A degenerate non-autonomous first-order hyperbolic model for size-structured populations with active and quiescent states

We develop a degenerate nonautonomous first-order hyperbolic partial differential equation model to characterize the dynamics of a size-structured population exhibiting both active and quiescent states. Our model extends the classical one-state size-structured population model. We establish the well-posedness of the model through a comparison principle. We analyze the long-term behavior of the solution by employing the upper-lower solution method. More precisely, we derive conditions on the model parameters that determine whether the population persists or goes extinct. We also numerically explore how these parameters affect the persistence of the population. We demonstrate both analytically and numerically that the two-state model can approximate the classical one-state model in several special cases. Furthermore, in these cases, we find that the autonomous counterparts of both models have approximately equal basic reproduction numbers.