AMER. ZOOL., 32:179-193 (1992)
Consider a Spherical Lizard: Animals,
Models, and Approximations'
MICHAEL P. O ' C O N N O R AND JAMES R. SPOTILA
Department of Bioscience and Biotechnology, Drexel University,
Philadelphia, Pennsylvania 19104
and
Savannah River Ecology Laboratory, Aiken, South Carolina 29801
SYNOPSIS. Ecologists and physiologists have used biophysical models to answer questions
and investigate hypotheses about animal biology for over 20 years, but many investigators
do not use such techniques because such modelling is perceived as an arcane art. Indeed,
there is no magic strategy to allow all ecologists to model any biophysical problem accurately
by means of simple recipes. In practice, biophysical ecology depends heavily on mathematical
and engineering principles. But, it need not be impenetrable. Here we discuss relatively
simple models that can be incorporated into many ecological studies. We also discuss some
of the important approximations and assumptions inherent in our treatments of radiative,
convective, evaporative, and conductive heat transfer. In so doing, we hope to encourage
the use of such models, and to engender an appreciation of when and under what conditions
predictions from such models are most likely to be misleading. Thus, we hope to help
ecologists to get into and, hopefully, out of trouble in biophysical ecology.
INTRODUCTION
One of the difficulties in making biophysical modelling, as a research tool, more
accessible to animal ecologists is that it
sometimes appears to be an arcane art. Biophysical ecology depends heavily on mathematics, has a jargon all its own, and, as in
any field, sometimes the fine points are
important. In biophysical models, those fine
points often seem more related to physics
and engineering than to biology. These factors have discouraged many animal ecologists from incorporating biophysical techniques into their own research. In this paper,
we do not intend to trivialize biophysical
models, claiming that all biophysical models are inherently simple; they are not. Nor
do we have a magic strategy that will allow
all ecologists to simply and accurately model
any biophysical phenomenon. Rather, we
suggest that relatively simple biophysical
models can sometimes be useful to ecologists (see Bartlett and Gates, 1967; Porter
and Gates, 1969; Spotila et al, 1972; Porter
and Tracy, 1974, 1983; Christian and Tracy,
1981; Christian et al, 1983; Bakken et al,
1985; Spotila et al, 1990). Then, we discuss
the inaccuracies, approximations, and
assumptions endemic to biophysical models to help outline what biophysical models
can and cannot do. In doing so, we will focus
primarily on models of thermal relations of
terrestrial organisms because a good deal of
research has focused on such models (Porter
and Gates, 1969; Spotila et al, 1972, 1973;
Porter etal, 1973; Bakken <?? al, 1974,1985;
Tracy, 1976, 1982; Crawford et al, 1983;
Porter, 1989) and because such models have
been so important to understanding the
ecology of terrestrial animals (Porter and
Gates, 1969; Spotila, 1972; Spotila et al,
1972, 1973; Porter et al, 1973; Reichert
and Tracy, 1975; Spotila and Gates, 1975;
Tracy, 1976; James and Porter, 1979; Porter and James, 1979; Christian and Tracy,
1981, 1985; Feder, 1982, 1983; Christian et
al, 1983, 1984; Kingsolver and Watt, 1983;
Turner and Tracy, 1983, 1986; Kingsolver
and Koehl, 1985; Stevenson, 1985a, b; Tracy
etal, 1986; Turner, 1987; Kingsolver, 1989;
Porter, 1989; Paladino et al, 1990; Casey,
1992; Grant and Porter, 1992).
MODELS AND MODELLING
In order to understand how approximations can affect the utility of models, one
must first understand that models have dif1
From the Workshop on Biophysical Ecology: Methods, Microclimates, and Models presented at the Annual ferent purposes. Nearly all biological modMeeting of the American Society of Zoologists, 27-30 els are intentional simplifications of comDecember 1989, at Boston, Massachusetts.
plex systems, intended to answer some
179
180
M. P. O'CONNOR AND J. R. SPOTILA
Models
Generality
Precision
Information required
FIG. 1. Relationships between the conceptual generality of biophysical models, the likely accuracy of their
predictions in particular situations, and the information required to "parameterize" the models.
question(s) about those systems. The questions asked with some biophysical models,
however, are heuristic in nature (e.g., how
do the importance of radiative and convective heat exchange differ between large and
small animals, will an animal be warmer in
environment A or environment B). Other
models are meant to provide reasonably
precise predictions about an animal (e.g.,
what will the body temperature of this animal be under these conditions). Most biophysical models lie somewhere on the continuum between these two extremes, with
the relative importance of heuristic value
and accurate prediction being determined
by the biological question under investigation.
As Levins (1968) has discussed, there is
some tension between the heuristic value
(generality) of a model and its ability to make
precise predictions for a particular animal
under particular conditions (accuracy, Fig.
1). Heuristic models tend to be simple and
depend on relatively simple, general
descriptions of processes, ignoring some of
the complexities of specific animals in specific environments (e.g., Porter and Gates,
1969; Spotila, 1972; Spotila et ai, 1973;
Mitchell, 1976; Stevenson, 1985a, b). Models designed to give precise predictions, on
the other hand, can be quite complex and
depend either on empirical observations
about processes (e.g., Nevins and Darwish,
1970; Tracy, 1976; Porter, 1989) or on large
amounts of data on the characteristics of
animals and environments (Grant and Porter, 1992). Heuristic models can be modified to give precise predictions, but in the
process, they become more complex and
come to require more data as inputs (Fig.
1).
When building a biophysical model, one
of the first decisions to be made is to what
extent the biological question to be answered
requires precise predictions and to what
extent less precise but more general predictions are desired. Our bias is that modelling
efforts should be motivated by biological
questions. Thus, the simplest, most general
model that answers the biological question
should be constructed. One advantage of
such an approach is that such models require
less technical expertise in biophysics, computation, and modelling strategies than more
"precise" models. Thus, our suggestion is
to approach our review of approximations
and gaps in biophysical methods with the
question "When can I get in trouble by using
this modelling technique?"
SOURCES OF ERROR
The simplifications and approximations
built into biophysical models introduce
inaccuracies or errors into the predictions
made by those models. Two of the more
important data about a given model are 1)
how far wrong the model predictions could
be, and 2) how those errors could be reduced.
In order to answer such questions, one might
first appreciate where the predictive errors
originate. The models used by biophysical
ecologists in general describe the exchange
of heat, mass, or momentum between an
animal and its environment. Thus, predictive uncertainties in biophysical models
stem from our descriptions of the animal,
its environment, and the transfer processes
(Table 1).
One of the most important assumptions
of biophysical models is that we can adequately describe the animal's environment.
For an animal free to move around its environment, we must be able to describe the
distribution of microhabitats in the environment. We know of no case in which we
can make a formal description of the distribution of microsites for any mobile terrestrial animal. Thermal ecologists have
responded either by grouping microsites into
a small number of classes, and then providing average or typical values for each class
MODELS AND APPROXIMATIONS
(Reichert and Tracy, 1975; Roughgarden
et ai, 1981; Scott et ai, 1982; Crawford
et ai, 1983; Christian et ai, 1983, 1984;
Porter, 1989; Grant and Porter, 1992), or
by scattering thermal mannequins about
the environment and using the distribution
of operative temperatures to describe the
environment (Bakken and Gates, 1975;
Peterson, 1987; Grant and Dunham, 1988;
O'Connor, 1989; Zimmerman and Tracy,
1989; Bakken, 1990). Both methods are
useful, but both describe the environment
imperfectly and thus represent potential
sources of error in models.
