Spatial structure and chaos in insect population dynamics

Abstract

MOST environments are spatially subdivided, or patchy, and there has been much interest in the relationship between the dynamics of populations at the local and regional (metapopulation) scales1. Here we study mathematical models for host-parasitoid interactions, where in each generation specified fractions (µN and µp, respectively) of the host and parasitoid subpopulations in each patch move to adjacent patches; in most previous work, the movement is not localized but is to any other patch2. These simple and biologically sensible models with limited diffusive dispersal exhibit a remarkable range of dynamic behaviour: the density of the host and parasitoid subpopulations in a two-dimensional array of patches may exhibit complex patterns of spiral waves or spatially chaotic variation, they may show static 'crystal lattice' patterns, or they may become extinct. This range of behaviour is obtained with the local dynamics being deterministically unstable, with a constant host reproductive rate and no density dependence in the movement patterns. The dynamics depend on the host reproductive rate, and on the values of the parameters µN and µp. The results are relatively insensitive to the details of the interactions; we get essentially the same results from the mathematically-explicit Nicholon–Bailey model of host-parasitoid interactions, and from a very general 'cellular automaton' model in which only qualitative rules are specified. We conclude that local movement in a patchy environment can help otherwise unstable host and parasitoid populations to persist together, but that the deterministically generated spatial patterns in population density can be exceedingly complex (and sometimes indistinguishable from random environmental fluctuations).