  We analyze a probabilistic cellular automaton describing the dynamics of
coexistence of a predator-prey system. The individuals of each species are
localized over the sites of a lattice and the local stochastic updating rules
are inspired on the processes of the Lotka-Volterra model. Two levels of
mean-field approximations are set up. The simple approximation is equivalent to
an extended patch model, a simple metapopulation model with patches colonized
by prey, patches colonized by predators and empty patches. This approximation
is capable of describing the limited available space for species occupancy. The
pair approximation is moreover able to describe two types of coexistence of
prey and predators: one where population densities are constant in time and
another displaying self-sustained time-oscillations of the population
densities. The oscillations are associated with limit cycles and arise through
a Hopf bifurcation. They are stable against changes in the initial conditions
and, in this sense, they differ from the Lotka-Volterra cycles which depend on
initial conditions. In this respect, the present model is biologically more
realistic than the Lotka-Volterra model.