Some habitats are quite complex, with
thermal characteristics that vary substantially on fine spatial and/or temporal scales.
Thus, air flows, particularly turbulent flows,
vegetative canopies, and soils (Rose, 1966;
Rosenberg, 1974; Gates, 1980;Vogel, 1981;
White, 1984) form environments that are
quite difficult to characterize accurately.
Heat exchange between these environments
and an animal can be, therefore, equally difficult to characterize accurately. Finally, the
temperatures and other thermal characteristics of many environments vary with time.
Thus, air temperature varies from minute
to minute {e.g., as a result of clouds obscuring the sun), from hour to hour, day to day,
season to season, and year to year. As air
temperature changes, so does the air's
capacity to hold water (the saturation vapor
density) and, thus, the potential for evaporative cooling. Further, different parts of
the environment warm and cool on somewhat different cycles. For example, on
cloudless days, solar radiation peaks at solar
noon, air temperatures peak 2-3 hr later,
and ground temperatures may peak still later
in the day (Gieger, 1965; Parton and Logan,
1981).
Having described the thermal environment, we must then describe the heat transfers between the organism and that environment. As with our descriptions of the
environment, our ability to describe heat
transfer processes is imperfect because the
processes are complex. Thus, we make a
number of simplifying assumptions to facilitate the analysis. One of the most obvious
assumptions is that the animal (or at least
181
TABLE 1. Potential sources of errors in biophysical
models.
Description of the environment
Variation in space
1) distribution of microhabitats
2) inherently complex habitats
effect of vegetative canopies
turbulent air flow conditions
soil properties
Variation in time—temperatures, humidity, wind
speed, radiation
Description of heat transfer processes
Simplifying assumptions
homogeneity of tissue and heat transfer at surface
linearity of thermal radiation and evaporation
Approximation of complex parameters—e.g., convection coefficients
Selection of heat transfer mechanisms to consider
some part of the animal) represents a thermally homogeneous object. The idea that
muscle, lung, liver, gut, bone, and blood can
be thought of as homogeneous, or that the
dorsum of an animal is thermally equivalent to its venter seems improbable at best.
Quite complex models of human (see examples in Hardy et al., 1970) and animal (Tracy,
1976; Bakken, 1981) heat transfer have been
built with this problem in mind. Unfortunately, such models are computationally
complex and, more importantly, require
much more detailed information about the
animal and its environment than is usually
available. Thus, we often model animals as
thermally homogeneous and radially symmetric simply because the models work well
enough to answer the questions asked (Porter and Gates, 1969; Spotila et ai, 1972,
1973; Tracy et ai, 1986). More complex
models can be built when the biological
questions require them (Tracy, 1976; Bakken, 1981; Grant and Porter, 1992). Nonetheless, we need to be aware that, no matter
how simple or complex the model is, it is
still an approximation with some inaccuracies.
A second kind of simplifying assumption
is the often made assumption of linearity.
If the amount of heat transferred between
animal and environment by some pathway
is proportional to the difference in temperature between the two objects, that mech-
182
M. P. O'CONNOR AND J. R. SPOTILA
describe convective heat transfer with a
convection
coefficient. In most cases, howo Evaporation (Penman)
7ever,
convection
coefficients can only be
+ Thermal Radiation (Taylor)
o"
6determined empirically, because they rep5resent the summation over the entire sur4face of the animal (or body part) of complex
UJ 3
processes including mixtures of diffusion and
3mass transfer of heat. As is often true in
2biology, and as Vogel (1981) has pointed
1out for the related concept of drag, deter0
mining convection coefficients under one set
body " alr <° )
of conditions and applying them under othFIG. 2. Errors in body temperature predictions of ers is not without some risk of error. A final
simple biophysical models due to linearizing assump- sort of simplifying assumption is the exclutions about thermal radiation (by Taylor series expan- sion of some heat transfer mechanisms from
sion) and evaporation (by Penman approximation). consideration in the model. Thus, conducBecause each approximation substitutes air temperature for the surface temperature of the animal, the errors tion to the ground is often ignored in energy
depend on the difference between air and body tem- balances for animals that hold their bodies
peratures. (Redrawn from Tracy et al., 1984)
up off the substrate, metabolic heat generation is often ignored in small ectotherms,
anism of heat transfer is said to be linear. and cutaneous evaporation is sometimes
However, not all heat transfer mechanisms ignored in desert lizards because these proare linear. Heat exchange by infrared (ther- cesses are thought to make only small conmal) radiation, for example, varies with the tributions to the animals' energy budgets.
difference between the fourth powers of the
In many cases, such exclusions and
absolute temperatures of the animal and its approximations can be justified either
surroundings. Heat exchange via evapora- because the error induced is truly small, or
tion depends, in an even more complicated, because the induced error is not important
non-linear manner, on the temperatures of to the biological questions motivating the
object and environment. Simple heat bal- model. Before such approximations are
ance models have traditionally assumed that made, however, one ought to carefully conthese non-linear heat transfer processes can sider the possible contributions of a heat
be approximated by linear mathematical transfer process to the energy balance of the
descriptions (Campbell, 1977; Tracy et al., model and the precision required of the
1984). The virtue of such linear approxi- model predictions.
mations is that they make both numerical
and analytic solutions of heat balance analMECHANISMS OF HEAT EXCHANGE
yses easier. In some cases, linear approxiIn an attempt to review the strengths and
mations are required to make solutions possible at all. The disadvantage is that, weaknesses of biophysical modelling and to
although these approximations work well point out possible pitfalls, we now turn to
for animals near environmental tempera- a more detailed review of specific heat transture, they introduce (sometimes large) errors fer mechanisms. We present them in the
into heat balances for animals whose body context of a steady-state, energy balance
temperatures can be far from air tempera- model for a terrestrial ectotherm, one of the
ture (Tracy et al., 1984; Fig. 2). The avail- common uses of such models (Porter and
ability of high speed microcomputers and Gates, 1969; Spotila et al., 1972, 1973; Porsophisticated equation processors are mak- ter et al., 1973, 1975; Porter and Tracy,
ing such approximations less important, but 1974, 1983; Tracy, 1976; Christian and
Tracy, 1981, 1985; Christian et al, 1983,
heuristic models still employ them.
A third "simplifying assumption" is the 1984; Porter, 1989; Paladino et al., 1990).
use of simple empirical approximations for The overall equation for such an energy balcomplex heat exchange processes. Thus, we ance is
Linearization of
rror due
ariziatlon
8-
T
T
C
183
MODELS AND APPROXIMATIONS
Qabs + M = R + C + \E + G
(1)
where
Qabs = radiation absorbed by the surface of
the animal (W/m2)
M = metabolic heat production (W/m2)
R = radiation emitted by the surface of
the animal (W/m2)
C = energy lost by convection (W/m2)
XE = heat lost by evaporation or gained
by condensation (W/m2)
G = heat lost by conduction (through
direct physical contact of the animal with soil, water, or substrate)
(W/m2)
A detailed presentation of each of the above
mechanisms of heat transfer is beyond the
scope of this paper. Readable accounts can
be found in Monteith, 1973; Campbell,
1977; Tracy, 1982; Spotila et at, 1991a. A
more complete, technical presentation can
be found in Gates (1980). What we hope to
present is a survey of the major gaps in our
knowledge about each mechanism of heat
transfer and the major difficulties in simulating heat transfer mechanisms. For each
mechanism of heat transfer, we will try to
point out both the gaps in our knowledge
about how to describe the environment, and
the difficulties in simulating the transfer of
heat between the animal and the environment.
