Asymptotic behavior and sharp estimates for spreading fronts in a cooperative system with free boundaries
Qian Qin, JinJing Jiao, Zhiguo Wang, Hua Nie
TL;DR
The paper analyzes a two-species cooperative reaction-diffusion system with two free boundaries modeling expanding invasion fronts. By combining upper/lower solutions, comparison principles, and a semi-wave framework, it establishes a spreading-vanishing dichotomy and derives sharp asymptotic front speeds governed by a semi-wave system, showing that spreading fronts share a common speed $s_{\mu,\rho}$ and that the densities converge locally to the corresponding semi-wave profile. It further demonstrates bounded deviations $h(t)-s_{\mu,\rho}t$ and $g(t)+s_{\mu,\rho}t$, and proves a sharp, frame-invariant description of the long-term population densities via the semi-wave solutions; the results provide precise, quantitative descriptions of multi-species invasion with free boundaries. The study extends the single-species free-boundary theory to two cooperators, yielding rigorous insights into front localization and asymptotic profiles with potential ecological and epidemiological applications.
Abstract
This paper investigates the dynamics of a reaction-diffusion system with two free boundaries, modeling the invasion of two cooperative species, where the free boundaries represent expanding fronts. We first analyze the long-term behavior of the system, showing that it follows a spreading-vanishing dichotomy: the two species either spread across the entire region or eventually die out. In the case of spreading, we determine the asymptotic spreading speed of the fronts by using a semi-wave system and provide sharp estimates for the moving fronts. Additionally, we show that the solution to the system converges to the corresponding semi-wave solution as time tends to infinity. These results contribute to a deeper understanding of the long-term dynamics of cooperative species in reaction-diffusion systems with free boundaries.