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

## TL;DR

This paper analyzes the spreading behavior of competing species modeled by the Lotka-Volterra system, establishing exact invasion speeds and convergence properties using a geometric optics approach, thus resolving an open question from 1997.

## Contribution

It introduces a geometric optics method to determine spreading speeds and invasion fronts in a multi-species competition model, providing new insights into nonlocal front propagation.

## Key findings

- Exact spreading speeds are derived for the species.
- Convergence to equilibrium states occurs between invasion fronts.
- One species spreads with a nonlocally pulled front.

## 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.