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the scaling exponents to fit the allometric scaling laws proposed by West and his colleagues (West etal. 1997,2002) (see also Section 3.4.5). They assume that tracheal tubes are space filling, that their cross-sectional area remains constant with divisions in the tubes, that the finest tubes are scale invariant, and that energy is minimized. Based on these and other assumptions they predict and show that metabolic rate should scale as m0 75, and that changes in cross-sectional area should scale as a quarter power of body mass too (West et al. 1997). These values are considerably different from some of those proposed by Kestler (1985), but all that is required for the null models to hold is that the scaling exponent for convection should be larger than that for diffusion.

Perhaps of more significance than the change in the relative contributions of gas exchange by diffusion and convection, is the body mass at which this change might take place. Kestler (1985) suggested that this changeover should take place at a body mass of approximately 1 g. This is likely to be the case if net convective respiratory water loss is always c.80 per cent of diffusive loss. However, if it is not, then it is not clear at what body size the changeover should take place. Despite the elapse of more than 15 years since Kestler (1985) published his analysis, little empirical work has been undertaken to investigate these theoretical predictions (but see Harrison 1997). Although comparisons of the relative proportions of cuticular and respiratory transpiration are commonly made in an effort to understand the significance of respiratory water loss, they are unlikely to reveal the way in which water balance has been modified by selection. Relative proportions might vary by as little as a few per cent despite major differences in the gas exchange mode and/or differences in cuticular versus repiratory transpiration (Chown 2002). Rather, an understanding of the way in which changes in respiratory mode might contribute to the conservation (or elimination) of water requires a comparative analysis of the absolute and relative contributions of each of the major routes of water loss across a suite of closely related taxa in the context of this theory. Doing this is relatively straightforward in insects that show discontinuous gas exchange cycles (DGCs) (see Section 3.4.2), and a comparative study of dung beetle water loss has illustrated the benefits of using a rigorous, analytical approach to separate cuticular and respiratory water loss (Chown and Davis 2003). The analytical methods developed by Gibbs and Johnson (2004), which allow calculation of cuticular water loss as the intercept of the intra-individual regression of water loss on metabolic rate irrespective of respiratory pattern, should make such comparisons more straightforward in insects that do not show discontinuous gas exchange.

Diffusive-convective gas exchange Based on the demonstration that diffusive gas exchange alone is unlikely in insect tracheal systems, Kestler (1985) developed a see-saw model for gas exchange, arguing that a pure diffusion-based system is unlikely to occur in insects, and that diffusive-convective or pure convective gas exchange is much more likely. He further developed Buck's theory of flow-diffusion and corrected the analysis, showing that during suction ventilation in insects (Section 3.4.2) oxygen shows cocurrent diffusive-convective gas exchange at the spiracles, described by the following equation:

where AS is spiracular cross-sectional area, v the flow velocity, px is the capacitance coefficient of gas x, ApS is spiracular partial pressure gradient for gas x, LS is the length of the spiracular tube, D 0 is the effective diffusion coefficient, and pA is the partial pressure of gas x at the opening of the tubular valve.

CO2, water vapour, and nitrogen show anticurrent diffusive-convective gas exchange, described as

M S—ANCU = AS ■ v ■ Py ■ (Apy x exp(—v ■ LS/Dy)

Kestler (1985) then examined these principles in the light of experimental work undertaken by himself on Periplaneta americana (Blattaria, Blattidae) and by others (mostly Schneiderman and his colleagues—see below) on the silkmoth

Hyalophora cecropia (Lepidoptera, Saturniidae), providing support for his theoretical analyses. Despite these analyses, the extent to which gas exchange might take place in insects by diffusion only remains contentious (Lighton 1996, 1998; Slama 1999). Nonetheless, Kestler's (1985) equations provide a foundation for examining this issue, and will make further exploration thereof more straightforward than otherwise might have been the case.

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