Asymptotic limits of the principal spectrum point of a nonlocal dispersal cooperative system and application to a two-stage structured population model

TL;DR

This work investigates the principal spectrum point $\lambda_p(\mu\circ\mathcal{K}+\mathcal{A})$ for a two-stage, nonlocal-dispersal cooperative system modeling juvenile and adult populations. By characterizing $\lambda_p$ via generalized principal eigenvalues and deriving its exact limits as dispersal rates $\mu_1,\mu_2$ vary, it reveals how local versus global dispersal shapes persistence. The paper also establishes the asymptotic profiles of positive steady states, showing convergence to kinetic equilibria for small dispersal and to spatially uniform, averaged-coefficient equilibria for large dispersal, with refined results for mixed regimes. Collectively, these results provide precise criteria for persistence/extinction and illuminate spatial distribution patterns under extreme dispersal, with implications for ecological management of stage-structured populations under nonlocal movement.

Abstract

This work examines the limits of the principal spectrum point, $λ_p$, of a nonlocal dispersal cooperative system with respect to the dispersal rates. In particular, we provide precise information on the sign of $λ_p$ as one of the dispersal rates is : (i) small while the other dispersal rate is arbitrary, and (ii) large while the other is either also large or fixed. We then apply our results to study the effects of dispersal rates on a two-stage structured nonlocal dispersal population model whose linearized system at the trivial solution results in a nonlocal dispersal cooperative system. The asymptotic profiles of the steady-state solutions with respect to the dispersal rates of the two-stage nonlocal dispersal population model are also obtained. Some biological interpretations of our results are discussed.

Theorems & Definitions (39)

• Definition 2.1: Principal Spectrum Point
• Theorem 2.1
• Theorem 2.2
• Remark 2.1
• Proposition 2.1
• Theorem 2.3
• Remark 2.2
• Proposition 2.2
• Theorem 2.4
• Theorem 2.5