Solar radiation
Shortwave, or solar, heat flow is conceptually very simple. Because animals do not
emit shortwave radiation, one needs to know
only the intensity of the solar radiation, the
absorptivity of the animal, and the surface
area of the animal that intercepts the solar
radiation to estimate heating via sunshine.
Solar radiation is composed primarily of
ultraviolet, visible, and near-infrared wavelengths and is usually measured with an
instrument such as an Eppley pyranometer.
There is also a thermal component to environmental radiation which can be treated
most easily as a net radiation problem (see
below). In this respect the approach presented here differs from that in some classic
treatments (e.g., Porter and Gates, 1969)
where all absorbed radiation is lumped into
a single term, called Qabs, as suggested in eq.
1. For purposes of modelling heat transfer
due to solar radiation we use eq. 2.
Qs = AvfsaS
(2)
where
Qs = rate of solar energy absorption by
the animal (W)
A = surface area of the animal (m2)
vf, = view factor, i.e., proportion of the
surface receiving solar radiation
absorptivity to solar radiation
a
(average over solar spectrum)
intensity of solar radiation (W/m2)
S
There are some geometric problems to consider (Monteith, 1973; Campbell, 1977;
Gates, 1980) but given relatively precise
tools to estimate the intensity of the radiation (see Gates, 1980; Rosenberg, 1974)
and using measured values for absorptivity
(spectral reflectometer, e.g., see Tracy, 1976;
Spotila etai, 1991a) and surface areas (photodermoplanimeter, e.g., see Halliday and
Hugo, 1963), the calculation of absorbed
radiation would seem straightforward.
Unfortunately, the shortwave radiative
environment is more difficult to describe
precisely than one might suspect. Both the
intensity of radiation and the absorptivity
of the animal's surface (skin, fur, feathers,
etc.) vary with the wavelength of light
through the infrared, visible, and ultraviolet
spectra. For an animal in the open on a
clear, cloudless day, we can describe the
spectrum and intensity of the radiation
striking the animal fairly well (McCullough
and Porter, 1971). But light shining directly
on the animal, that scattered by the atmosphere, that reflected from the ground or
vegetation, and that which has passed
through clouds or vegetation all have different spectra. Further, on cloudy days or
for animals in complex environments {e.g.,
in vegetation or among rocks), the intensity
of direct, scattered, reflected, and transmitted light varies unpredictably with time and
with the animal's movements. The various
components of solar radiation can be measured (see Rosenberg, 1974; Gates, 1980),
but for an animal in a complex environment, the measurements can only represent
"typical" or "average" values and are
184
M. P. O'CONNOR AND J. R. SPOTILA
unlikely to cover the range of values that
the animal might experience. Thus, models
of solar radiative heating yield predictions
that depend on measured or predicted typical values of radiant intensity. If the animal
under consideration lives in a complex habitat, some effort to determine the effects of
shading (or light flecks) can be valuable,
especially in heliothermic animals (Scott et
al., 1982; Standora, 1982; Christian et al,
1983, 1984; Grant and Dunham, 1988).
Thermal (longwave infrared) radiation
As with shortwave radiation, mechanistic
descriptions of infrared heat transfer
between an animal and its environment
appear quite simple (eq. 3), and deal with
the same three factors: the area absorbing
(and emitting) the radiation (A), the absorptivity of the animal's surface to the radiation
(e), and the intensity of the radiation (<r(T4
- T4)).
R =
-
Tr4)
(3)
where
R = rate of heat transfer by longwave
thermal radiation (W)
surface area of the animal (m2)
A
view factor, proportion of surface
area exchanging longwave radiation
with the environment
longwave infrared emissivity of skin
6
the Stefan-Boltzmann constant
a
(5.673 x 10-8 W m"2 K"4)
body surface temperature (K)
T r = equivalent radiant temperature of the
environment (K)
There are two difficulties in simulating
heat transfer by thermal radiation. First,
thermal radiation is a non-linear mechanism of heat transfer, making the actual calculations more difficult. Second, and more
troublesome, it can be difficult to determine
the proper environmental radiative temperature (Tr).
When necessary, the non-linear intensity
of the thermal radiation can be approximated by a linear expression (i.e., "linearized," in modellingjargon) called the Taylor
expansion (eq. 4).
= Tr4 + 4 Tr3(Ts - Tr)
(4)
where
Ts = body surface temperature (K)
Tr = equivalent radiant temperature of the
environment (K)
Fortunately, under many circumstances, the
errors in estimated body temperature
induced by using the Taylor approximation
are small (Tracy et al., 1984; Fig. 2). The
animals in which the linearization errors will
be largest are those for whom thermal radiation is a major avenue of heat exchange
(e.g., dry-skinned animals standing in burrows, where solar radiation is excluded and
air movement is minimal). Fortunately, in
such cases, ectothermic animals tend to have
body temperatures close to that of their
environments, in which case, the Taylor
approximation yields small errors (Fig. 2).
The environmental radiative temperature (Tr) is difficult to evaluate because it is
a composite of the temperatures of all of the
parts of the animal's environment from
which thermal radiation can reach the animal. Those parts of the environment include
the ground, any vegetative or rocky features
nearby, and the sky (for animals in the open).
Although measurements of thermal radiation (using a net radiometer see Rosenberg,
1974; Gates, 1980) can provide measurements of the "equivalent" radiative temperature at some point in the environment,
as with solar radiation, those measurements
can only describe typical or average values.
Convection
Although expressions for convective heat
transfer appear quite simple (eq. 5), that
appearance conceals a fairly complex process.
, - TJ
C=
(5)
where
C = rate of heat loss by convection (W)
A = surface area of the animal (m2)
hc = convection coefficient (W m~2 K"1)
T r = surface temperature of the animal
Ta = fluid temperature (°Q
Convective heat loss includes a mixture of
diffusive heat transfer through a fluid (e.g.,
air) and mass transport of heat by the fluid
185
MODELS AND APPROXIMATIONS
(Gates, 1980; Vogel, 1981; White, 1984).
Conditions:
Very near the animal, diffusion is the dom- o
Ta = 25°C
wind speed = 2 m/s
inant mode of heat transfer with increasing
solar radiation = 800 W/m2
mass transport at larger distances from the
surface of the animal. Furthermore, the distance from the skin at which mass transport o a
Error In convection
„_„
coefficient
begins to dominate over diffusion (the
s
boundary layer thickness) is larger at the 2
iii
downstream edge of a body than at the
upstream edge (see Schlieren photographs
0
.0001 .001
.01
.1
10
1 0 0 1000 10000
Mass (kg)
Monteith, 1973; Gates, 1980). Thus, the
convection coefficient (h,.) is an empirically, FIG. 3. Errors in body temperature predictions due
rather than theoretically, determined value to errors in estimated convection coefficients (e.g., due
that integrates the heat loss over the entire to use of Mitchell's [1976] approximation, eq. 6) under
body and depends on the wind speed, shape, specified conditions.
and orientation of the animal. In order to
avoid the use of huge tables of convection ilar errors in estimation of convection coefcoefficients, a number of approximations are ficients result in larger errors in estimated
available (see summary in Gates, 1980). body temperature.
