Invariant Ratios Vs. Dimensionless Ratios

Abstract

In their Report “The illusion of invariant quantities in life histories” (19 Aug., p. 1236), S. Nee et al. show that purely statistical methods may lead us to conclude that certain life-history ratios are invariant when, in fact, they are not invariant at all, but the statistical procedure—in which one regresses X + c on X —causes one to think so. Some confusion may have also occurred in this field because of the difference between a dimensionless ratio and an invariant one. A simple case, originally attributable to Beverton and Holt ([1][1]), can illustrate the point. If an organism grows according to the von Bertalanffy form L ( t ) = Lω (1 - e − kt ), where t is age, L ω is asymptotic size, k is the growth rate, survival to age t is e -Mt (where M is the rate of mortality), and fitness with maturity at age t is e -Mt L ( t ) b (where b is the allometric parameter connecting size and fecundity), then it is an exercise in introductory calculus to show that the optimal age of maturity is t * = (1/ k ) log [( M + bk )/ M ] and that the relative size at maturity is L ( t *)/ L ω = b /[ b + ( M / k )]. The ratio M / k is dimensionless but need not be invariant. However, for any two species in which this ratio is the same, the relative size at maturity will be the same. I suggest that it might be more productive for us to follow the example of fluid mechanics and replace the notion of invariants by explicit dimensionless numbers. Define, for example, the “Beverton number” ν B = M/k so that L ( t *)/ L ω = b /( b + ν B ). Then we conclude that for species in which ν B →, relative size at maturity will be very small, whereas for those species in which ν B → 0, relative size at maturity will be close to asymptotic size. Life-history invariants may be elusive, but dimensionless numbers and their life-history consequences are not. 1. 1.[↵][2]1. R.J. H. Beverton, 2. S. J. Holt , CIBA Found. Colloq. Ageing 54, 142 (1959). [OpenUrl][3] [1]: #ref-1 [2]: #xref-ref-1-1 View reference 1. in text [3]: {openurl}?query=rft.jtitle%253DCIBA%2BFound.%2BColloq.%2BAgeing%26rft.volume%253D54%26rft.spage%253D142%26rft.atitle%253DCIBA%2BFOUND%2BCOLLOQ%2BAGEING%26rft.genre%253Darticle%26rft_val_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Ajournal%26ctx_ver%253DZ39.88-2004%26url_ver%253DZ39.88-2004%26url_ctx_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Actx