Longtime behaviors of a reducible cooperative system with nonlocal diffusions and free boundaries

TL;DR

This paper investigates the long-term dynamics of a reducible cooperative system with nonlocal diffusion and free boundaries, revealing complex behaviors due to multiple steady states and conditions for accelerated spreading.

Contribution

It provides a comprehensive classification of the longtime behaviors of the system, including steady states, spreading speeds, and conditions for accelerated spreading, which extends previous irreducible models.

Findings

Multiple nonnegative steady states identified.

Complete classification of longtime behaviors achieved.

Conditions for accelerated spreading determined.

Abstract

This paper aims at understanding the longtime behaviors of a reducible cooperative system with nonlocal diffusions and different free boundaries, describing the interactions of two mutually beneficial species. Compared with the irreducible and monostable cooperative system, the system we care about here has many nonnegative steady states, leading to much different and rich longtime behaviors. Moreover, since the possible nonnegative steady states on half space are non-constant, we need to employ more detailed analysis to understand the corresponding steady state problems which in turn helps us to derive a complete classification for the longtime behaviors of our problem. The spreading speeds of free boundaries and more accurate limits of $(u,v)$ as $t\to\infty$ are also discussed, and accelerated spreading can happen if some threshold conditions are violated by kernel functions.