Perhaps the most useful of those approxiFurther complicating the modelling of
mations is Mitchell's (1976), which assumes convective heat exchange is the complexity
the animal is a sphere of equivalent mass
of the convective environment. Even for
(eq. 6).
animals in the open on smooth, fiat surfaces, wind speed increases and air temperhc = 6.77V 06 D-° 4
(6) ature varies with increasing distance off the
ground (Rose, 1966; Rosenberg, 1974;
D = (volume)1/3
(7) Gates, 1980; Fig. 4). Typically, modelers
where
use air temperature and wind speed at the
animal's
mid-torso height as average valD = the characteristic dimension of the
ues.
Simulations
comparing this approach
animal (m)
to
more
complex
integrations
of wind speed
V = wind speed (m/sec)
and air temperatures suggested that body
In trials against real animals, this approxi- temperatures predicted using different
mation predicted convection coefficients approaches do not vary by more than 0.1°C
within 5% of measured values, except for (O'Connor, unpublished data). Variations
long spindly animals such as snakes when in wind velocity, irregularities in the ground
the error can be up to 15%. Under many profile, obstructions (vegetation, rocks, etc.),
conditions, even these large-sounding errors and turbulent air flows all make the conproduce only modest errors in predicted vective environment much harder to presteady-state body temperatures (Fig. 3). dict in anything but an "average" sense.
Paradoxically, the importance of accurate
estimates of convective heat loss appears to Evaporation
Heat loss by evaporation (or gain by conbe greatest in large animals (Fig. 3), which
are said to be convectively decoupled (Por- densation), because of the large amount of
ter and Gates, 1969). Under the high-radi- heat required to vaporize a volume of water,
ation, low-conduction conditions of the can be a major pathway of heat transfer in
simulations shown in Figure 3, the heat load animals. Evaporative water loss and, thus,
imposed on ectotherms by solar radiation heat loss is a two step process. First, water
is dissipated to a large extent by convective or water vapor must cross whatever cutaheat loss. The smaller convection coeffi- neous barriers separate the tissues from the
cients of larger animals require larger body- atmosphere. Then, water vapor is lost to the
air temperature gradients to balance similar atmosphere by convection. The process is
radiant heat loads (eqs. 1 and 5). Thus sim- analogous to the flow of electricity through
V-
n
15%
186
M. P. O'CONNOR AND J. R. SPOTILA
A1r and Ground
Temperature
Wind Speed
" I
punoj
J
hel ght abov
y
y
y
y
y
y
I
\
. \
\
\
\
• .V.
Temperature
FIG. 4. Idealized vertical profiles of wind speed and air temperature above a substrate wanned by solar radiation
or by conduction of heat from deeper layers of the soil.
two resistors in series. The total resistance
of two electrical resistors in series is the sum
of the individual resistance. Similarly the
total resistance to evaporative water loss is
the sum of dermal and atmospheric resistances to evaporation (eq. 8).
r = r, + re
(8)
where
r = total resistance to water vapor transport (sec/m)
Tj = internal (cutaneous) resistance to
vapor transport (sec/m)
re = external (boundary layer) resistance
to vapor transport (sec/m)
In many cases, either the cutaneous (e.g.,
many desert lizards—Lillywhite and Maderson, 1982; Mautz, 1982; Tracy, 1982) or
atmospheric (e.g., many amphibians—Lillywhite and Licht, 1975; Spotila and Berman, 1976; Tracy, 1976) resistance to evaporation accounts for nearly all of the total
resistance, but in some cases (e.g., phyllomedusine frogs—Shoemaker and McClanahan, 1975;McClanahan# al., 1978; Shoemaker et al., 1987, 1989) both cutaneous
and atmospheric resistances must be considered (eq. 9).
E c = AAp/r = A
ps(Ts) - RHp.(TJ
r, + re
(9)
where
Ec = evaporative water loss (kg/sec)
A = surface area (m2)
Ap = change in vapor density from tissues to air (kg/m3)
r = total resistance to water vapor
transport (s/m)
Ps(Ts) = saturated water vapor density at
tissue temperature (kg/m3)
Pa(Ta) = saturated water vapor density at
air temperature (kg/m3)
RH = atmospheric relative humidity (0
to 1)
T, = internal (cutaneous) resistance to
vapor transport (s/m)
rc = external (boundary layer) resistance to vapor transport (s/m)
Atmospheric resistance is due to convective processes, and the external resistance
to evaporation is related to the thermal convection coefficient by the Lewis Rule (Gates,
1980; Spotila et al., 1991a). Cutaneous
resistance to evaporation is a physiological
characteristic of the animal, may be under
physiological control (Shoemaker et al.,
1987, 1989), and must be determined
empirically. Thus, once the cutaneous resistance has been measured, most of the difficulties in modelling evaporation are similar to those of modelling convective heat
exchange mentioned above. The external
MODELS AND APPROXIMATIONS
resistance to evaporation is related to the
thermal convection coefficient (eq. 6) by the
Lewis Rule (eq. 10, see Gates, 1980; Spotila
etai, 1991a).
re = 0.93rh =
(10)
where
rc = external resistance to water vapor
transfer (s/m)
rh = external resistance to heat transfer
(s/m)
hc = heat transfer (convection) coefficient
(W m-2 °C->)
p = density of air (kg/m3)
cp = specific heat of air at constant pressure (J kg-' °C-')
pcp « 1,200 J m - 3 0 C - '
In modelling evaporation, then, one must
either measure the convection coefficient
empirically (under the appropriate conditions of wind speed and direction, see Spotila and Berman, 1976) or depend on the
same sorts of approximations used to estimate thermal convection coefficients. One
additional complication is that the water
vapor densities (ps in eq. 9) that act as the
driving potential for evaporation are not
linearly related to temperature. Thus, evaporative water and heat loss is not linearly
related to the difference between air and
body temperature. One approach to estimating evaporation is to "linearize" the
relationship between temperature and vapor
pressure using the Penman approximation
(eq. 11, Penman, 1958; Tracy et al., 1984).
Ps(Ts) = ps(Ta) + s(Ts -
(11)
where
ps(Ts) = saturation vapor density at skin
temperature (kg/m3)
Ps(Ta) = saturation vapor density at air
temperature (kg/m3)
Ts = skin temperature (K)
Ta = air temperature (K)
s = slope of the saturation vapor density vs. temperature plot (varies
with temperature) (kg m~3 K~')
Like other linearizing assumptions, the Penman approximation works well when the
difference between the air and skin temper-
187
atures is small. As the difference between
air temperature and that at the surface of
the animal increases, however, the errors in
the body temperatures predicted using the
approximation increase dramatically (Fig.
2). A second approach that is becoming more
popular as microcomputer speeds increase
and inexpensive equation solving software
(e.g., TK! Solver, MathCAD, MAPLE)
improves, is to avoid linearizing approximations and allow the computer to solve
the more complicated non-linear equations.
When numerical solutions are adequate, this
approach has the advantage of being more
accurate. For heuristic models, however, this
must be balanced against the fact that nonlinear equations can be nearly impossible to
solve in analytic form. This is one of the
cases in which heuristic simplicity and
numerical accuracy can come squarely and
routinely into conflict. If one opts for the
simpler linearized model, the numerical
predictions should be checked against nonlinear models for divergence. Bear in mind,
however, that the differences in predictions
between linear and non-linear models may
be small compared to the differences in predictions produced by different reasonable
assumptions about the animal (e.g., convection coefficients) or the environments
(wind speed, air temperature, humidity).
