Michel Baguette and Nicolas Schtickzelle. 2006. Negative relationship between dispersal distance and demography in butterfly metapopulations. Ecology 87:648–654.

Appendix B. Explanation of the density dependence calculations.

A density dependence function has already been estimated for P. eunomia (Schtickzelle and Baguette 2004), B. aquilonaris (Baguette and Schtickzelle 2003) and E. aurinia (Schtickzelle et al., in press). The same methodology was applied here to obtain the same function for the two other species. Basic data were time series of yearly population sizes (sexes pooled). For E. e. bayensis and L. achine, the series being available for several close local populations, population sizes and patch areas were summed over the populations. We found the pooling justified even if some within-metapopulation (between local populations) differences in density-dependent processes may exist (some differences in intensity and form of density dependence were reported between JRC and JRH for E. e. bayensis :McLaughlin et al. 2002) for two reasons. (1) In all cases synchrony between the lumped populations was found to be high (probably due to a similar regional climate). (2) We were only interested in obtaining an average density dependence function at the metapopulation level, which would be comparable between the metapopulations, to illustrate metapopulation-specific patterns, not to precisely unravel detailed density-dependent processes at a local scale.

To be comparable between species, population sizes were transformed into densities by dividing them by the habitat area (summed over the lumped populations). The yearly population growth rate (R_t) was then computed as the ratio

(B.1)

where D_t and D_t-1 are population densities at year t and t-1 respectively.

The chosen density dependence function, the Ricker equation, is classically formulated as:

(B.2)

with r the intrinsic population growth rate and K the carrying capacity (Hastings 1997). If we pose R_max=e^r, it may be reformulated as

(B.3)

with R_max the maximum growth rate and K the carrying capacity. According to this function, R_max corresponds to the growth rate at very low population density (N_t-1 = 0) and K corresponds to R = 1 (population stability) respectively (Akçakaya et al. 1999). The existence of such a negative exponential relationship was tested, and parameters R_max and K estimated for each data set using the SAS NLIN procedure (SAS Institute 1999).

Table 1 gives a summary of population growth and dispersal parameters as observed in the five metapopulations. In all five cases, the relationship between the population growth rate and density of the previous year clearly and significantly fits the negative exponential function expressed by Eq. B.3 (Fig. 1 in the main text) even if the longer time series (E. e. bayensis) shows more variability at low population size. The E. e. bayensis metapopulation presents an extreme population size of around 1150 individuals/ha, the closest other ones being around 450 individuals/ha); the removal of this outlier  neither altered the quality of the fit of the negative exponential function nor changed its parameter estimates significantly (R_max = 2.26 instead of 2.19, K = 304 instead of 385 individuals/ha). The parameters of the density dependence function (maximum growth rate R_max and carrying capacity K) are variable in the different metapopulations but are clearly in the same order of magnitude: the maximum growth rate varies between 1.24 and 3.11 while the carrying capacity ranges from 152 to 723 butterflies/ha (Fig. 1 in the main text) for Fritillaries, and somewhat lower (78 individuals/ha) for L. achine. L. achine’s suitable habitat being a narrow strip at glade edges, it is likely that the estimation of patch area as glade area is an overestimation, which might be responsible for this lower estimation of equilibrium density.

See Appendix C for a list of references.