Age-density dependent population dispersal in RN

We discuss an N-dimensional model for diffusion of age dependent populations. Let ρ(x,t,a) be the population density at time t, age a and spatial position x, and let u(x,t)=∫0Aρ(x,t,a) da be the total population at t and x. The directed dispersal model to be discussed, with diffusion depending on the gradient of the total population, isρt+ρa=k div (ρ∇u)−μ(a,u)ρ,ρ(x,t,0)=∫0Aβ(a,u)ρ(x,t,a) da,u(x,t)=∫0Aρ(x,t,a) da ρ(x,0,a)=ρ0(x,a)⩾0.With some assumptions on the form of the death and birth modulus, this system is reduced to a mixed parabolic–hyperbolic system containing the Porous Medium equation in RN. We discuss the existence, regularity and localization of the solution of this reduced system.