Thus, something akin to a sensitivity analysis is in order. The objective of such analyses is to determine which sorts of simplifications, assumptions, and approximations
are likely to lead to large errors and which
to only small errors. For instance, if solar
radiation accounts for most of the heat
entering the animal and evaporation for
most of the heat leaving the animal, the
predicted body temperatures will likely
depend more strongly on approximations
affecting those heat exchanges than on those
affecting thermal radiation (Spotila et al.,
1991a).
Conduction to the ground
Modelling heat or mass exchange between
an animal and the ground is one of the more
difficult tasks a biophysical ecologist can
attempt. Although at the simplest level the
estimation of conductive heat exchange
188
M. P. O'CONNOR AND J. R. SPOTILA
ers are themselves difficult to model (Porter
and Gates, 1969;CenaandMonteith, 1975a,
b, c, Gates, 1980).
Contact resistance arises from imperfect
contact between the animal and the ground.
Perfect contact is nearly impossible, and
substantial contact resistances are found
even when two smooth metal bars abut one
another. Conduction across a contact resisTb-T,
increases as the area making good con(12) tance
G-IQA,
tact and the smoothness of the contact (even
on a microscopic scale) increase (White,
1984). Conduction of heat away from the
where
animal
in the substrate, as with tissue conG = conduction to the substrate (W)
duction
not only on the characterlo
Kf = conductivity of the skin (Wm- C-') istics of depends
the
substrate
on the geometry
Ag = area of the animal
in contact with the of the problem (see but
Tracy,
1976, 1982;
2
substrate (m )
Gates,
1980;
White,
1984).
It
is often difTb = body temperature of the animal (°C) ficult to know which of these three
processes
Tg = temperature of the substrate (°C)
will
be
the
most
restrictive
and,
hence,
ratedf = thickness of the skin (m)
limiting step, and different models assume
Furthermore, there are no simple, empirical that most of the resistance lies in the skin
rules to circumvent those approximations (Spotila et al., 1973), the tissue as a whole
as existed for the equally complicated case (Tracy, 1976), or external to the tissues
of convection. Heat loss by conduction to (Ackerman, 1980, 1981a, b; Ackerman et
the ground is a three part process. First, heat al., 1985). Each of these models may be
must be transferred to the venter through appropriate under different circumstances,
the tissues of the animal from where it was depending upon the relative conductances
generated (e.g., metabolism) or where it of skin, other tissues, and the ground. Conentered the animal (solar radiation heating tact resistances, although possibly importhe dorsum). Second, the heat energy must tant, are nearly impossible to measure in
cross the interface between the animal and ecologically relevant settings and are usually
the ground through what heat transfer engi- ignored.
neers know as a contact resistance. Finally,
Describing the ground to or from which
the heat must be carried away by conduc- heat is conducted can be tricky. First, the
tion through the substrate, affecting the thermal characteristics of the ground depend
temperature of the substrate near the animal not only on the type of substrate (sands,
in the process. None of these processes is clays, silt stone, rock, wood) but also on the
terribly easy to model.
condition of the substrate (e.g., hydration,
Transfer of heat within the tissues of the compaction, temperature; Rosenberg, 1974).
animal is perhaps the simplest of the three Because soil composition and hydration can
processes. Nonetheless, tissue heat transfer, vary over small spatial scales, and hydration
even assuming the bloodflowdoes not sig- varies with time, it is often difficult to
nificantly affect the rate of "conduction," describe the thermal properties of soils predepends not only on the thermal character- cisely. Secondly, irregularity in shading by
istics of the tissue (i.e., conductivity) but on vegetation and clouds and variation due to
the shape and size of the animal (Bakken, uneven surface features lead to variation in
1981; Tracy, 1976; Gates, 1980). In endo- substrate temperatures on very small spatial
therms, subepidermal lipids and extraepi- and temporal scales. Finally, the very fact
dermal fur or feathers provide layers of that an animal is sitting on a substrate
resistance to conduction, making the char- changes the thermal environment of that
acteristics of underlying layers less impor- substrate. Heat is either added or removed
tant. But the conductances of fur and feath- by conduction, while convection and radiappears quite simple (eq. 12), a plethora of
approximations are buried in that apparent
simplicity. Eq. 12, for instance, is valid only
if heat conduction through the skin is the
rate limiting step rather than conduction
through the animal, conduction through the
substrate or some combination of these
steps.
189
MODELS AND APPROXIMATIONS
ation are, at least partly, blocked. Thus, the
temperature of the soil a few centimeters
from an animal may differ from the temperature of the soil on which the animal
rests. This is intuitive to anyone who has
stood barefoot on a hot beach.
Thus, conduction to the soil can be modelled only approximately, and only using
average temperatures and conductivities
over some spatial and temporal scale. Fortunately, many animals have only minimal
areas of contact with the ground (e.g., the
tips of limbs) or rest on substrates with low
thermal conductivity (e.g., leaf litter, many
woods) that limit the rate of conductive heat
transfer.
Metabolism
In ectotherms, metabolic rates are usually
small enough in comparison to the rate of
heat transfer across the animal's skin to be
ignored (Bartholomew, 1982; Tracy, 1982;
Spotila et ai, 1991a). In large ectotherms,
however, peripheral tissues of the animal
may provide sufficient insulation to allow
the maintenance of significant core to skin
temperature gradients (Carey and Teal,
1969a, b; Carey et ai, 1971, 1982; Block,
1986, 1987; Paladino et ai, 1990). In endotherms, with their combination of higher
metabolic rates and better insulation,
metabolism cannot be ignored. Metabolic
heat is usually assumed to be generated in
a homogeneous core surrounded by insulative layers of tissue. Unless the question
of interest is the distribution of temperature
at different points in the body (Heinrick,
1974,1976,1979), such an approach is usually sufficient to answer questions about heat
exchange with the environment. The major
problem in simulating the thermal effects of
metabolic rate is that metabolic rate is nonlinearly related to body temperature (Q10
effect) and subject to physiological and
behavioral control. Thus, we usually pick a
level of metabolism and compute an energy
balance based on that assumption, then
recompute the energy balance for different
assumptions about metabolism.
Transient energy balance models
All of our discussion so far has centered
on steady-state energy balance models, i.e.,
those models in which the animal is assumed
to have reached a thermal steady-state with
its environment. This assumption is rarely
strictly true, but models making such
assumptions have been so useful that transient energy balance models, those in which
animals may store heat energy (warm up)
or lose energy (cool down), have found little
application in general ecology. One of the
major reasons that transient models are
rarely used is that they are much more computationally complex than steady-state
models. Steady-state models can be formulated with algebraic, if sometimes nonlinear, equations. Transient models, on the
other hand, are formulated as differential
(or difference) equations which are more difficult to solve analytically or numerically.
