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Fig. 4. Number of adult females predicted by simulation models of the Serengeti woodlands population under three different scenarios. (A) Each territory is occupied by only one adult female; all adult daughters disperse. (B) Lions live in prides that must contain >3 females to be viable, but cubs do not suffer higher mortality in excessively large prides. (C) Prides must be >3 females to be viable, and cubs born in large prides suffer higher mortality. (B) and (C) illustrate representative runs of the stochastic models (see figs. S1 and S2 for output from 10 runs of each model), whereas the model shown in (A) is deterministic. (D) Frequency distribution of annual changes in female population size averaged over 100 runs of model (B), open bars, and model (C), black bars. Model (B) generated an average of only 1.08 annual population changes larger than 30% (because of the disease epidemic), whereas model (C) correctly predicted an average of 4.08 such changes (the epidemic plus three upward shifts).

% annual change in population etza

sity dependence: Cubs in large prides had similar mortality as those in medium-sized prides. The final model imposed both a threshold minimum viable pride size and the observed levels of cub mortality in excessively large prides.

In a solitary species, gradual changes in the environment in the Serengeti woodlands produce a continuous response in adult population size (Fig. 4A), because females can be added one at a time as the number of potential territories increases. In a social species with a threshold minimum group size but lacking within-group density dependence, adult population growth is less continuous, but the shifts between equilibria are still gradual because daughters can always be added to preexisting prides (Fig. 4B). With both a threshold minimum pride size and within-group density dependence, however, adult population growth is abrupt, and the model often generates the kind of saltatory equilibrium observed in the empirical data (Fig. 4, C and D). With an upper limit on pride size, moderate-sized prides require exceptional circumstances to rear large cohorts of daughters, and this is the only scenario that accurately predicts a delayed (but abrupt) recovery from the 1994 CDV outbreak (see also figs. S1 and S2).

Lion social structure imposes a coarsegrained tempo on population change that is further amplified by synchronous recruitment of large cohorts by multiple prides and stabilized by within-group density dependence. Until now, population models have assumed that population trends could be predicted by extrapolation from the survival and repro

The incorporation of a small quantity of impurities can drastically change the mechanical strength of metals. Auger electron spectros-

duction of individuals. However, a more complete understanding of population dynamics can only be achieved by incorporating the impact of social organization and family structure on the population as a whole.

References and Notes

1. V. C. Wynne-Edwards, Animal Dispersion in Relation to Social Behaviour (Oliver & Boyd, Edinburgh, 1962).

3. C. J. Krebs, Ecology: The Experimental Analysis of Distribution and Abundance (Harper & Row, New York, 1972).

4. A. R. E. Sinclair, in Serengeti: Dynamics of an Ecosystem, A. R. E. Sinclair, M. Norton-Griffiths, Eds. (Univ. of Chicago Press, Chicago, 1979), pp. 82-103.

5. C. Packer, A. E. Pusey, L. Eberly, Science 293, 690 (2001).

6. K. E. McComb, C. Packer, A. E. Pusey, Anim. Behav. 47, 379 (1994).

7. R. Heinsohn, C. Packer, Science 269, 1260 (1995).

8. C. Packer et al., in Reproductive Success, T. H. Clutton-Brock, Ed. (Univ. of Chicago Press, Chicago, 1988), pp. 363-383.

9. C. Packer, D. Gilbert, A. E. Pusey, S. J. O'Brien, Nature 351, 562 (1991).

10. K. G. Van Orsdol, J. P. Hanby, J. D. Bygott, J. Zool. 206, 97 (1985).

11. H. Kruuk, D. W. MacDonald, in Behavioural Ecology, R. M. Sibley, R. H. Smith, Eds. (Blackwell, Oxford, 1985), pp. 521-536.

12. J. P. Hanby, J. D. Bygott, C. Packer, in Serengeti II: Research, Management and Conservation of an Ecosystem, P. Arcese, A. R. E. Sinclair, Eds. (Univ. of Chicago Press, Chicago, 1995), pp. 315-331.

13. Time-series correlations between the change in the number of adults (age >2 years) in a given month and the change in the number of prides during that same month were highest when "prides" were defined as groups containing four adult females. Plains: N = 369 pride months, r = 0.374, P < 0.0001; woodlands: N = 437 pride months, r = 0.249, P < 0.0001. Autocorrelations within each time series were not significant; best fits were found with a zero time lag between the number of adults and the number of prides.

14. R. Heinsohn, C. Packer, A. E. Pusey, Proc. R. Soc. London Ser. B 263, 475 (1996).

15. An optimal segmentation method (21) determined the copy studies, together with various tensile tests, show that the sulfur (S)-induced embrittlement of nickel (Ni) is clearly asso-

number of segments and the date of the change points for each population. This method segments a data series so as to minimize the total sum-of-squares deviations by using the mean and sum of squares for each segment (and assuming a normal distribution and constant variance). The minimum number of equilibria for each habitat was determined by a dynamic programming algorithm that measured the improvement in the sum of squares with each additional segment. A cumulative sum (CUSUM) technique (22) confirmed the number and date of the change points by detecting persistent shifts from a known mean in a time series (table S1).

16. J. P. Hanby, J. D. Bygott, in Serengeti: Dynamics of an Ecosystem, A. R. E. Sinclair, M. Norton-Griffiths, Eds. (Univ. of Chicago Press, Chicago, 1979), pp. 249-262.

17. J. G. C. Hopcraft, A. R. E. Sinclair, C. Packer, J. Anim. Ecol., in press.

18. M. E. Roelke-Parker et al., Nature 379, 441 (1996).

19. B. Kissui, C. Packer, Proc. R. Soc. London Ser. B 271, 1867 (2004).

20. C. Packer et al., Conserv. Biol. 5, 219 (1991).

22. D. M. Hawkins, D. Olwell, Cumulative Sum Control Charts and Charting for Quality Improvement (Springer, New York, 1998).

23. Supported by NSF Long-Term Research in Environmental Biology grants DEB-9903416 and DEB-0343960, NSF Biocomplexity grant BE-0308486, the Canadian Natural Sciences and Engineering Research Council, and the Frankfurt Zoological Society. We thank H. Brink, J. Fryxell, D. M. Hawkins, G. Sharam, K. Skinner, I. Taylor, P. West, and K. Whitman for advice and assistance and the Tanzanian Wildlife Research Institute and Tan-zanian National Parks for permission to conduct research. This paper is the outcome of a working group on the Biocomplexity of the Serengeti hosted by the National Center for Ecological Analysis and Synthesis from 2001 to 2003.

Supporting Online Material

www.sciencemag.org/cgi/content/full/307/5708/390/

DC1

SOM Text

Table S1

Figs. S1 and S2

10 September 2004; accepted 11 November 2004

10.1126/science.1105122

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