On competition models under allee effect: Asymptotic behavior and traveling waves

Wei Feng; Michael Freeze; Xin Lu

doi:10.3934/cpaa.2020256

Article Contents

2020, Volume 19, Issue 12: 5609-5626. Doi: 10.3934/cpaa.2020256

This issue Previous Article Solutions of nonlocal problem with critical exponent Next Article

On competition models under allee effect: Asymptotic behavior and traveling waves

Department of Mathematics and Statistics, University of North Carolina at Wilmington, Wilmington, NC 28403

^*Corresponding author
^*Corresponding author

Received: January 31, 2020

Revised: May 31, 2020

Early access: October 2020

Published: December 2020

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Abstract

In this article, we study a reaction-diffusion model on infinite spatial domain for two competing biological species ($ u $ and $ v $). Under one-side Allee effect on $ u $-species, the model demonstrates complexity on its coexistence and $ u $-dominance steady states. The conditions for persistence, permanence and competitive exclusion of the species are obtained through analysis on asymptotic behavior of the solutions and stability of the steady states, including the attraction regions and convergent rates depending on the biological parameters. When the Allee effect constant $ K $ is large relative to other biological parameters, the asymptotic stability of the $ v $-dominance state $ (0,\:1) $ indicates the competitive exclusion of the $ u $-species. Applying upper-lower solution method, we further prove that for a family of wave speeds with specific minimum wave speed determined by several biological parameters (including the magnitude of the $ u $-dominance states), there exist traveling wave solutions flowing from the $ u $-dominance states to the $ v $-dominance state. The asymptotic rates of the traveling waves at $ \xi \rightarrow \mp \infty $ are also explicitly calculated. Finally, numerical simulations are presented to illustrate the theoretical results and population dynamics of coexistence or dominance-shifting.

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Mathematics Subject Classification: Primary: 35B35, 35B40, 35C07; Secondary: 35K57, 35Q92.

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Figure 1. Permanence in the competition model under Allee effect

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Figure 2. Asymptotic stability of the steady state $ (0,1) $, $ v $ species dominance

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Figure 3. Traveling wave front connecting $ (u_s^{(1)},0) $ to $ (0,1) $, competitive exclusion of $ u $-species

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Figure 4. Traveling wave front connecting $ (u_s^{(2)},0) $ to $ (0,1) $, competitive exclusion of $ u $-species

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