behavior for a periodic Lotka–Volterra reaction–diffusion system with strong competition

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Abstract

This paper is concerned with the long time behavior of bounded solutions to a two-species time-periodic Lotka–Volterra reaction–diffusion system with strong competition. It is well known that solutions of the Cauchy problem of this system with front-like initial values converge to a bistable periodic traveling front. One may ask naturally how solutions of such time-periodic systems with other types of initial data evolve as time increases. In this paper, by transforming the system into a cooperative system on \([\textbf{0},\textbf{1}]\), we first show that if the bounded initial value \(\varvec{\varphi }(x)\) has compact support and equals \(\textbf{1}\) for a sufficiently large x-level, then solutions converge to a pair of diverging periodic traveling fronts. As a by-product, we obtain a sufficient condition for solutions to spread to \(\textbf{1}\). We also prove that if the two species are initially absent from the right half-line \(x>0\) and the slower one dominates the faster one on \(x<0\), then solutions approach a propagating terrace, which means that several invasion speeds can be observed.

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Acknowledgements

We are very grateful to the anonymous referee for careful reading and helpful suggestions which led to an improvement of our original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, Shaanxi, People’s Republic of China

    Liyan Pang & Shi-Liang Wu

  2. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    Shigui Ruan

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  1. Liyan Pang
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Correspondence to Shi-Liang Wu.

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Communicated by P. Rabinowitz.

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S.-L. Wu: Research was partially supported by the NSF of China (No. 12171381) and Natural Science Basic Research Program of Shaanxi (No. 2020JC-24).

S. Ruan: Research was partially supported by National Science Foundation (DMS-1853622 and DMS-2052648).

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Pang, L., Wu, SL. & Ruan, S. Long time behavior for a periodic Lotka–Volterra reaction–diffusion system with strong competition. Calc. Var. 62, 99 (2023). https://doi.org/10.1007/s00526-023-02436-3

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