# Invasion waves in the presence of a mutualist

## TL;DR

This study analyzes invasion waves in a three-species diffusive model with mutualism, deriving conditions for linear determinacy and explicit invasion speeds, revealing how mutualists influence invasion dynamics.

## Contribution

It extends the 2-species Lotka-Volterra model to include mutualism, providing new conditions and formulas for invasion speeds in a three-species context.

## Key findings

- Mutualist can increase invasion speed when benefiting the invader.
- Explicit formulas for invasion speeds are derived under certain conditions.
- Mutualist's effect on invasion speed depends on which species benefits.

## Abstract

This paper studies invasion waves in the diffusive Competitor-Competitor-Mutualist model generalizing the 2-species Lotka-Volterra model studied by Weinberger et al. The mutualist may benefit the invading or the resident species producing two different types of invasions. Sufficient conditions for linear determinacy are derived in both cases, and when they hold, explicit formulas for linear spreading speeds of the invasions are obtained by linearizing the model. While in the first case the linear speed is increased by the mutualist, it is unaffected in the second case. Mathematical methods are based on converting the model into a cooperative reaction-diffusion system.