Better computers and more available software are making it possible to compute
numerical solutions, but analytic solutions
remain so elusive that biophysical ecologists have yet to establish a recognized
vocabulary for describing models of heating
and cooling. Furthermore, in biological
terms, masses and capacitances (which can
be difficult to quantify) are as important as
conductances in transient models. So, with
the exception of studies of rates of heating
and cooling (Cowles, 1958; Bartholomew
and Tucker, 1964; Grigg etai, 1979; Turner
and Tracy, 1983, 1985; Fraser and Grigg,
1984) and some predictions about the temperatures of dinosaurs (e.g., Spotila et ai,
1973, 1991*; Spotila, 1980; Tracy et ai,
1986; Turner and Tracy, 1986), very little
work has focused on transient energy balance models (but see Stevenson, 1985a, b;
Turner, 1987). Thus, we omit a fuller discussion of transient models at this time.
SOME ADVICE
Much of our discussion of heat transfer
has been about the difficulties of modelling.
It may seem that nothing should be modelled. We do not advocate such a position.
Rather, we hope to point out where and
under which conditions models can be weak
and how their utility is circumscribed. It is
the job of the modeller to try to generate
models appropriate to the questions asked,
and to know the weaknesses of those models
just as researchers must use appropriate sta-
190
M. P. O'CONNOR AND J. R. SPOTILA
tistics and know something about when tests
might lead them astray. It is our conviction
that most questions about the biophysics of
ecology can be approached by modelling
when useful and that many of these problems can be modelled by most ecologists.
On a more positive note, we offer the following advice to those who have a question
that they think a model can help them
understand. 1) Try to build the model. If
the effort does nothing else, it will usually
help define the parts of the question more
clearly. It is often useful to modify an older
model built for a similar purpose. The older
model may not answer the question at hand,
but it can serve as a good starting point and
has the advantage of showing how someone
else thought out the problem. 2) Build a
simple model first. The simple model can
be expanded as needed. This strategy has
several advantages. Simple models are just
plain easier to build. It is easier to keep track
of all of the parts of a simple model than is
the case with complex models. Furthermore, one can intuit the behavior of simple
models more easily than that of complicated
models. For complicated problems, the
model can be built up of small parts each
of which is built and debugged separately.
This is a page right out of the structured
programmer's handbook. It is easier to do
several small things well and put them
together when you know they work than to
build a huge model with many parts. Furthermore, as above, it is easier to debug the
smaller parts separately. Models, like computer programs, always require debugging.
One other advantage of this process is that
the debugging process can be very instructive if the parts of the model can be kept
straight. Most of our intuitions about the
physical processes that biophysical ecologists model are limited. When models start
behaving in unexpected ways, one must
always ask whether the problem is with the
model or with our expectations. In an edifying proportion of instances, the models
simply refine our intuitions. In other
instances empirical data can lead to a refinement of the model. 3) Check the model frequently during construction by picking simple or extreme cases whose behavior you
know or can guess. 4) Complex thermal
environments demand some special attention during model building. For example,
in estimating convection coefficients, it is
important to use the wind speeds and air
temperatures found where the animal is
located, not at 1.5 m above the ground where
they are typically measured (see Bakken,
1990). Similarly, if conduction to the ground
is an important mode of heat transfer, think
carefully about the ground temperatures and
soil properties you use. 5) Look for any
apparent behavioral or physiological thermoregulatory adaptations in your study animal. They may point to important modes
of heat transfer and/or important ways in
which the animal modifies heat transfer.
Variations in color, orientation to the sun,
shuttling between microhabitats, waterproof coatings, panting, countercurrent heat
exchangers, and large flat surfaces (e.g.,
wings, fins, ears, sails of pelycosaurs) have
all been shown to affect the biophysical ecology of animals. 6) To repeat a classic piece
of advice, remember that model predictions
are just that. Divergence between predictions and empirical data can indicate problems with the model or differences between
what we assumed about the animal and reality. The former problem requires a refined
model, the latter and more exciting possibility may require rethinking of the model
or experiments to determine what important process was missed in building the
model.
Finally, do not be shy about talking with
biophysical ecologists. Most are genuinely
happy tofindnew applications for their craft,
willing to collaborate when appropriate, and,
contrary to rumor, rarely cannibalistic.
ACKNOWLEDGMENTS
The manuscript was prepared with support from grants from the Department of
Energy (DE-FG02-88ER 60727) and the
National Science Foundation (DCB-9019780). Travel and lodging funds were provided by the American Society of Zoologists.
REFERENCES
Ackerman, R. A. 1980. Physiological and ecological
aspects of gas exchange by sea turtle eggs. Amer.
Zool. 20:575-584.
MODELS AND APPROXIMATIONS
Ackerman, R. A. 1981a. Growth and gas exchange
ofembryonic sea turtles (Chelonia, Caretta). Copeia
1981:757-765.
Ackerman, R. A. 19816. Oxygen consumption by sea
turtles (Chelonia, Caretta) eggs during development. Physiol. Zool. 54:316-324.
Ackerman, R. A., R. C. Seagrave, R. Dmi'el, and A.
Ar. 1985. Water and heat exchange between
parchment-shelled reptile eggs and their surroundings. Copeia 1985:703-711.
Bakken, G. S. 1981. A two-dimensional operativetemperature model for thermal energy management by animals. J. Thermal Biol. 6:23-30.
Bakken, G. S. 1990. Arboreal perch properties and
the operative temperature experienced by small
animals. Ecology 70:922-930.
Bakken, G. S. and D. M. Gates. 1975. Heat-transfer
analysis of animals: Some implications for field
ecology, physiology, and evolution. In D. M. Gates
and R. Schmerl (eds.), Perspectives in biophysical
ecology, pp. 255-290. Springer-Verlag, New York.
Bakken, G. S., D. M. Gates, T. H. Strunk, and M.
Kleiber. 1974. Linearized heat transfer relations
in biology. Science 183:976-978.
Bakken, G. S., W. R. Santee, and D. J. Erskine. 1985.
Operative and standard operative temperature:
Tools for thermal energetic studies. Amer. Zool.
25:933-944.
Bartholomew, G. A. 1982. Physiological control of
body temperature. In C. Gans and F. H. Pough
(eds.), Biology of the Reptilia, Vol. 12, pp. 167211. Academic Press, London.
Bartholomew, G. A. and V. A. Tucker. 1964. Size,
body temperature, thermal conductance, oxygen
consumption, and heart rate in Australian varanid
lizards. Physiol. Zool. 37:341-354.
Bartlett, P. N. and D. M. Gates. 1967. The energy
budget of a lizard on a tree trunk. Ecology 48:316322.
Block, B. A. 1986. Structure of the brain and eye
heater tissue in marlins, sailfish, and spearfishes.
J. Morph. 190:169-189.
Block, B. A. 1987. Strategies for regulating brain and
eye temperatures: A thermogenic tissue in fish. In
P. Dejours, L. Bolis, C. R. Taylor, and E. R. Weibel
(eds.), Comparative physiology: Life in water and
on land, pp. 401^120. Fidia Res. Series, IX-Liviana Press, Padova (Springer-Verlag, New York).
Campbell, G. S. 1977. An introduction to environmental biophysics. Springer-Verlag, New York.
Carey, F. G. and J. M. Teal. 1969a. Mako and porbeagle: Warm-bodied sharks. Comp. Biochem.
Physiol. 28:199-204.
Carey, F. G. and J. M. Teal. 19696. Regulation of
body temperature by the bluefin tuna. Comp. Biochem. and Physiol. 28:205-213.
Carey, F. G., J. W. Kanwisher, O. Brazier, G. Gabrielson, J. G. Casey, and H. L. Pratt. 1982. Temperature and activities of a white shark, Carcharodon carcharias. Copeia 1982:254-260.
