Nonlinearity and the Moran effect

The study of synchronization phenomena in ecology is important because it helps to explain interactions between population dynamics and extrinsic environmental variation1,2,3,4,5. Grenfell et al.1 have examined synchronized fluctuations in the sizes of two populations of feral sheep which, although situated on close but isolated islands, were nevertheless strongly correlated (observed value of the population correlation, rp, 0.685). Using a nonlinear threshold model, they argue that this level of population correlation could only be explained if environmental stochasticity was correlated between the islands, with the environmental correlation, re, higher than 0.9 “on average” (Fig. 1a). This unusually high environmental correlation is far greater than would be predicted by the Moran effect2, which states that the population correlation will equal the environmental correlation in a linear system. Grenfell et al.1 imply that a simple nonlinearity in population growth can mask or even destroy the Moran effect1,3,4. Here we show that these surprising results are an artefact of the techniques used to measure noise correlations and synchronization. Figure 1 Simulation results using the nonlinear threshold model with density-dependent switching between two independent environmental noise components, as described by Grenfell et al.1. a, Correlation between island sheep populations, rp , as a function of the model's noise parameters ra and rb (Fig. 3a of ref. 1). The model1 is set up so that ra and r b represent the correlation of the environmental noise affecting the two sheep populations when they are both either above or below threshold, respectively (see c for colour calibration). According to ref. 1, rp equals the observed value of ro = 0.658 only if the environmental correlations r a and rb are larger than “0.9 on average”. b, True environmental correlation, re, as a function of the model parameters ra and rb. To calculate re, we stored all environmental shocks received by each model sheep population and then simply determined their correlation: hence re is viewed as the correlation coefficient of shocks directly experienced by the two model sheep populations. The surface for re is almost identical to that of rp in a, demonstrating a Moran effect. c, Colour-coding of correlations depicted in a and b. d, Analysis of systems in which model environmental correlation parameters r = ra= rb (that is, systems along the diagonal of parameter space in a and b ). Red curve, cut through surface in a above the line r = ra = rb. The correlation between sheep populations rp is plotted against the model's environmental correlation parameter r; the deviation from the reference diagonal (black) seems to indicate the absence of a Moran effect, as argued in ref. 1. Blue curve, rp vs re, where re (horizontal axis) is the true environmental correlation (see b ). Note that this line is very close to the reference diagonal (black), where rp = r e, as predicted by the Moran effect. Magenta line, effects of small sample size (n = 18) due to statistical scattering in an ensemble of 1,000 simulation runs: for each value of environmental correlation r e (horizontal axis), 5% of the simulations have rp values larger than the value plotted. Green horizontal line, the measured correlation between the two observed sheep populations is ro = 0.685. The point of intersection of the green and magenta lines yields the minimum value of the ‘true environmental correlation’, r e, for which we can expect to find the observed sheep correlation by chance alone (that is, in at least 1 in 20 runs, or 5%, when r e = 0.3). Further details are available from the authors.