The spreading fronts in a mutualistic model with delay

TL;DR

This paper analyzes a mutualistic ecological model with delay, focusing on the behavior of spreading fronts described by free boundaries, and identifies conditions for different spreading outcomes.

Contribution

It establishes the existence, uniqueness, and asymptotic behavior of solutions for a delayed mutualistic model with free boundaries, revealing different spreading scenarios.

Findings

Free boundaries tend to finite or infinite limits monotonically.

Strong inter-specific competition leads to global slow solutions with unbounded boundaries.

Weak competition results in blowup or fast global solutions.

Abstract

This article is concerned with a system of semilinear parabolic equations with two free boundaries describing the spreading fronts of the invasive species in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that two free boundaries tend monotonically to finite or infinite at the same time, and the free boundary problem admits a global slow solution with unbounded free boundaries if the inter-specific competitions are strong, while if the inter-specific competitions are weak, there exist the blowup solution and global fast solution.