Carey, F. G., J. M. Teal, J. W. Kanwisher, K. V. Lawson, and J. S. Beckett. 1971. Warm-bodied fish.
Amer. Zool. 11:135-144.
Casey, T. M. 1992. Biophysical ecology and heat
exchange in insects. Amer. Zool. 32:225-237.
191
Cena, K. and J. L. Monteith. 1975a. Transfer processes in animal coats: I. Radiative processes. Proc.
R. Soc, London 188:377-393.
Cena, K. and J. L. Monteith. 19756. Transfer processes in animal coats: II. Conduction and convection. Proc. R. Soc, London 188:395-411.
Cena, K. and J. L. Monteith. 1975c. Transfer processes in animal coats: HI. Water vapour diffusion.
Proc. R. Soc., London 188:413-423.
Christian, K. A. and C. R. Tracy. 1981. The effect
of the thermal environment on the ability of hatchling Galapagos land iguanas to avoid predation
during dispersal. Oecologia 49:218-223.
Christian, K. A. and C. R. Tracy. 1985. Measuring
air temperatures in field studies. J. Thermal Biol.
10:55-56.
Christian, K. A., C. R. Tracy, and W. P. Porter. 1983.
Seasonal shifts in body temperature and use of
microhabitats by Galapagos land iguanas (Conolophus pallidus). Ecology 64:463-468.
Christian, K. A., C. R. Tracy, and W. P. Porter. 1984.
Physiological and ecological consequences of
sleeping site selection by the Galapagos land iguana
(Conolophus pallidus). Ecology 65:752-758.
Cowles, R. B. 1958. Possible origin of dermal temperature regulation. Evolution 12:347-357.
Crawford, K. M., J. R. Spotila, and E. A. Standora.
1983. Operative environmental temperatures and
basking behavior of the turtle, Pseudemys scripta.
Ecology 64:989-999.
Feder, M. E. 1982. Thermal ecology of neotropical
lungless salamanders (Amphibia: Plethodontidae): Environmental temperatures and behavioral
responses. Ecology 63:1665-1674.
Feder, M. E. 1983. Integrating the ecology and physiology of plethodontid salamanders. Herpetologka 39:291-310.
Fraser, S. and G. C. Grigg. 1984. Control of thermal
conductance is insignificant to thermoregulation
in small reptiles. Physiol. Zool. 57:392-400.
Gates, D. M. 1980. Biophysical ecology. SpringerVerlag, New York.
Gieger, K. S. 1965. The climate near the ground.
Harvard University Press, Cambridge, Massachusetts.
Grant, B. W. and A. E. Dunham. 1988. Thermally
imposed time constraints on the activity of the
desert lizard Sceloporus merriami. Ecology 69:167—
176.
Grant, B.W. and W. P. Porter. 1992. Modeling global
macroclimate constraints on ectotherm energy
budgets. Amer. Zool. 32:154-178.
Grigg, G. C , C. R. Drane, and G. P. Courtice. 1979.
Time constants of heating and cooling in the eastern water dragon, Physignathus leseurii, and some
generalizations about heating and cooling in reptiles. J. Therm. Biol. 4:95-103.
Halliday, E. C. and T. J. Hugo. 1963. Photodermoplanimeter. J. Appl. Physiol. 18:1285.
Hardy, J. D., A. P. Gagge, and J. A. J. Stolwijk. 1970.
Physiological and behavioral temperature regulation. Charles C. Thomas Publishers, Springfield,
Illinois.
Heinrich, B. 1974. Thermoregulation in endothermic
insects. Science 185:747-756.
192
M. P. O'CONNOR AND J. R. SPOTILA
Heinrich, B. 1976. Heat exchange in relation to blood
flow between thorax and abdomen in bumblebees.
J. Exper. Biol. 64:561-585.
Heinrich, B. 1979. Keeping a cool head: Honeybee
thermoregulation. Science 205:1269-1271.
James, F. C. and W. P. Porter. 1979. Behavior-microclimate relationships in the African rainbow lizard, Agama agama. Copeia 1979:585-593.
Kingsolver, J. G. 1989. Weather and population
dynamics of insects: Integrating physiological and
population ecology. Physiol. Zool. 62:314-334.
Kingsolver, J. G. and M. A. R. Koehl. 1985. Aerodynamics, thermoregulation, and the evolution of
insect wings: Differential scaling and evolutionary
change. Evolution 39:488-504.
Kingsolver, J. G. and W. B. Watt. 1983. Thermoregulatory strategies in Colias butterflies: Thermal
stress and the limits to adaptation in temporally
varying environments. Am. Natur. 121:32-55.
Levins, R. 1968. Evolution in changing environments, Some theoretical explorations. Princeton
Univ. Press, Princeton, New Jersey.
Lillywhite, H. B. and P. Licht. 1975. A comparative
study of integumentary mucous secretions in
amphibians. Comp. Biochm. Physiol. A 51 A:937941.
Lillywhite, H. B. and P. F. A. Maderson. 1982. Skin
structure and permeability. In C. Gans and F. H.
Pough (eds.), Biology of the Reptilia, Vol. 12, pp.
397-442. Academic Press, London.
Mautz, W. J. 1982. Patterns of evaporative water
loss. In C. Gans and F. H. Pough (eds.), Biology
of the Reptilia, Vol. 12, pp. 443-481. Academic
Press, London.
McClanahan, L. L., J. N. Stinner, and V. H. Shoemaker. 1978. Skin lipids, water loss, and energy
metabolism in a South American tree frog (Phyllomedusa sauvagei). Physiol. Zool. 51:179-187.
McCullough, E. M. and W. P. Porter. 1971. Computing clear day solar radiation spectra for the
terrestrial ecological environment. Ecology 52:
1008-1015.
Mitchell, J. W. 1976. Heat transfer from spheres and
other animal shapes. Biophys. J. 16:561-569.
Monteith, J. L. 1973. Principles of environmental
physics. Edward Arnold Ltd., London.
Nevins, R. G. and M. A. Darwish. 1970. Heat transfer through subcutaneous tissue as heat generating
porous material. In J. D. Hardy, A. P. Gagge, and
J. A. J. Stolwijk (eds.), Physiological and behavioral temperature regulation, pp. 281-301. Charles
C. Thomas Publishers, Springfield, Illinois.
O'Connor, M. P. 1989. Thermoregulation in anuran
amphibians: Physiology, biophysics, and ecology.
Ph.D. Diss., Colorado State University, Fort Collins, Colorado.
Paladino, F. V., M. P. O'Connor, and J. R. Spotila.
1990. Metabolism of leatherback turtles, gigantothermy, and thermoregulation of dinosaurs.
Nature 344:858-860.
Parton, W. J. and J. A. Logan. 1981. A model for
diurnal variation in soil and air temperature. Agric.
Meteorol. 23:205-216.
Penman, H. L. 1958. Natural evaporation from open
water, bare soils, and grass. Proc. R. Soc, London
A 193:120-140.
Peterson, C. R. 1987. Daily variation in the body
temperatures of free-ranging garter snakes. Ecology 68:160-169.
Porter, W. P. 1989. New animal models and experiments for calculating growth potential at different
elevations. Physiol. Zool. 62:286-313.
Porter, W. P. and D. M. Gates. 1969. Thermodynamic equilibria of animals with environment.
