Non-linear Age-Dependent Population Dynamics with Spatial Diffusion

Abstract

Several models in population dynamics are governed by reaction-diffusion equations or parabolic equations. In this work, we present a population model containing both age-structure and spatial diffusion.

Our model is:

$$ \left\{ \begin{array}{ll} \displaystyle {\frac{\partial u}{\partial t} + \frac{\partial u}{\partial a}} = \varDelta u - \mu _n(a)u(x,t,a) - \mu _e( P(x,t)) u(x,t,a). \\ P(x,t)= \displaystyle { \int ^\infty _0} u(x,t,a) da . \end{array} \right. $$

where u(x, t, a) is a positive function which represents the density in both age (a) and space (x). P(x, t) represents the total population at position x.

Existence and uniqueness results are obtained, and also the asymptotic behavior of the solution is studied.