Age-density dependent population dispersal in \(\mathbb{R}^N\) (Q1806784)

We discuss an \(N\)-dimensional model for diffusion of age dependent populations. Let \(\rho(x,t,a)\) be the population density at time \(t\), age \(a\) and spatial position \(x\), and let \(u(x,t)=\int^A_0\rho(x,t,a)\text{d}a\) 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, is \[ \rho_t+\rho_a=k \text{div} (\rho\nabla u)-\mu(a,u)\rho,\quad\rho(x,t,0)=\int^A_0\beta(a,u)\rho(x,t,a)\text{d}a, \] \[ u(x,t)=\int^A_0\rho(x,t,a)\text{d}a,\quad\rho(x,0,a)=\rho_0(x,a)\geq 0. \] 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 \(\mathbb{R}^N\). We discuss the existence, regularity and localization of the solution of this reduced system.