Models of Diffusion-Mediated Gas Exchange in

Models of Diffusion-Mediated Gas Exchange in Animal Burrows Author(s): Philip C. Withers Source: The American Naturalist, Vol. 112, No. 988 (Nov. - Dec., 1978), pp. 1101-1112 Published by: The University of Chicago Press for The American Society of Naturalists Stable URL: http://www.jstor.org/stable/2460351 Accessed: 09-10-2017 23:56 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms

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Vol. 112, No. 988 The American Naturalist November-December 1978

MODELS OF DIFFUSION-MEDIATED GAS EXCHANGE IN ANIMAL BURROWS PHILIP C. WITHERS* Department of Biology, University of California, Los Angeles, California 90024

Almost every phylum of animals includes species which crawl or burrow through a

substrate or construct elaborate shelters (Wood 1880; Hancocks 1973; von Frisch

1974), and such an existence often entails a considerable barrier against gaseous diffusion. Though the distribution of small burrowing animals is probably not limited by the diffusion gradients required to maintain their aerobic metabolism

(Kevan 1962), larger burrowing animals, and particularly endotherms with their high mass-specific metabolic rates, may be limited in distribution since the lower surface/volume of larger burrows reduces gaseous exchange between the animal and the external medium.

In fact, diffusion has been shown to be insufficient to support the metabolism of many fossorial endotherms without the establishment of extreme 02 and CO2 gradients. The CO2 concentration in mammalian burrows may be as high as

10%-15% (Kennerley 1964; Studier and Baca 1968; Studier and Proctor 1971; Williams and Rausch 1973). High concentrations of CO2 have also been measured in bird nests (White et al., in prep.), termite mounds (see Noiroit 1970), bee swarms (Simpson 1961; Nagy and Stallone 1976), and in nests of turtle eggs (Prange and Ackerman 1974). The complexity and interactions of parameters which determine the gaseous

environment within animal burrows (such as edaphic properties, burrow morphol-

ogy, and biological variables) make experimental study difficult. Studier and Baca (1968) and Soholt (1974) reported the effects of the number of burrow occupants, their metabolic rate, and soil moisture content on the gas concentrations in mamma-

lian burrows. However, little is known concerning the importance of other edaphic and biotic variables in determining the gas composition in burrows. The significance

of the body mass of the burrow occupant, which is important through surface/volume effects, and differences between subaquatic and subterranean burrows have not been systematically investigated. The present study incorporates

edaphic and biotic variables into general models of diffusional gas exchange in a variety of animal burrows and investigates the significance of altering each of those variables. * Present address: Department of Zoology, Duke University, Durham, North Carolina 27706. Amer. Natur. 1978. Vol. 112, pp. 1101-1112. ? 1978 by The University of Chicago. 0003-0147/78/1288-0003$01.30

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1102 THE AMERICAN NATURALIST METHODS

Models of Subterranean Animals

Subterranean and subaquatic animals can be divided for convenience into three

categories: (1) species which bore tunnels through impermeable media such as rock or wood, or construct shelters of impermeable media such as compacted mud (model

I); (2) small animals which move through porous substrata without constructing a tunnel (model II); and (3) large animals which construct a nest chamber with tunnels for access to the surface (model III). The general burrow models assume that there is a solitary occupant, that the

interstitial spaces of the substrate have 02 and CO2 concentrations equivalent to the ambient medium, and that the metabolic rate of the animal is equal to the standard

metabolic rate predicted from allometric relationships (see below). The effects of altering these assumptions upon the predicted gaseous conditions will be considered in the following discussion.

The gaseous environments associated with the above models are analyzed for steady-state and non-steady-state conditions by application of diffusion theory (Crank 1956) and analogous equations for heat exchange (Carslaw and Jaeger 1959; Kreith 1973). Metabolic Requirements

The metabolic rates of different animals can be predicted from allometric relation-

ships of the form VO2,m = aM' where V02,m is the aerobic metabolic rate (ml 02

min-'), M is body mass (g), and b is a constant, which is usually about 0.75. The

constant a indicates the magnitude of aerobic metabolism of a 1-g animal; the value

of a for unicells is 0.000029 ml 02 min-' and is 0.0023 and 0.064 ml 02 min- for ectotherms and mammals, respectively (Hemmingsen 1960).

