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June  2020, 40(6): 3683-3714. doi: 10.3934/dcds.2020050

## 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

Received  April 2019 Revised  September 2019 Published  October 2019

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

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.

Citation: Qian Liu, Shuang Liu, King-Yeung Lam. Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3683-3714. doi: 10.3934/dcds.2020050
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##### References:
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 $v(0, x) = \chi_{[-20, 0]}$
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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