On principal eigenvalues of linear time-periodic parabolic systems: symmetric mutation case

TL;DR

The paper analyzes the principal eigenvalue $\lambda(\omega,\rho)$ of a linear time-periodic parabolic system with symmetric mutation, detailing how spatio-temporal heterogeneity shapes persistence thresholds. It establishes a Hamilton–Jacobi limit governing the origin regime, provides monotonicity results with precise conditions, and classifies the complete topology of level sets in the $\omega$–$\rho$ plane, unveiling rich, non-monotone diffusion effects in multi-phenotype systems. The results generalize scalar-periodic findings to fully coupled systems and have implications for dispersal-driven growth and persistence-extinction transitions in heterogeneous environments. The methodology blends asymptotic analysis, viscosity solutions, and eigenvalue comparisons to reveal a coherent picture of how frequency and diffusion interact in structured populations.

Abstract

The paper is concerned with the effect of the spatio-temporal heterogeneity on the principal eigenvalue of some linear time-periodic parabolic system. Various asymptotic behaviors of the principal eigenvalue and its monotonicity, as a function of the diffusion rate and frequency, are first derived. In particular, some singular behaviors of the principal eigenvalues are observed when both diffusion rate and frequency approach zero, with some scalar time-periodic Hamilton-Jacobi equation as the limiting equation. Furthermore, we completely classify the topological structures of the level sets for the principal eigenvalues in the plane of frequency and diffusion rate. Our results not only generalize most of the findings in [S. Liu and Y. Lou, J. Funct. Anal., 282 (2022), 109338] for scalar periodic-parabolic operators, but also reveal more rich global information, for time-periodic parabolic systems, on the dependence of the principal eigenvalues upon the spatio-temporal heterogeneity.

Figures (4)

• Figure 1: Illustrations of the persistence region $\mathbb{E}$ in the $\rho$-$\omega$ plane for the essentially positive and fully coupled mutation matrix ${\bf M}$, which is marked by the shaded areas, while the blank areas correspond to the region where $\lambda(\omega,\rho)\geq 0$.
• Figure 2: The illustrations of the quantities defined in \ref{['def_underlineC']} and \ref{['def_underlineC2']}, which are some double limit values of the principal eigenvalue $\lambda(\omega,\rho)$ when $\omega$ tends to $0$ or $+\infty$ and/or $\rho$ tends to $0$ or $+\infty$.
• Figure 3: A sketch of the level sets of principal eigenvalue $\lambda(\omega,\rho)$ in $\rho$-$\omega$ plane as shown in Theorem \ref{['liu-levelset']}, which are illustrated by the bounded or unbounded curves presenting the graphs of function $\omega=\omega_{\ell}(\rho)$ for the cases $\underline{C}< C_*<C_*^+<\underline{C}^+ <\overline{C}$ and $\underline{C}< C_*<\underline{C}^+<C_*^+< \overline{C}$.
• Figure 4: Graphs of $\lambda(\omega,\rho)$ as functions of $\rho$, with fixed $\omega$, for the cases given in Theorem \ref{['liucor-1']}. These figures are for illustration purpose only, as the exact shapes of the graphs could be more complex.

Theorems & Definitions (44)

• Theorem 1.1: BH2020
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Corollary 2.3