Analysis of a parabolic-hyperbolic hybrid population model

TL;DR

This paper investigates the global dynamics of a hybrid parabolic-hyperbolic population model, establishing well-posedness, spectral properties, and the role of the reproductive rate as a stability threshold.

Contribution

It introduces a comprehensive analysis of the model's stability criteria, linking the reproductive rate to the stability of equilibria and providing explicit spectral characterizations.

Findings

The model solutions are globally well-posed and asymptotically smooth.

The reproductive rate $\\mathcal{R}_{0}$ determines stability thresholds.

When $\mathcal{R}_{0}<1$, the trivial equilibrium is stable; when $\mathcal{R}_{0}>1$, a positive stable equilibrium exists.

Abstract

This paper is concerned with the global dynamics of a hybrid parabolic-hyperbolic model describing populations with distinct dispersal and sedentary stages. We first establish the global well-posedness of solutions, prove a comparison principle, and demonstrate the asymptotic smoothness of the solution semiflow. Through the spectral analysis of the linearized system, we derive and characterize the net reproductive rate $\mathcal{R}_{0}$. Furthermore, an explicit relationship between $\mathcal{R}_{0}$ and the principal eigenvalue of the linearized system is analyzed. Under appropriate monotonicity assumptions, we show that $\mathcal{R}_{0}$ serves as a threshold parameter that completely determines the stability of steady states of the system. More precisely, when $\mathcal{R}_{0}<1$, the trivial equilibrium is globally asymptotical stable, while when $\mathcal{R}_{0}>1$, the system is uniformly persistent and there is a positive equilibrium which is unique and globally asymptotical stable.