Table 1.
Density dependence in the gray-sided vole
| Hypothesis | Parameter restrictions | Model parameters (SE) | AIC | R2, % | |||
|---|---|---|---|---|---|---|---|
| 1 | b1 = b2 = 0 | −0.66Xcr(t − 1) | −0.67Xcr(t − 2) | 112.09 | 45.0 | ||
| (0.11) | (0.11) | ||||||
| 2 | a1 = b1 and a2 = b2 | −0.68Xv(t − 1) | −0.64Xv(t − 2) | 110.07 | 47.3 | ||
| (0.10) | (0.10) | ||||||
| 3a | b1 = 0 and a2 = b2 | −0.69Xcr(t − 1) | −0.65Xv(t − 2) | 110.86 | 46.4 | ||
| (0.11) | (0.11) | ||||||
| 3b | a1 = b1 and b2 = 0 | −0.69Xv(t − 1) | −0.71Xcr(t − 2) | 105.64 | 51.9 | ||
| (0.10) | (0.10) | ||||||
| 3c | b2 = 0 | −0.69Xcr(t − 1) | −0.71Xcr(t − 2) | −0.67Xov(t − 1) | 107.64 | 51.9 | |
| (0.10) | (0.11) | (0.26) | |||||
| Full model | None | −0.69Xcr(t − 1) | −0.70Xcr(t − 2) | −0.64Xov(t − 1) | −0.08Xov(t − 2) | 109.57 | 52.0 |
| (0.10) | (0.11) | (0.29) | (0.33) | ||||
Based on Eq. 1 in the main text, hypotheses 1–3 are fitted to the time-series data shown in Fig. 1b and c. Xcr is the log density of gray-sided vole; Xv is the log density of all voles together; and Xov = Xv − Xcr is a measure of the density of other voles (this measurement is not identical to the log-transformed density of other voles and was chosen to preserve additivity on a log scale). The full model has no restrictions on the parameters. The model presented for hypothesis 3c is the best choice among models that include some b parameters different from the corresponding a parameters. The best model (i.e., the one with the lowest AIC) corresponds to hypothesis 3b, which says that direct regulation depends on the total vole density, whereas delayed regulation depends only on gray-sided vole density. Standard errors (SE) of regression parameters are given in parentheses. Except b2 in the full model, all estimated parameters are significantly different from zero at P < 0.01 (t test).