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Article Contents

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

• * Corresponding author: King-Yeung Lam

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.

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

 Citation:

• 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 $v(0, x) = \chi_{[-20, 0]}$

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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