On first-order hyperbolic partial differential equations with two internal variables modeling population dynamics of two physiological structures

In this paper we develop fundamental theories for a scalar first-order hyperbolic partial differential equation with two internal variables which models single-species population dynamics with two physiological structures such as age–age, age–maturation, age–size, and age–stage. Classical techniques of treating structured models with a single internal variable are generalized to study the double physiologically structured model. First, the semigroup is defined based on the solutions and its infinitesimal generator is determined. Then, the compactness of solution trajectories is established. Finally, spectrum theory is employed to investigate stability of the zero steady state and asynchronous exponential growth of solutions is studied when the zero steady state is unstable.