May 21, 2018, Monday
University of Colorado at Boulder Search A to Z Campus Map University of Colorado at Boulder CU 
Search Links

MBW:Models of Diffusion-Mediated Gas Exchange in Animal Burrows

From MathBio

Jump to: navigation, search

Article review by Matthew Cullen

P. C. Withers. Models of diffusion-mediated gas exchange in animal burrows. Am. Naturalist, 112:1101-1112, 1978.


In this paper the question of how burrowing species respire is addressed. Burrows tend to be a strong barrier against gaseous diffusion, which begs the question, is diffusion sufficient to meet a species’ oxygen demands. This paper addresses that question using 3 separate models to account for different types of burrows. Furthermore, it looks at differences between terrestrial diffusion and aqueous diffusion. In the end, it was determined that subaquatic ectotherms (cold-blooded organisms) in burrows constructed in impermeable media could not survive on diffusion alone unless they are extremely small in size. Subterranean ectotherms however, could survive on diffusion alone assuming there are not a large number of them in the same burrow. Subterranean endotherms (warm-blooded organisms) were found to have relatively low concentrations of O_{2} and high concentrations of CO_{2} in their burrows. The most sensitive factors were found to be soil porosity and soil moisture (mainly due to its effect on soil porosity). Interestingly enough, depth and dimension of the burrow were found to be of little significance.


Almost all phyla of animals have at least one species that burrows or constructs shelters. Such structures seem to be strong barriers against gaseous diffusion, which could pose a problem for larger burrowing animals with high metabolic rates. In this study burrowing species will be divided in to 3 categories; species that dig tunnels through impermeable materials, species that move through porous substrata without constructing tunnels, and larger animals that construct nest chambers with tunnels that lead to the surface.

The amount of O_{2} and animal needs can be determined by its metabolic rate. The metabolic rate can be approximated by allometric relationships of the form {\dot  {V}}O_{{2,m}}=aM^{b} where {\dot  {V}}O_{{2,m}} is the aerobic metabolic rate (mlO_{2}/min), M is the body mass (g), and b is a constant usually around 0.75. a indicates the magnitude of aerobic metabolism of a 1-g animal. This varies depending on species, but tends to be greater for larger animals. While metabolic rate is dependent on many environmental variables, it will be shown that these can be easily accounted for in the standard models.

It will be assumed that all animals are perfect spheres and that the radius of the burrow (r_{b}) will be 1.25 times the radius of the animal (r_{a}) or r_{b}=1.25r_{a}. We will also assume that the animal has a density of 1g/(cm^3) leading to a metabolic rate of {\dot  {V}}O_{{2,m}}=a\left[{\frac  {4}{3}}\pi (0.8r_{b})^{3}\right]^{{0.75}}

For the case of small animals that move through substrata without constructing tunnels, gaseous exchange through soil must be considered. Gaseous exchange may occur through passive diffusion, or through flow of the medium. Passive flows may be the effect of wind/water currents, fluctuations in temperature or pressure, or a pressure gradient. Most gaseous exchange in terrestrial substrates is most likely due to passive diffusion because the concentrations of O_{2} and CO_{2} within soil are usually similar to ambient values. Aquatic media gas exchange is primarily caused by mass flow due to diffusion being much slower in aquatic substrates, causing high concentrations of O_{2} to be found only in the top few centimeters of soil.

The porosity of a substrata (f) is the ratio of non-solid to solid volume and is defined as f={\frac  {V_{a}+V_{w}}{V_{a}+V_{w}+V_{s}}} where V_{a} is the percent volume of air, V_{w} is the percent volume of water, and V_{s} is the percent solid volume. porosity can vary from about 0.2 (fine sand) to 0.6(course gravel). Because diffusion of O_{2} in air is approximately 320,000 times greater than that of diffusion through water, it is sometimes more useful to look at the air-filled porosity(f_{a}). Air-filled porosity is the ratio of the volume of air to the total volume or f_{a}=V_{a}/100.

Using f, it is possible to approximate the diffusion constant through a porous substrata (D_{s}) by D_{s}=D_{m}f^{{3/2}} With these values determined, it is now possible to build and implement our models.

Steady-State Models

Due to the highly different circumstances regarding the 3 types of burrows to be modeled, it is necessary to use a different equation in each case. In all of these models, it is assumed that oxygen is entering the burrow at the same rate that it is being metabolized, or {\dot  {V}}_{c}={\dot  {V}}O_{{2,m}} where {\dot  {V}}_{c} is the diffusion rate. Thus, we must find an r_{b} and a \Delta u for which this condition is met.

Model 1

This equation models the diffusion of air through a cylindrical tunnel bored in to a impermeable media.

