Longtime behaviors of a reducible cooperative system with nonlocal diffusions and free boundaries
Lei Li, Mingxin Wang
TL;DR
The paper analyzes a reducible cooperative system with nonlocal diffusion and two free boundaries to capture interactions between two mutually beneficial species. It establishes global well-posedness and a spreading–vanishing dichotomy, with long-time behavior characterized by half-space and bounded-domain steady states depending on whether each species spreads. When spreading occurs for both species, the system converges to a jointly positive half-space state on expanding regions, with front speeds governed by semi-wave thresholds; if one species vanishes, the surviving component approaches its corresponding half-space steady state. A key contribution is a general lemma for steady-state problems with nonincreasing kernels that yields sharper spreading estimates and improved convergence results, including accelerated spreading when tail conditions fail.
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.