  Stochastic, spatially extended models for predator-prey interaction display
spatio-temporal structures that are not captured by the Lotka-Volterra
mean-field rate equations. These spreading activity fronts reflect persistent
correlations between predators and prey that can be analyzed through
field-theoretic methods. Introducing local restrictions on the prey population
induces a predator extinction threshold, with the critical dynamics at this
continuous active-to-absorbing state transition governed by the scaling
exponents of directed percolation. Novel features in biologically motivated
model variants include the stabilizing effect of a periodically varying
carrying capacity that describes seasonally oscillating resource availability;
enhanced mean species densities and local fluctuations caused by spatially
varying reaction rates; and intriguing evolutionary dynamics emerging when
variable interaction rates are affixed to individuals combined with trait
inheritance to their offspring. The basic susceptible-infected-susceptible and
susceptible-infected-recovered models for infectious disease spreading near
their epidemic thresholds are respectively captured by the directed and dynamic
isotropic percolation universality classes. Systems with three cyclically
competing species akin to spatial rock-paper-scissors games may display
striking spiral patterns, yet conservation laws can prevent such noise-induced
structure formation. In diffusively coupled inhomogeneous settings, one may
observe the stabilization of vulnerable ecologies prone to finite-size
extinction or fixation due to immigration waves emanating from the interfaces.