{\dot  {V}}_{c}={\dot  {V}}O_{{2,m}}={\frac  {\pi r_{b}^{2}D_{m}\Delta u}{l_{b}}}

Where \Delta u is the absolute difference in concentration between the ambient air and the air in the burrow (atm), and l_{b} is the length of the burrow. In order for this to be a steady state, we must find values of r_{b} and \Delta u that satisfy the condition {\dot  {V}}_{c}={\dot  {V}}O_{{2,m}}

Model 2

This equation models diffusion through a sphere of radius r_{n} a distance d from a planar surface, or a perfectly spherical chamber a distance d into a porous substrata.

{\dot  {V}}_{{sph}}={\dot  {V}}O_{{2,m}}={\frac  {4\pi r_{n}D_{s}\Delta u\;d}{2d+r_{b}}}

It is assumed that r_{n}=2r_{b}=2.5r_{a}. It is interesting to note that assuming d>r_{b} we have {\frac  {d}{2d+r_{b}}}\approx {\frac  {1}{2}} and thus diffusion is relatively independent of d.

Model 3

The diffusion of O_{2} in to a spherical chamber with many entrances is approximately equal to the sum of the diffusion through each entrance and the diffusion through the rest of the sphere. The diffusion through the remainder of the sphere can be calculated using a slightly altered version of model 2.

{\dot  {V}}_{{nc}}={\dot  {V}}_{{sph}}\cdot {\frac  {4r_{n}^{2}-(n\cdot r_{b}^{2})}{4r_{n}^{2}}}

where {\dot  {V}}_{{nc}} is the diffusion through the walls of the nest chamber, and n is the number of tunnels.

We are unable to use model 1 to estimate the diffusion through each tunnel because in model one it was assumed that the media that was bored in to was impermeable. Here it is not. In a cylindrical burrow which lies parallel to a planar surface there are 2 types of diffusion which must be accounted for; radial and longitudinal. They are modeled in the following equations

{\dot  {V}}_{{b,r}}=\pi r_{b}^{2}D_{m}\Delta u\;\mu \tanh \left({\frac  {\mu l_{b}}{2}}\right)

{\dot  {V}}_{{b,l}}={\frac  {\pi r_{b}^{2}D_{m}\Delta u\;\mu }{\sinh \left(\mu l_{b}\right)}}


\mu ^{2}={\frac  {2D_{s}}{r_{b}^{2}D_{m}\cosh ^{{-1}}\left({\frac  {d}{l_{b}}}\right)}}

Thus the total diffusion is the sum of the parts; {\dot  {V}}_{{nc}}+{\dot  {V}}_{{b,r}}+{\dot  {V}}_{{b,l}}={\dot  {V}}O_{{2,m}}


Now that models have been derived, they will be implemented with realistic values. Here is a chart of the values to be used.

Model Parameters.jpg

We will look at 5 species for each model; aquatic and terrestrial unicellular, aquatic and terrestrial ectothermic, and terrestrial endothermic. We will not look at aquatic endotherms due to the rareness of such a species, and the unreasonable concentration difference needed to support it. When the above parameters were applied to the first model, the following was the result.

First Model.jpg

It is clear that for all species, a larger tunnel radius (and thus animal) will require a larger concentration difference to support it. We can also notice that all terrestrial require less of a concentration difference than any aquatic animals. This is due the diffusion constant of O_{2} in water is drastically lower than in air. Lastly, note that the vertical order of the species is due to their metabolic rate, with unicellular organisms having the lowest and endotherms having the highest. Next we will look at a similar plot that uses the 2nd model

Second Model.jpg

From this plot we can notice that though the slope of all lines is larger, most all \Delta u are smaller than their counterparts in model 1. This leads to the interesting conclusion that a suitably small animal can respire more easily when encased in a porous substrate than when in an impermeable burrow with an open tunnel. Last we will look at the result from the last model

Last Model.jpg

From this plot we can see that this structure allows for the best diffusion. This is to be expected, seeing as this model combines the diffusion methods of the 2 previous models (spherical and burrow longitudinal) and adds an additional one (burrow radial). So far we have seen how only one parameter r_{b} affects the necissary concentration difference. Next we will look at how each parameter affects \Delta u and which is the most sensitive.

Sensitivity analysis

It is important to know the parameter sensitivity in a complex model so that the primary processes can be identified. In our case the sensitivity in model 3 will be tested as it includes the previous 2 models and is the most complex. To do this we will consider a nominal \Delta {\hat  {u}}. We will let \Delta {\hat  {u}} be determined by the values in the previous chart and fixing r_{b}=1{\text{cm}}. Once this value is obtained, we can vary each parameter individually and look at the relative change, or {\frac  {\Delta u}{\Delta {\hat  {u}}}}.

