  In this paper we consider a size-structured population model where
individuals may be recruited into the population at different sizes. First and
second order finite difference schemes are developed to approximate the
solution of the mathematical model. The convergence of the approximations to a
unique weak solution with bounded total variation is proved. We then show that
as the distribution of the new recruits become concentrated at the smallest
size, the weak solution of the distributed states-at-birth model converges to
the weak solution of the classical Gurtin-McCamy-type size-structured model in
the weak$^*$ topology. Numerical simulations are provided to demonstrate the
achievement of the desired accuracy of the two methods for smooth solutions as
well as the superior performance of the second-order method in resolving
solution-discontinuities. Finally we provide an example where supercritical
Hopf-bifurcation occurs in the limiting single state-at-birth model and we
apply the second-order numerical scheme to show that such bifurcation occurs in
the distributed model as well.
