  In this paper, we investigated the global attractivity of the positive
constant steady state solution of the mature population $w(t,x)$ governed by
the age-structured model: \begin{equation*} \left\{\begin{array}{ll}
\frac{\partial u}{\partial t}+\frac{\partial u}{\partial a}=D(a)\frac{\partial
^2 u}{\partial x^2} - d(a)u, & t\geq t_0\geq A_l,\;a\geq 0,\; 0< x< \pi,\\
w(t,x)=\int_r^{A_l}u(t,a,x)da,& t\geq t_0\geq A_l,\; 0<x<\pi,\\
u(t,0,x)=f(w(t,x)), & t\geq t_0\geq A_l,\; 0<x<\pi,\\
u_x(t,a,0)=u_x(t,a,\pi)=0,\;& t\geq t_0\geq A_l,\; a \geq 0, \end{array}
\right. \end{equation*} when the diffusion rate $D(a)$ and the death rate
$d(a)$ are age dependent, and when the birth function $f(w)$ is nonmonotone. We
also presented some illustrative examples.
