  We are concerned with a class of degenerate diffusion equations with time
delay describing population dynamics with age structure. In our recent study
[{\em Nonlinearity}, 33 (2020), 4013--4029], we established the existence and
uniqueness of critical traveling wave for the time-delayed degenerate diffusion
equations, and obtained the reducing mechanism of time delay on critical wave
speed. In this paper, we now are able to show the asymptotic spreading speed
and its coincidence with the critical wave speed $c^*(m,r)$ of sharp wave, and
prove that the initial perturbation or the boundary of the compact support of
the solution propagates at the critical wave speed $c^*(m,r)$ for the
time-delayed degenerate diffusion equations. Remarkably, different from the
existing studies related to spreading speeds, the time delay and the degenerate
diffusion lead to some essential difficulties in the analysis of the spreading
speed, because the time-delay makes the critical speed of traveling waves slow
down, and the degenerate diffusion causes the loss of regularity for the
solutions. By a phase transform technique combined with the monotone method, we
can determine the asymptotic spreading speed. Furthermore, we propose a
brand-new sharp-profile-based difference scheme to handle large variation of
degenerate diffusion $(u^m)_{xx}$ near the sharp edge and carry out some
numerical simulations which perfectly confirm our theoretical results.
