Ecological Archives E088-020-A1

Richard P. Duncan, David M. Forsyth, and Jim Hone. 2007. Testing the metabolic theory of ecology: allometric scaling exponents in mammals. Ecology 88:324–333.

Appendix A. Derivation of the relationship between rm and alpha.

Savage et al. (2004) derive a relationship between rm and alpha. Here we present an alternative simpler derivation of the relationship.

The starting point of the modelling is the two-stage Lotka equation describing the relationship between the finite population growth rate ( = Nt+1/Nt), and demographic parameters, age at first reproduction (, years), annual fecundity (b, female young/female/year), survival from birth to age at first reproduction (l), annual adult survival (s) and age at last reproduction (, years). The model (Lande 1988) is:

Rearranging the equation to retain on the left-hand side and moving all other terms to the right hand side, and then taking natural logarithms of both sides of the equation, and then taking logarithms to base 10 of both sides of the equation and rearranging the equation to retain only log r = log(ln) on the left hand side, gives:

which describes a negative relationship between log r and log α with a slope of –1. Note that the second logarithmic transformation could be to either base 10 or base e. We used base 10 because it is the usual form of the relationship between rm and mass. The maximum value of r is rm, which occurs as survival rates (l and s) approach 1.0, and fecundity (b) is at a high rate, associated with abundant food and no predators, parasites or competitors.

LITERATURE CITED

Lande, R. 1988. Demographic models of the northern spotted owl (Strix occidentalis caurina). Oecologia 75:601–607.

Savage, V. M., J. F. Gillooly, J. H. Brown, G. B. West, and E. L. Charnov. 2004. Effects of body size and temperature on population growth. American Naturalist 163:429–441.



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