Spreading speeds of diffusive age-structured equations in slowly and rapidly varying media

In this paper, we investigate the effects of the spatial period in determining the spreading speed of age-structured reaction-diffusion equations in spatially periodic media. We establish the asymptotic behavior of the spreading speed when the spatial period tends to zero and infinity, respectively. By comparing the spreading properties between the age-structured models and the classic Fisher-KPP reaction-diffusion equations, we show that the introduction of the age structure leads to more complicated spreading dynamics. Specifically, in rapidly oscillating media, there does not exist a definitive relationship between their spreading speeds. In contrast, in slowly oscillating media, the two spreading speeds coincide and can be characterized by a family of periodic Hamilton-Jacobi equations, with the zero order term determined by an age-structured equation parameterized by the spatial variable. Finally, we present an explicit formula for the limiting spreading speed in patchy environments, via constructing the viscosity solutions of the associated Hamilton-Jacobi equation.