reply: Nonlinearity and the Moran effect

Abstract

Grenfell et al. reply — The Moran effect refers to systems of population dynamics that are linear: under these circumstances, the long-term correlation between population densities will be the same as the correlation between the random environmental perturbations. The Soay sheep exhibit significant nonlinearity in their density dependence (Fig. 2a of ref. 1). At low populations, numbers tend to increase exponentially, with mean growth rate r=0.24, whereas at high densities (above a threshold of 1,172 animals), the population tends to decline, with mean r=−0.29. Thus, when populations on two adjacent islands are both above their thresholds, both will tend to decline, and when both are below their thresholds, both will tend to increase.

Main

This immediately introduces a positive correlation between population dynamics on adjacent islands in the absence of any environmental forcing. When one population is above the threshold and one is below, the expectation is of no short-term correlation because the trends will tend to cancel out. Adding noise to the system has two effects. Correlated noise tends to push the dynamics into synchrony, because both populations tend to crash to low densities during the same years (for example, those with the most extreme winter weather). The nonlinearity means, however, that the same stochastic event could drive one population below the threshold but leave another population above it (for example, if initial population densities are sufficiently different). In this case, the two populations would experience different regimes of density dependence during the same year and synchrony would be reduced. Intuitively, then, in the presence of nonlinear density dependence, environmental forcing has to be stronger if it is to drive the populations into synchrony and keep them there.

Blasius and Stone have pointed out two problems with our analysis. First, they show that we had the two populations experiencing different — hence uncorrelated — noise during periods when the two populations were on opposite sides of the threshold. Correcting this mistake reduces the level of noise correlation (r_n) required to produce the observed level of population correlation (r_p) (r_p=0.685) from r_n>0.9 to around 0.8 for large samples. This means that the extra-Moran effect is reduced, but not abolished.

Their second point is that, with realistically short time series (such as our 18 points), variability in the inter-population correlation coefficient generates a relatively high expectation of observing correlations higher than Moran, leading to inflated type-I errors. We have carried out further calculations with the corrected model that agree qualitatively with this. However, even short model simulations show the imprint of nonlinearity in their aggregate correlation structure — a strong downward bias in population correlation for a given level of noise correlation, compared to various linear null models (Fig. 1; for more details, see ref. 2). Thus, the impact of nonlinearity on population correlation is apparent in the collective behaviour of short simulations, as well as in individual realizations of the model's long-term dynamics.

Figure 1: Scatter plot (blue) of inter-island population correlation (r _p) against true noise correlation (r_n) for 3,000 simulations (each of 18 time points) of the SETAR model, defined as in Table 1 of ref. 1, with the correction in noise realization proposed by Blasius and Stone.

The black line indicates where population correlation equals the noise correlation (the expectation of the Moran effect). See ref. 2 for more details.

There are several important directions for studies on the interactions between noise and determinism in population dynamics. Most important is an increase in the realism of the underlying model. The inclusion of age- and sex-structure effects is essential, because we know that animals of different ages and sexes experience markedly different patterns of mortality³. A further improvement would incorporate threshold density as a random variable rather than a constant (it is intraspecific competition for food that underlies the density dependence, and food supply determines the sheep density at which competition kicks in). This would allow for island-to-island differences in the response of food availability to environmental noise, so that islands with the same population densities could experience different density-dependence regimes in the same year.

The ability to detect extra-Moran correlations depends critically on the balance between noise and density dependence (B. Blasius and L. Stone, personal communication), so any model refinement that explains more variability in terms of population processes will increase our powers of evaluation. Technically, developments in nonlinear time-series analysis need to encompass estimation of age⁴ and spatial heterogeneities, as well as the dissection of process noise from measurement error. This is particularly important for the relatively short time series found in ecology, where even linear time-series models can produce a complex range of correlation behaviours².

A thorough understanding of spatial dynamics can only come about once the interaction between correlated noise and nonlinear density dependence is understood through long-term ecological studies, combined with models.