Christer Solbreck and Anthony R. Ives. 2007. Density dependence vs. independence, and irregular population dynamics of a swallow-wort fruit fly. Ecology 88:1466–1475.

Appendix A. Predicting population-level larval survivorship from within-fruit density dependence.

Analyses provided in the main text show strongly density-dependent larval survivorship at the population level (Fig. 2A), and strongly density-dependent survivorship of larvae from individual fruit (Fig. 3). Here we address whether the population-level pattern can be explained from information obtained at the level of individual fruit. Even though the same data were used for both population-level and individual-level analyses, the data were used in very different ways. Specifically, for the individual-level analyses, we pooled fruit from all samples across all 12 years, and we investigated the effect of fruit size on survivorship. Here, we ask whether the unexplained variation in population-level larval survivorship can be explained using information about survivorship of larvae within individual fruit.

Predicting the population-level survivorship from data on within-fruit survivorship requires first determining the fruit size and distribution of egg marks per fruit in a given year, and then averaging over all fruit to obtain the population-level survivorship. For each of the 12 years of data, we computed the mean numbers of seeds per fruit, . From Eq. 2 it is then possible to calculate the asymptotic mean number of larvae per fruit, b(). Specifically, to translate from individual fruit to the collection of fruit in a given year, it is necessary to know the distribution of eggs among fruit. Therefore, we fit a negative binomial distribution to the number of eggs per fruit for each of the 12 years of data, and we then computed a common value of the aggregation parameter k = 5.66 (Southwood 1978). Letting p(i|E(t),k) be the probability from this negative binomial distribution that a fruit has i eggs given the mean number of eggs per fruit is E(t), the predicted mean number of larvae, , is

(A.1)

where we have used the generating function for the negative binomial distribution (Feller 1968) to convert the summation into an integral. From this, the predicted egg–larval survivorship is /E(t).

Figure A1A plots the predicted survivorship, /E(t), calculated both from the population-level data (Eq. 1 in the text) and from Eq. A.1 using data from individual fruit vs. the observed survivorship, L(t)/E(t), computed from population-level data. Because Eq. 1 does not incorporate fruit size, to predict survival from individual fruit data (Eq. A.1) we assumed that the mean fruit sizes in each year were the same. The correlation between the observed and predicted survivorship from the population-level data (marked by o’s) was 0.60, while that between observed survivorship that predicted from individual fruit data (marked by x’s) was 0.67 (Table A1).

TABLE A1. Correlations between observed and predicted larval survivorship.

Although the calculations of population-level survivorship in the main text did not include fruit size, we can incorporate fruit size using the model

(A.2)

where is the mean fruit size in year t as in Eq. A.1. Incorporating annual variation in fruit size increases the correlation between observed and predicted survivorship to 0.77 for predictions based on both population-level data (Eq. A.2) and individual-level data (Eq. 1) (Fig. A1B, Table A1).

In the analyses with fruit size, the strong correlation between predictions from population-level and individual-level analyses (0.97, Table A1) indicates that the population-level predictions are well-explained by density-dependent larval survivorship acting at the scale of individual fruit. Because the individual-level model aggregated fruit samples across years, this demonstrates that the processes underlying density-dependent larval survivorship do not change from year to year. Nonetheless, there is density-independent larval survivorship that is not explained by the density-dependent models; the correlations between predicted and observed survivorships are high (0.77), but not perfect. We investigated the possibility that density-independent parasitism (Fig. 2B) might explain the density-independent larval survivorship by performing the analyses described above but using the sum of larvae and parasitoids; the correlation between observed and predicted larval survivorship factoring out parasitism remained 0.77.

In summary, density-dependent variation in population-level larval survivorship is well-explained by density-dependent survivorship of larvae within individual fruit after accounting for fruit size. This density dependence is consistent from one year to the next. Furthermore, part of the unexplained variance in population-level survivorship seen in Fig. 2A can be explained by variation in fruit size among years. Nonetheless, even after accounting for fruit size, the correlation between observed and predicted survivorship is only 0.77, so there remains considerable density-independent variation in survivorship. We do not know the source of this unexplained density-independent survivorship.

FIG. A1. Predicted vs. observed larval survivorship. Observed survivorships, L(t)/E(t), were computed directly from the population-level data for the 12 years, 1991–2002. (A) Predicted larval survivorship, /E(t), calculated from Eq. 2 using population-level data (o’s) and calculated from Eq. A.1 using individual fruit data under the assumption that fruit size did not vary among years (x’s). (B) Predicted larval survivorship, /E(t), calculated from Eq. A.2 incorporating year-to-year variation in mean fruit size (o’s) and calculated from Eq. A.1 using individual fruit data under the assumption that fruit size did not vary among years (x’s).

LITERATURE CITED

Feller, W. 1968. An introduction to probability theory and its applications. Volume 1. John Wiley and Sons, New York, New York, USA.

Southwood, T. R. E. 1978. Ecological methods. Chapman and Hall, New York, New York, USA.