Local and Ensemble Dynamics of Linked Populations

Abstract

In a recent issue of the Journal of Animal Ecology, Bascompte & Sole (1994) investigated the effect of introducing immigration between a number of populations each represented by a simple one-step map. They conclude that this linkage generally serves to destabilize individual populations, increasing their propensity to exhibit chaotic fluctuations, but tending to reduce the variance of the ensemble. Although not mentioned by Bascompte & Sole, there have been several other recent papers on linkage between potentially chaotic populations (e.g. McCallum 1992; Gonzalez-Andujar & Perry 1993a,b; Stone 1993). These papers suggest that linkage between simple maps tends to decrease the likelihood of chaotic fluctuations, rather than increase it, as Bascompte & Sole suggest. These studies use a smaller number of populations than considered by Bascompte & Sole or simply consider immigration to be a time-independent constant. Even so, they suggest that Bascompte & Soles conclusion about local dynamics should be tempered. While linkage does have an effect on local dynamics, it is not clear that there is a general rule governing the nature of this effect. For the simplest case, where immigration is a constant, Ruxton (1993) provides an example where introducing low levels of immigration can induce chaos in a model population which would show a regular cycle if isolated. However, higher levels of immigration recover a simple cycle again. Bascompte & Sole also conclude that ensemble dynamics are 'stabilized' by the addition of linkage, in that the variance decreases and the resultant total population resembles a steady state with added noise. This appears at odds with my own observation of 10 linked maps where the global dynamics commonly appear as a simple cycle (Ruxton 1994). However, a reconciliation of these two, apparently contradictory, observations may be found in the work of de Roos, McCauley & Wilson (1991) who considered a predator-prey interaction on a discrete lattice. If movement takes place only between adjacent sites then they observed a noisy signal around a roughly constant population. However, when all lattice sites were equally available for movement, then large amplitude cycles were obtained. A similar situation appears to apply to our linked populations. I allow dispersal to all populations equally and obtain regular cycles whereas Bascompte and Sole allow movement only between neighbouring populations and obtain a 'noisy equilibrium'. Hence, although Bascompte and Sole are correct in concluding that the number of subpopulations has an important effect on the behaviour of the ensemble, the degree of linkage between individual populations should also be emphasized as an equally important ecological parameter. One of the most developed areas for exploration of the way linkage between populations affects dynamics lies in the field of human childhood diseases. Here, in contrast to most other areas, firmly grounded theoretical models can be combined with a considerable amount of data (e.g. Sugihara, Grenfell & May 1990; Grenfell 1992).