Long time behavior for a periodic Lotka–Volterra reaction–diffusion system with strong competition
Long time behavior for a periodic Lotka–Volterra reaction–diffusion system with strong competition
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DOI：
10.1007/s00526-023-02436-3
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发表时间：
2023-02
影响因子：
2.1
通讯作者：
Liyan Pang;Shiliang Wu;S. Ruan
中科院分区：
文献类型：
--
作者：
Liyan Pang;Shiliang Wu;S. Ruan
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, we first show that if the bounded initial value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\varphi }(x)$$\end{document} has compact support and equalsfor a sufficiently largex-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. We also prove that if the two species are initially absent from the right half-lineand the slower one dominates the faster one on, then solutions approach a propagating terrace, which means that several invasion speeds can be observed.
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, we first show that if the bounded initial value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\varphi }(x)$$\end{document} has compact support and equalsfor a sufficiently largex-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. We also prove that if the two species are initially absent from the right half-lineand the slower one dominates the faster one on, then solutions approach a propagating terrace, which means that several invasion speeds can be observed.