Analysis of a parabolic–hyperbolic hybrid population model: an integrated semigroup approach

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

. Furthermore, an explicit relationship between

R 0 ${\mathcal{R}}_{0}$

and the principal eigenvalue of the linearized system is analyzed. Under appropriate monotonicity assumptions, we show that

R 0 ${\mathcal{R}}_{0}$

serves as a threshold parameter that completely determines the stability of steady states of the system. More precisely, when

R 0 < 1 ${\mathcal{R}}_{0}{< }1$

, the trivial equilibrium is globally asymptotically stable, while when

R 0 > 1 ${\mathcal{R}}_{0}{ >}1$

, the system is uniformly persistent and there is a positive equilibrium which is unique and globally asymptotically stable.