Reaction-advection-diffusion competition models under lethal boundary conditions

Kwangjoong Kim; Wonhyung Choi; Inkyung Ahn

doi:10.3934/dcdsb.2021250

Article Contents

2022, Volume 27, Issue 9: 4749-4767. Doi: 10.3934/dcdsb.2021250

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Reaction-advection-diffusion competition models under lethal boundary conditions

1.
College of General Education, Kookmin University, 77, Jeongneung-Ro, Seoul, 02707, Republic of Korea
2.
Department of Mathematics, Korea University, 145, Anam-Ro, Seoul, 02841, Republic of Korea
3.
Department of Mathematics, Korea University, 2511, Sejong-Ro, Sejong, 30019, Republic of Korea

^* Corresponding author: Inkyung Ahn
^* Corresponding author: Inkyung Ahn

Received: January 2021

Revised: June 2021

Early access: October 2021

Published: September 2022

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Abstract

In this study, we consider a Lotka–Volterra reaction–diffusion–advection model for two competing species under homogeneous Dirichlet boundary conditions, describing a hostile environment at the boundary. In particular, we deal with the case in which one species diffuses at a constant rate, whereas the other species has a constant rate diffusion rate with a directed movement toward a better habitat in a heterogeneous environment with a lethal boundary. By analyzing linearized eigenvalue problems from the system, we conclude that the species dispersion in the advection direction is not always beneficial, and survival may be determined by the convexity of the environment. Further, we obtain the coexistence of steady-states to the system under the instability conditions of two semi-trivial solutions and the uniqueness of the coexistence steady states, implying the global asymptotic stability of the positive steady-state.

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Mathematics Subject Classification: Primary: 35J60, 35J60, 92D25.

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Figure 1. Case $ \Delta m > 0 $: (a) Instabilities of $ (0,\theta_\nu) $ when $ \alpha = 0 $, (b) Stability of $ (0,\theta_\nu) $ when $ \alpha = 0.1 $ ($ m(x) = 1-0.4\sin(\pi x), \mu = 0.01, \nu = 0.02 $)

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Figure 2. Case $ \Delta m < 0 $ : (a) Stabilities of $ (0,\theta_\nu) $ when $ \alpha = 0 $, (b) Instability of $ (0,\theta_\nu) $ when $ \alpha = 0.1 $ ($ m(x) = 0.5+0.4\sin(\pi x), \mu = 0.02, \nu = 0.01 $)

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Figure 3. $ \Delta m $ changes its sign : (a) Stability of $ (0,\theta_\nu) $ when $ \alpha = 0 $, (b) Instability of $ (0,\theta_\nu) $ when $ \alpha = 0.5 $ ($ m(x) = 0.8+0.2\cos(4\pi x), \mu = 0.02, \nu = 0.01 $)

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