  In this paper, by incorporating the general delay to the reaction term in the
memory-based diffusive system, we propose a diffusive system with memory delay
and general delay (e.g., digestion, gestation, hunting, migration and
maturation delays, etc.). We first derive an algorithm for calculating the
normal form of Hopf bifurcation in the proposed system. The developed algorithm
for calculating the normal form of Hopf bifurcation can be used to investigate
the direction and stability of Hopf bifurcation. As a real application, we
consider a diffusive predator-prey model with ratio-dependent Holling type-3
functional response, which includes with memory and gestation delays. The Hopf
bifurcation analysis without gestation delay is first studied, then the Hopf
bifurcation analysis with memory and gestation delays is studied. By using the
developed algorithm for calculating the normal form of Hopf bifurcation, the
supercritical and stable spatially homogeneous periodic solutions induced
jointly by memory and general delays are found. The stable spatially
homogeneous periodic solutions are also found by the numerical simulations
which confirms our analytic result.