The parameters the will be varied are; burrow length, soil porosity, metabolic rate, depth of nest chamber, number of occupants, and soil moisture. For the parameters burrow length, metabolic rate, and depth of nest chamber variation can be performed easily by multiplying by a scaler factor. This factor will then vary between 0 and 4 yielding a parameter that varies between 0 and 4 times its nominal value. Unfortunately, for the parameters of soil porosity, number of occupants, and soil moisture more care must be taking in their variation.

Soil porosity can not be increased past f_{a}=1. Such a value would be physically impossible. Thus, to compensate that parameter will only be increased by a maximum factor of 2 and will be held there even as other parameters increase by larger factors. Soil moisture must also be treated careful.

Soil moisture affects the system by changing the porosity via the relation f_{a}={\hat  {f}}_{a}-sm where {\hat  {f}}_{a} is the nominal porosity and sm is the percent soil moisture. Because it was not considered in the nominal value, the nominal value of {\hat  {sm}}=0. It is obvious that a negative soil moisture is impossible, thus it can not be decreased from its nominal value. Instead it will be held at 0 for all factors less than 1. Then it will increase linearly from 0% to 30% past the nominal point. This should yield sufficient variation.

The last parameter to consider is the number of occupants. It affects the the Total metabolism in the following way

{\dot  {V}}O_{{2,m}}(1)=aM^{{0.75}}

{\dot  {V}}O_{{2,m}}(n)=an(M/n)^{{0.75}}


{\frac  {{\dot  {V}}O_{{2,m}}(n)}{{\dot  {V}}O_{{2,m}}(1)}}={\frac  {an(M/n)^{{0.75}}}{aM^{{0.75}}}}=n^{{0.25}}


{\dot  {V}}O_{{2,m}}(n)=n^{{0.25}}\cdot {\dot  {V}}O_{{2,m}}(1)

where n is the number of occupants. Now that we understand how this parameter will affect the system we must consider a good way to vary it. First we must realize that {\hat  {n}}=1 and our standard way of varying would cause n to range from 0 to 4. This is insufficient to model the real world as there are collonies of burrowing animals (termites for example) who number in the 10,000s. Instead we will use a logarithmic factor that varies from 10^{{-1}} to 10^{3}. This will give a more realistic interpretation.

Now that variation of all parameters is understood, their relative change on \Delta u was graphed in the following plot


From this plot we can see that the most sensitive parameters are soil porosity, number of occupants, metabolic rate, and soil moisture. we can also see that burrow length and the amount of burrows have a relatively small effect. Using this information we can deduce what the what the most important components of the burrow are to O_{2} diffusion.

Soil moisture and soil porosity can be considered to be paired, as soil moisture has a direct effect upon soil porosity. The mechanism that these parameters main affect is the ability of O_{2} to diffuse through the soil. This informs us that the diffusion through the soil and in to the nest chamber plays a large role in this system.

Like the parameters above, metabolic rate and number of occupants can be considered paired because they both affect {\dot  {V}}O_{{2,m}}. The above plot implies that as metabolic rate or number of occupants increases the gradient needed to support the organisms increases as well. This follows intuition as an increase in either parameter will lead to an increase in {\dot  {V}}O_{{2,m}} requiring that the organisms receive more O_{2}.

It is interesting to note that the depth of the nest chamber and the length of the burrow had virtually no effect. The lack of dependence on depth implies that there is a small concentration gradient between the surface any reasonable depth below it. Thus, the depth of a nest chamber will have little effect on how much oxygen can diffuse in to it. It seems odd that the length of the burrow would have so little effect on the concentration gradient. One possible explanation is that the diffusion through the tunnels in to the nest chamber is relatively small compared to the diffusion through the chamber it's self. This idea can be supported if we look at previous findings.

A comparison of model 1 to model 2 shows that a smaller concentration gradient is needed for animals surrounded by porous substrata rather than animals living in impermeable burrows. Thus, for a constant concentration gradient more oxygen would diffuse through the soil than would diffuse down the burrow. This agrees with our sensitivity findings that a reduction to the porosity of the soil would cause a greater effect on the concentration gradient than an increase of burrow length. Thus, the dimensions of a nest built in a permeable material would be near irrelevant.


In this study we have seen how diffusion of O_{2} in to a burrow is affected by a variety of components. We have seen how 2 of the largest factors in each model was the size of the animal, and if the burrow was terrestrial or aquatic. Aquatic burrows were found to be so poorly suited to diffusion that only micro-organisms could rely on it alone, while other aquatic species would have to use some form of mass action. It was found that burrowing endotherms require high concentration gradients to support their greater metabolic rate, and are also limited in size by it. Furthermore large colonies of burrowing animals were predicted to require some form of mass flow to support their increased metabolic needs.

External Links

P. C. Withers. Models of diffusion-mediated gas exchange in animal burrows. Am. Naturalist, 112:1101-1112, 1978. [1]