  In contrast to the neutral population cycles of the deterministic mean-field
Lotka--Volterra rate equations, including spatial structure and stochastic
noise in models for predator-prey interactions yields complex spatio-temporal
structures associated with long-lived erratic population oscillations.
Environmental variability in the form of quenched spatial randomness in the
predation rates results in more localized activity patches. Population
fluctuations in rare favorable regions in turn cause a remarkable increase in
the asymptotic densities of both predators and prey. Very intriguing features
are found when variable interaction rates are affixed to individual particles
rather than lattice sites. Stochastic dynamics with demographic variability in
conjunction with inheritable predation efficiencies generate non-trivial time
evolution for the predation rate distributions, yet with overall essentially
neutral optimization.
