Appendix B. Derivation of the relationship between and B.
Here we use the model described in West et al. (2001) and Gillooly et al. (2002) to derive a direct relationship between basal metabolic rate and age at first reproduction.
West et al. (2001) use the following equation to describe the change in body mass (m) of an organism through time (t)
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Where B is the basal metabolic rate of the whole organism at time t, Bc is the metabolic rate of a single cell, mc is the mass of a cell and Ec is the metabolic energy required to create a single cell. Following (West et al. 2002), we assume that cellular metabolic rate scales with body mass at maturity (M), such that:
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Where Nc is the total number of cells.
Hence:
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Integrating gives:
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where c is the constant of integration.
Setting initial conditions at time t0 = 0 as , then
and:
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The onset of adulthood is assumed to be at (see (Gillooly et al. 2002), Box 2) which occurs at time t = α, so that at maturity:
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Hence, the theory predicts that the relationship between age at first reproduction and mass-corrected basal metabolic rate should have a scaling exponent of -1.
LITERATURE CITED
Gillooly, J. F., E. L. Charnov, G. B. West, V. M. Savage, and J. H. Brown. 2002. Effects of size and temperature on developmental time. Nature 417:7073.
West, G. B., J. H. Brown, and B. J. Enquist. 2001. A general model for ontogenetic growth. Nature 413:628631.
West, G. B., W. H. Woodruff, and J. H. Brown. 2002. Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals. Proceedings of the National Academy of Sciences USA 99:24732478.