  Discrete time, spatially extended models play an important role in ecology,
modelling population dynamics of species ranging from micro-organisms to birds.
An important question is how 'bottom up', individual-based models can be
approximated by 'top down' models of dynamics. Here, we study a class of
spatially explicit individual-based models with contest competition: where
species compete for space in local cells and then disperse to nearby cells. We
start by describing simulations of the model, which exhibit large-scale
discrete oscillations and characterise these oscillations by measuring spatial
correlations. We then develop two new approximate descriptions of the resulting
spatial population dynamics. The first is based on local interactions of the
individuals and allows us to give a difference equation approximation of the
system over small dispersal distances. The second approximates the long-range
interactions of the individual-based model. These approximations capture
demographic stochasticity from the individual-based model and show that
dispersal stabilizes population dynamics. We calculate extinction probability
for the individual-based model and show convergence between the local
approximation and the non-spatial global approximation of the individual-based
model as dispersal distance and population size simultaneously tend to
infinity. Our results provide new approximate analytical descriptions of a
complex bottom-up model and deepen understanding of spatial population
dynamics.
