  We propose two models of the evolution of a pair of competing populations.
Both are lattice based. The first is a compromise between fully spatial models,
which do not appear amenable to analytic results, and interacting particle
system models, which do not, at present, incorporate all of the competitive
strategies that a population might adopt. The second is a simplification of the
first, in which competition is only supposed to act within lattice sites and
the total population size within each lattice point is a constant. In a special
case, this second model is dual to a branching annihilating random walk. For
each model, using a comparison with oriented percolation, we show that for
certain parameter values, both populations will coexist for all time with
positive probability. As a corollary, we deduce survival for all time of
branching annihilating random walk for sufficiently large branching rates. We
also present a number of conjectures relating to the r\^{o}le of space in the
survival probabilities for the two populations.
