Analysis of a parabolic-hyperbolic hybrid population model

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.