The eigentheory for nonlocal cooperative-advective system and its role in the study of free boundary system for directional epidemic models

TL;DR

This work addresses the directional spread of epidemics by modeling nonlocal cooperative diffusion with advection and free boundaries. The authors develop a rigorous spectral framework for the associated linearized operator, proving the existence and simplicity of a principal eigenvalue and linking it to the basic reproduction number $R_0$ via the next-generation operator, $R_0=\rho(\mathscr J(-\mathscr T)^{-1})$. They establish well-posedness for the nonlinear free-boundary problem and prove a sharp spreading–vanishing dichotomy, with threshold criteria depending on $R_0$, the initial habitat size $h_0$, and the boundary-expansion rate $\mu$. The results provide a mathematically rigorous threshold theory for nonlocal cooperative–advective systems with free boundaries, informing epidemic and ecological invasion dynamics under heterogeneous transport and nonlocal dispersal.

Abstract

In this paper, we propose and analyze a nonlocal cooperative reaction--diffusion system with free boundaries and drift terms, motivated by directional epidemic spread. Lacking a variational structure but requiring sharper regularity of solutions, the model poses substantial analytical challenges compared with previous works~\cite{Du,Berestycki2016a,Berestycki2016b,Cao2019,NguyenVo2022,Tang2024a,Tang2024b}. We first establish the well-posedness of the local problem and the global existence and uniqueness of classical solutions in $C^1$ space. We then study the associated nonlocal eigenvalue problem, proving the existence, simplicity, qualitative properties, and asymptotic behavior of the principal eigenvalue. The analysis employs Fredholm theory, the Crandall--Rabinowitz bifurcation theorem, and Hadamard-type derivative formulas to describe its parameter dependence and connection with the basic reproduction number~$R_0$. Building on this spectral characterization, we show that the system admits a \emph{sharp vanishing--spreading dichotomy} in its long-term dynamics. When $R_0\le1$, all solutions vanish; for $R_0>1$, the outcome depends on the initial domain size~$h_0$ and the free-boundary expansion rate~$μ$. There exists a critical habitat length~$\mathcal L^\ast$ such that if $h_0<\mathcal L^\ast$, a threshold $\widehatμ>0$ separates vanishing ($μ\in(0,\widehatμ]$) from spreading ($μ>\widehatμ$). In the spreading regime, solutions converge to the unique positive steady state, while in the vanishing regime they decay uniformly to zero. These results provide a rigorous framework for the threshold dynamics of cooperative--advective nonlocal systems and offer mathematical insight for further studies in epidemic modeling, ecological invasion, and population dynamics.

Theorems & Definitions (52)

• Theorem 1: Global existence and uniqueness
• Theorem 2: Principal eigenvalue
• Theorem 3: Qualitative structure and parameter dependence of the principal eigenvalue
• Theorem 4: Vanishing-Spreading
• Lemma 5: Maximum Principle
• Lemma 6: Comparison Principle
• proof
• Lemma 7
• proof
• Lemma 8