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Dynamical behavior of solutions of a reaction–diffusion–advection model with a free boundary

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Abstract

This paper is devoted to study the population dynamics of a single species in a one-dimensional environment which is modeled by a reaction–diffusion–advection equation with free boundary condition. We find three critical values \(c_0\), 2 and \(\beta ^*\) for the advection coefficient \(-\beta \) with \(\beta ^*>2>c_0>0\), which play key roles in the dynamics, and prove that a spreading-vanishing dichotomy result holds when \(-2<\beta \leqslant c_0\); a small spreading-vanishing dichotomy result holds when \(c_0<\beta <2\); a virtual spreading-transition-vanishing trichotomy result holds when \(2\leqslant \beta <\beta ^*\); only vanishing happens when \(\beta \geqslant \beta ^*\); a virtual vanishing-transition-vanishing trichotomy result holds when \(\beta \leqslant -2\). When spreading or small spreading or virtual spreading happens for a solution, we make use of the traveling semi-wave solutions to give a estimate for the asymptotic spreading speed and asymptotic profile of the right front.

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Acknowledgements

The authors would like to thank the two anonymous referees for their suggestions and comments which helped to improve the presentation of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250358, China

    Ningkui Sun &  Di Zhang

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  1. Ningkui Sun
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  2. Di Zhang
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Correspondence to Ningkui Sun.

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This research was partly supported by the NSF of China (No. 11801330), Shandong Provincial Natural Science Foundation of China (No. ZR2023YQ002), and the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions (No. 2021KJ037).

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Sun, N., Di Zhang Dynamical behavior of solutions of a reaction–diffusion–advection model with a free boundary. Z. Angew. Math. Phys. 75, 40 (2024). https://doi.org/10.1007/s00033-023-02183-7

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  • DOI: https://doi.org/10.1007/s00033-023-02183-7

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