Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Qian Liu; Shuang Liu; King-Yeung Lam

doi:10.3934/dcds.2020050

Article Contents

2020, Volume 40, Issue 6: 3683-3714. Doi: 10.3934/dcds.2020050 open access

This issue Previous Article Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion Next Article A functional approach towards eigenvalue problems associated with incompressible flow

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

1.
Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China
2.
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

^* Corresponding author: King-Yeung Lam
^* Corresponding author: King-Yeung Lam

Received: March 31, 2019

Revised: August 31, 2019

Published: March 2020

The last author is partially supported by NSF grant DMS-1853561

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Abstract

We establish spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states between successive invasion fronts. Our method is inspired by the geometric optics approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised by Shigesada et al. in 1997, and shows that one of the species spreads to the right with a nonlocally pulled front.

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Mathematics Subject Classification: Primary: 35K58, 35B40; Secondary: 35D40.

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Figure 1. Asymptotic behaviors of the solutions to (1) with $ a = 0.6, \, b = 0.5, \, r = 1 $, and $ d = 1.5 $ in $ \rm(a) $, $ d = 1 $ in $ \rm(b) $, $ d = 0.5 $ in $ \rm(c) $, where the initial data are chosen as $ u(0, x) = \chi_{[-1000, 0]} $ and $ v(0, x) = \chi_{[-20, 0]} $

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Figure 2. The graphs of $ x_i(t)/t $ ($ i = 1, 2, 3 $) with $ a = 0.6, \, b = 0.5, \, r = 1 $ and $ d = 1.5 $ where the initial data are chosen as $ u(0, x) = \chi_{[-1000, 0]} $ and $ v(0, x) = \chi_{[-20, 0]} $

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