  We study diffusion-reaction processes on periodic square planar lattices and
simple cubic (sc) lattices. Considered first is a single diffusing reactant
undergoing an irreversible reaction upon first encounter with a stationary
co-reactant ["one-walker (1W) problem"]. We then generalize this scenario to
allow for a competing reaction, i.e., instantaneous trapping of the diffusing
reactant with probability $s$ at any vacant site before interacting with a
(stationary) co-reactant at a target site. We determine the mean walklength of
the diffusing reactant until irreversible reaction occurs. We use generating
functions and the theory of finite Markov processes, as well as MC simulations.
To investigate the dependence of walklength on lattice size we compute the
first, finite size corrections to the Green function of the sc lattice, and
provide a Pad\'e approximation for this quantity. Finally, we consider the case
where both reactant and co-reactant undergo synchronous nearest-neighbor
displacements ["two-walker (2W) problem"]. In this case, reactant and
co-reactant can individually be trapped with probability $s$ at any vacant
lattice site, or can undergo an irreversible reaction on first encounter at any
site. When $s=0$ we find that, both for the 1W and the 2W problem, for lattices
with (approximately) the same number of sites, the mean walklength is smaller
(and hence the reaction efficiency greater) in $d=3$ than in $d=2$. Increasing
$s$ tends to reduce differences in system dimensionality, and distinctions
between the 1W problem and the 2W problem. Our model provides a good starting
point to develop studies on the efficiency of apparently diverse
diffusion-reaction processes, such as diffusion on a partially poisoned
catalytic substrate or photosynthetic trapping of excitations.
