  We consider a reaction-diffusion process with retardation. The particles,
immersed in traps initially, remain inactive until another particle is
annihilated spontaneously with a rate $\lambda$ at a certain point $\vec x$. In
that case the traps within a sphere of radius $R(t)= v t^{\alpha}$ around $\vec
x$ will be activated and a particle is released with a rate $\mu$. Due to the
competition between both reactions the system evolves three different time
regimes. While in the initial time interval the diffusive process dominates the
behavior of the system, there appears a transient regime, where the system
shows a driveling wave solution which tends to a non-trivial stationary
solution for $v \to 0$. In that regime one observes a very slow decay of the
concentration. In the final long time regime a crossover to an exponentially
decaying process is observed. In case of $\lambda = \mu$ the concentration is a
conserved quantity whereas for $\mu > \lambda$ the total particle number tends
to zero after a finite time. The mean square displacement offers an anomalous
diffusive behavior where the dynamic exponent is determined by the exponent
$\alpha $. In one dimension the model can be solved exactly. In higher
dimension we find approximative analytical results in very good agreement with
numerical solutions. The situation could be applied for the development of a
bacterial colony or a gene-pool.
