# Response to Comments on "On the Regulation of Populations of Mammals, Birds, Fish, and Insects"

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Science  24 Feb 2006:
Vol. 311, Issue 5764, pp. 1100
DOI: 10.1126/science.1121565

## Abstract

The technical comments by Getz and Lloyd-Smith, Ross, and Doncaster focus on specific aspects of our analysis and estimation and do not demonstrate any results opposing our key conclusion—that, contrary to what was previously believed, the relation between a population's growth rate (pgr) and its density is generally concave.

We analyzed the relation between the size of a population and its rate of growth, in 1780 time series of mammals, birds, fish, and insects (1). Nonlinear regression was used to fit a discrete version of the theta-logistic equation (see Eq. 2 below). The parameter θ takes values less than 1, equal to 1, or greater than 1, depending on whether the relation is concave, linear, or convex viewed from above. We reported that the relation is concave in 78% of species and that there is little variation among the major taxonomic groups. Here, we respond to technical comments by Ross (2), Getz and Lloyd-Smith (3), and Doncaster (4).

Ross's comments (2) are predicated on the assertion that per capita population growth rate, pgr, must be finite when N = 0. However, pgr is a per capita rate defined as 1/NdN/dt, so when N = 0 the value of pgr involves division by zero. Thus, there is no reason to suppose that pgr is finite when N = 0. In reality, growth rates can only be calculated for populations that consist of at least one individual. This is important, because there is a mathematical singularity in the theta-logistic equation at N = 0 such that for nonpositive values of θ, r0 not only becomes infinite but also switches from +∞ to –∞ at N = 0. This is described and illustrated in figure S2 of our Supporting Online Material (SOM) for (1). Further discussion of the effects on population growth of nonpositive values of θ can be found in (5). The accuracy of figure 2 in (1), which more plausibly assumes that pgr is finite for some N ≥ 1, is easily checked by numerical calculation. Ross (2) also objects to our description of our estimation procedure as nonlinear least squares. We described it thus because it minimizes the residual sums of squares from the nonlinear model by using a search over a suitably fine grid of values for the nonlinear parameter θ.

Getz and Lloyd-Smith (3) argue that we should have interpreted our results in terms of the theta-Ricker equation $Math$(1) instead of the discretization of the theta-logistic equation $Math$(2) but these are mathematically equivalent. The authors claim that our estimates are biased because our method gives a different estimate than that of (6) when applied to a particular Accipiter nisus data set but, contrary to their assertion, (6) analyzed a different data set. Their other criticisms are based on misunderstanding of the nature of regression modeling. Regression models allow prediction of the mean y variate for each of the x variates within the range of observed x variates. These models are not valid beyond the range of their observations, yet Getz and Lloyd-Smith (3) criticize one of our estimates of θ because extrapolation of the fitted equation beyond the range of observations yields unrealistic values of pgr. They further criticize our model because it predicts that abundance converges on carrying capacity, but this again misunderstands the nature of regression modeling, which predicts mean pgr for each N but does not predict the scatter of the data about the fitted regression line. Getz and Lloyd-Smith (1) also assert that we summarily dismissed incorporating trophic interactions using time delays. However, our report explicitly stated that “it would be interesting to explore the possibility of including time lags.” There is no reason to suppose that any of the criticisms posed by Getz and Lloyd-Smith would affect our conclusions about concavity and convexity. Nevertheless, we support their call for investigation of the effects on concavity of migration, trophic interactions, and autocorrelations in environmental factors.

Finally, Doncaster (4) is right to point out that the effect of measurement error is to erode signal and that when all signal is lost, θ takes the value 0 and then pgr varies as the logarithm of population density with slope –1. Indeed, we noted this in our report (1). However, we also showed that none of the taxonomic classes except the insects have slope –1. Doncaster (2) uses a different equation to model dynamics, so it is not surprising that some of his results and interpretations differ from those of our study. We agree with Doncaster that the strength of density dependence [also known as return rate (7)] is informative about population regulation and hope that others will take up the challenge of estimating its value.

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