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Principal spectral theory and asymptotic analysis for time-periodic cooperative systems with temporally nonlocal dispersal

Hao Wu, Wan-Tong Li, Jian-Wen Sun, Hoang-Hung Vo

TL;DR

This work develops a comprehensive principal spectral theory for time-periodic cooperative systems with nonlocal dispersal. It introduces resolvent-positive operator techniques to derive existence criteria for a principal eigenvalue, and constructs smooth upper and lower matrix-valued approximations that converge to the original operator, ensuring the principal spectrum point can be used as a threshold for nonlinear dynamics. It also establishes variational characterizations and monotonicity properties of the generalized principal eigenvalues, and analyzes the asymptotic behavior of the principal spectrum point with respect to dispersal rate, dispersal range, and frequency under general assumptions. The results are applied to models of Zika virus transmission and stem cell regeneration, illustrating how the theory informs global dynamics and parameter sensitivity in complex nonlocal, time-periodic systems.

Abstract

This paper investigates the principal spectral theory and the asymptotic behavior of the principal spectrum point for a class of time-periodic cooperative systems with nonlocal dispersal operators, incorporating both coupled and uncoupled nonlocal terms. By applying the theory of resolvent positive operators and their perturbations, we first establish criteria for the existence of the principal eigenvalue. We then construct sequences of smooth upper and lower approximating matrix-valued functions, each of whose corresponding operators satisfies the principal eigenvalue existence condition. This approximation framework allows the principal spectrum point to effectively substitute for the principal eigenvalue in characterizing the global dynamics of the nonlinear system. Moreover, it facilitates the study of the asymptotic behavior of the principal spectrum point with respect to parameters under fairly general assumptions. Subsequently, for systems with both coupled and uncoupled nonlocal terms, we analyze the asymptotic behavior of the principal spectrum point in terms of the dispersal rate, dispersal range, and frequency. Finally, we illustrate the applicability of our theoretical results through a Zika virus model and a stem cell model.

Principal spectral theory and asymptotic analysis for time-periodic cooperative systems with temporally nonlocal dispersal

TL;DR

This work develops a comprehensive principal spectral theory for time-periodic cooperative systems with nonlocal dispersal. It introduces resolvent-positive operator techniques to derive existence criteria for a principal eigenvalue, and constructs smooth upper and lower matrix-valued approximations that converge to the original operator, ensuring the principal spectrum point can be used as a threshold for nonlinear dynamics. It also establishes variational characterizations and monotonicity properties of the generalized principal eigenvalues, and analyzes the asymptotic behavior of the principal spectrum point with respect to dispersal rate, dispersal range, and frequency under general assumptions. The results are applied to models of Zika virus transmission and stem cell regeneration, illustrating how the theory informs global dynamics and parameter sensitivity in complex nonlocal, time-periodic systems.

Abstract

This paper investigates the principal spectral theory and the asymptotic behavior of the principal spectrum point for a class of time-periodic cooperative systems with nonlocal dispersal operators, incorporating both coupled and uncoupled nonlocal terms. By applying the theory of resolvent positive operators and their perturbations, we first establish criteria for the existence of the principal eigenvalue. We then construct sequences of smooth upper and lower approximating matrix-valued functions, each of whose corresponding operators satisfies the principal eigenvalue existence condition. This approximation framework allows the principal spectrum point to effectively substitute for the principal eigenvalue in characterizing the global dynamics of the nonlinear system. Moreover, it facilitates the study of the asymptotic behavior of the principal spectrum point with respect to parameters under fairly general assumptions. Subsequently, for systems with both coupled and uncoupled nonlocal terms, we analyze the asymptotic behavior of the principal spectrum point in terms of the dispersal rate, dispersal range, and frequency. Finally, we illustrate the applicability of our theoretical results through a Zika virus model and a stem cell model.
Paper Structure (11 sections, 20 theorems, 109 equations)

This paper contains 11 sections, 20 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Background and main contributions
  3. Notations and basic definitions
  4. Main results
  5. Principal spectral theory
  6. The existence of the principal eigenvalue
  7. Approximation of the principal spectrum point
  8. Properties of the principal spectrum point
  9. Asymptotic behavior of the principal spectrum point
  10. Asymptotic behavior with respect to dispersal rate
  11. Asymptotic behavior with respect to dispersal range

Key Result

Lemma 1.2

For any given $x\in\bar{\Omega}$, the eigenvalue problem admits a principal eigenvalue $\lambda_{A}(x)$ with a positive eigenfunction $\varphi_{A}(x,t)$. Moreover, $\lambda_{A}(x)$ and $\varphi_{A}(x,t)$ are as smooth in $x$ as $A(x,t)$ in $x$, and when $A(x,t)\equiv A(x)$, $\lambda_{A}(x)$ is the largest real part of the eigenvalues of the matrix $A(x)$.

Theorems & Definitions (40)

  • Definition 1.1
  • Lemma 1.2
  • Theorem 1.3: Existence of the principal eigenvalue
  • Theorem 1.4
  • Theorem 1.5: Approximating the principal spectrum point
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8: Asymptotic behavior with respect to the dispersal rate
  • Remark 1.9
  • Theorem 1.10: Asymptotic behavior with respect to the dispersal range
  • ...and 30 more