The Spatial Dynamics of Host--Parasitoid Systems

1. We consider models for host--parasitoid interactions in spatially patchy environments, where in each generation specified fractions of the host and parasitoid subpopulations in each patch move to adjacent patches. In most previous work of this general kind, the movement is not localized in this way, but involves `global' mixing of the populations prior to dispersal. 2. A remarkable range of dynamical behaviour is exhibited by a mathematically explicit model with constant host reproductive rate, deterministically unstable local dynamics and dispersing hosts and parasitoids that only move to nearest-neighbour patches in a density-independent way. The density of the host and parasitoid subpopulations in a two-dimensional array of patches may exhibit complex patterns of spiral waves, spatial chaos, a so-called static `crystal lattice' pattern, or they may become extinct. The probability of extinction rises rapidly when the number of patches present decreases below some characteristic arena size which varies with the scale of the spatial dynamics. 3. The different types of spatial dynamics that are observed depends critically on the fractions of hosts and adult parasitoids that disperse in each generation from the patches in which they emerged. Low rates of host dispersal tend to lead to chaotic patterns unless this rate is very low and parasitoid dispersal rates very high, in which case `crystal lattice' patterns may occur. Intermediate to high rates of host dispersal tend to result in spiral patterns. The effect of varying host rates of increase, within-patch parasitism that is inherently stabilizing and random patch-to-patch variation are also discussed. 4. The results are relatively insensitive to the details of the interaction. Thus, a similar range of behaviour (spirals, chaos and crystal lattices) is discernible from a very general `cellular automaton' model in which only qualitative categories of patch densities are specified together with a very simple set of `transition rules'. The diffusive dispersal of the explicit model is parallelled by making the new state in each generation depend, not only on the current state of the given `cell', but also on the states of a specified set of neighbouring cells.