Localization of age-dependent anti-crowding populations

In this work we prove the existence of solutions and study the localization and nonlocalization of the population in the Gurtin-MacCamy model with age-dependence and diffusion \begin{align*} \frac{{\partial \rho }}{{\partial t}} + \frac{{\partial \rho }}{{\partial a}} &= {(\rho {u_x})_x} - \mu (a, u)\rho, \\ \rho (x, t, 0) &= \int_0^\infty {\beta (a, u)\rho (x, t, a) da},\\ \rho (x, 0, a) &= {\rho _0}(x, a) \ge 0, \end{align*} where \[ u(x, t) = \int _0^\infty {\rho (x, t, a) da}\] and \[ \beta (a, u) = \beta (u)\sum \limits _{k = 1}^n {{b_k}{a^k}{e^{ - \alpha a}}, \qquad \mu (a, u) = \mu (u)}.\]