Metabolic rate is dependent upon many environmental variables such as temperature and the level of activity. Although the models incorporate Hemmingsen's equations for standard metabolic rate, it will be shown that the concentration gradient required to maintain metabolism is in direct proportion to the absolute metabolic rate, and it is therefore simple to allow for changes in temperature or activity in the standard models. All animals will be assumed, unless otherwise stated, to be spheres, and since the

burrow must be somewhat larger than the animal, I will assume that the radius of the

burrow (rb) is 1.25 times the radius of the animal (ra). The metabolic rate of such an

animal is, assuming its density is 1 g cm- 3, V02m = a[ir(0.8rb)3]0.75. The assumptions that rb = 1.25ra and density is 1 g cm-3 have a slight but quantifiable effect on V02,m-

Diffusion Theory Gaseous exchange between an animal's burrow and the external medium may occur through passive diffusion, or through actively or passively induced flow of the medium past the animal. Passive mass flow may be caused by wind or water current

penetration into the substrate, fluctuations in ambient temperature or pressure, shifts


DIFFUSION-MEDIATED GAS EXCHANGE 1103

in a water table, a pressure gradient if the respiratory quotient is less than one, or

through viscous entrainment by the external medium (Collis-George 1959; Glinka 1963; Webb and Theodor 1972; Vogel and Bretz 1972; Vogel et al. 1973; Vogel

1974). Most gas exchange in terrestrial substrates is probably through passive diffusion (Evans 1966), but exchange in aquatic media is primarily due to mass flow (Braefield 1964).

The general equation for diffusion is aQ/at = ADm aU/ax, where aQ/at is the quantity of material diffusing per unit time (cm3 s-'), Dm is the diffusion constant (cm2 atm-' min-1), A is the area for diffusion (cm2), aU is the concentration difference (atm), and ax is the diffusion distance (cm).

Soil Diffusion The concentrations of 02 and CO2 in the interstitial soil spaces are normally

similar to ambient values (Glinka 1963; Hillel 1971), but diffusion is much slower in aquatic substrates, and high concentrations of 02 are generally found only in the top

few centimeters of soil (Braefield 1964; Sassaman and Magnum 1972). The possible overestimation of diffusion in subaquatic anaerobic substrates is of little importance,

as it will be shown that virtually no subaquatic burrowing species can rely upon diffusion for gas exchange even under the most favorable circumstances.

The porosity (f) of substrata is the ratio of nonsolid to solid volume, i.e.,

f = (Va + Vw)/(V, + Vw + Vs), where T' is the percent volume of air in the substrate, V is the percent water volume, and Vs is the percent solid volume. Porosity varies from about 0.2 (fine sand) to 0.6 for coarse gravel (Hilgard 1906). The air-filled porosity

(fa) is the ratio of air volume to total volume, i.e.,fa = Vi/100. The air-filled porosity is more important than fin determining the diffusion rate since diffusion constant gases are greater in air (e.g., for 02, Dm = 11 cm2 atm-1 min-') than in water (DM = 0.000034 cm2 atm- 1 min- 1). The diffusion constant for porous substrata (Ds) is dependent upon the physical configuration of the pores, e.g., pore tortuosity and

internal surface area (Hillel 1971). An empirical relationship between Ds and f Ds = Dm f 3/2 (Marshall 1959). Diffusion constants are essentially independent of temperature over the biological temperature range. However, the solubility of 02 and CO2 in water is dependent upon temperature, and this could influence the rates of 02 depletion and CO2

accumulation in interstitial water spaces under non-steady-state conditions.

RESULTS

Steady-State Diffusion

Model 1.-The quantity of gas diffusing along a cylindrical burrow in an impermeable medium is VC = (irr72 Dm Au)/lb = V02,m where Vc is the rate of diffusion (cm3 min- 1), Au is the absolute difference in concentration between the ambient air and the air in the burrow (atm), and lb is the burrow length (cm) (Crank 1956). The values

of Au required to support VO2,m are dependent upon whether the animal is

subaquatic or subterrestrial and upon body mass (fig. 1, left). The Au is greater for


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