Spatial organization and mobility effects in collective hunting and defense strategies of predator-prey systems

There is a myriad of strategies that predators utilize to increase
their rate of success. Among them, preys may be attacked in a cooperative,
coordinated way, these actions being
correlated in space and time. The number of known examples
of coordinated hunting, whether intra or interspecies, has increased in the last years
and examples include hawks, crocodiles, spiders, etc.
Although there are some additional costs, hunting or defending in group may bring
several benefits for predators and preys, respectively, what have been widely studied.
Despite these mounting evidences, much less attention has been dedicated to model such behavior.

This problem has been recently
considered within a game theoretical framework in which the abundances of preys
and predators were assumed constant and only the fraction of
those populations using either an individual or collective
strategy evolves. Lett et al (2004 Theor. Pop. Biol. {\bf 65} 263)
considered a mean field approach in which these densities
are described by Lotka-Volterra-like equations, taking
into account some of the advantages and
disadvantages for both preys and predators choosing a grouping
strategy. More specifically, it is assumed that grouping lowers the
risk of predation at the cost of increasing the competition for
resources, while predators have a greater probability of success
at the expense of having to share the prey with others.

We present a spatial version of this model that locates individuals or
groups on a lattice and study it in the limits of both low and high population
viscosity (with or without diffusion, respectively), and compare these results
with the mean field predictions. Of particular interest is the coexistence
region with both grouped and individual predators and prey persist within
the population. When compared with the mean field case, fundamental differences
appear and are strongly affected by finite size effects.