Ecological Monographs 39:245-270.
Porter, W. P. and F. C. James. 1979. Behavioral
implications of mechanistic ecology. II: The African rainbow lizard, Agama agama. Copeia 1979:
594-619.
Porter, W. P., J. W. Mitchell, W. A. Beckman, and C.
B. DeWitt. 1973. Behavioral implications of
mechanistic ecology: Thermal and behavioral
modeling of desert ectotherms and their microenvironments. Oecologia 13:1-54.
Porter, W. P., J. W. Mitchell, W. A. Beckman, and C.
R. Tracy. 1975. Environmental constraints on
some predator-prey interactions. In D. M. Gates
and R. Schmerl (eds.), Perspective in biophysical
ecology, pp. 347-363. Springer-Verlag, New York.
Porter, W. P. and C. R. Tracy. 1974. Modeling the
effects of temperature changes on the ecology of
the garter snake and leopard frog. In J. W. Gibbons
and R. R. Scharitz (eds.), Thermal ecology, pp.
594-609. Technical Information Center, Office of
Information Services, U.S. Atomic Energy Commission.
Porter, W. P. and C. R. Tracy. 1983. Biophysical
analyses of energetics, time-space utilization, and
distributional limits. In R. B. Huey, E. R. Pianka,
and T. W. Schoener (eds.), Lizard ecology, Studies
of a model organism, pp. 55-83. Harvard University Press, Cambridge.
Riechert, S. E. and C. R. Tracy. 1975. Thermal balance and prey availability: Bases for a model relating web-site characteristics to spider reproductive
success. Ecology 56:265-284.
Rose, C. W. 1966. Agricultural physics. Pergammon
Press, London.
Rosenberg, N. J. 1974. Microclimate: The biological
environment. Wiley, New York.
Roughgarden, J., W. P. Porter, and D. Heckel. 1981.
Resource partitioning of space and its relationship
to body temperature in Anolis populations. Oecologia 50:256-264.
Scott, J. R., C. R. Tracy, and D. Pettus. 1982. A
biophysical analysis of daily and seasonal utilization of climate space by a montane snake. Ecology 63:482^193.
Shoemaker, V. H., M. A. Baker, and J. P. Loveridge.
1989. Effect of water balance on thermoregulation
in waterproof frogs (Chiromantis and Phyllomedusa). Physiol. Zool. 62:133-146.
Shoemaker, V. H. and L. L. McClanahan. 1975.
Evaporative water loss, nitrogen excretion and
osmoregulation in phyllomedusine frogs. J. Comp.
Physiol. 100:331-345.
Shoemaker, V. H., L. L. McClanahan, P. C. Withers,
S. S. Hiliman, and R. C. Drewes. 1987. Ther-
MODELS AND APPROXIMATIONS
moregulatory response to heat in the water proof
frogs Phyllomedusa and Chiromantis. Physiol.
Zool. 60:365-372.
Spotila, J. R. 1972. Role of temperature and water
in the ecology of lungless salamanders. Ecol.
Monogr. 42:95-125.
Spotila, J. R. 1980. Constraints of body size and
environment on the temperature regulation of
dinosaurs. In R. D. K. Thomas and E. C. Olson
(eds.), A cold look at warm-blooded dinosaurs, pp.
233-252. American Association for the Advancement of Sciences Symposium 28. Westview Press,
Boulder, Colorado.
Spotila, J. R. and E. N. Berman. 1976. Determination
of skin resistance and the role of the skin in controlling water loss in amphibians and reptiles.
Comp. Biochem. Physiol. 55:407-411.
Spotila, J. R., R. E. Foley, and E. A. Standora. 1990.
Thermoregulation and climate space of the slider
turtle. In J. W. Gibbons (ed.), Life history and
ecology of the slider turtle, Chapter 22, pp. 288298. Smithsonian Institution Press, Washington,
D.C.
Spotila, J. R. and D. M. Gates. 1975. Body size,
insulation and optimum body temperatures of
homeotherms. In D. M. Gates and R. Schmerl
(eds.), Perspectives in biophysical ecology, pp. 291 301. Springer-Verlag, New York.
Spotila, J. R., P. W. Lommen, G. S. Bakken, and D.
M. Gates. 1973. A mathematical model for body
temperatures of large reptiles: Implications for
dinosaur ecology. Am. Nat. 107:391-404.
Spotila, J. R., M. P. O'Connor, and G. S. Bakken.
1991a. Biophysics of heat and mass transfer in
amphibians. In M. E. Feder and W. W. Burggren
(eds.), Ecological physiology of Amphibia, Chap.
4. University of Chicago Press, Chicago, Illinois.
Spotila, J. R., M. P. O'Connor, P. Dodson, and F. V.
Paladino. 19916. Hot and cold running dinosaurs: Body size, metabolism and migration. Modem geology 16:203-227.
Spotila, J. R., O. H. Soule, and D. M. Gates. 1972.
The biophysical ecology of the alligator: Heat
energy budgets and climate spaces. Ecology 53:
1094-1102.
Standora, E. A. 1982. A telemetric study of the thermoregulatory behavior and climate-space of freeranging yellow-bellied turtles, Pseudemys scripta.
Ph.D. Diss., University of Georgia, Athens, Georgia.
193
Stevenson, R. D. 1985a. Body size and limits to the
daily range of body temperature in terrestrial ectotherms. Am. Nat. 125:102-112.
Stevenson, R. D. 19856. The relative importance of
behavioral and physiological adjustments controlling body temperature in terrestrial ectotherms. Am. Nat. 126:362-386.
Tracy, C. R. 1976. A model of the dynamic exchanges
of water and energy between a terrestrial amphibian and its environment. Ecol. Monogr. 46:293326.
Tracy, C. R. 1982. Biophysical modeling in reptilian
physiology and ecology. In C. Gans and F. H.
Pough (eds.), Biology of the Reptilia, Vol. 12, pp.
275-321. Academic Press, London.
Tracy, C. R., J. S. Turner, and R. B. Huey. 1986. A
biophysical analysis of possible thermoregulatory
adaptations in sailed pelycosaurs. In N. Hotton,
III, P. D. MacLean, J. J. Roth, and E. C. Roth
(eds.), The ecology and biology of mammal-like
reptiles, pp. 195-206. Smithsonian Institution
Press, Washington, D.C.
Tracy, C. R. et al. 1984. Errors resulting from linear
approximations in energy balance equations. J.
Therm. Biol. 9:261-264.
Turner, J. S. 1987. On the transient temperatures of
ectotherms. J. Therm. Biol. 12:207-214.
Turner, J. S. and C. R. Tracy. 1983. Blood flow to
appendages and the control of heat exchange in
American alligators. Physiol. Zool. 56:195-200.
Turner, J. S. and C. R. Tracy. 1985. Body size and
the control of heat exchange in alligators. J. Therm.
Biol. 10:9-11.
Turner, J. S. and C. R. Tracy. 1986. Body size, homeothermy and the control of heat exchange in
mammal-like reptiles. In N. Hotton, III, P. D.
MacLean, J. J. Roth, and E. C. Roth (eds.), The
ecology and biology of mammal-like reptiles, pp.
185-194. Smithsonian Institution Press, Washington, D.C.
Vogel, S. 1981. Life in moving fluids: The physical
biology of flow. Princeton University Press,
Princeton, New Jersey.
White, F. M. 1984. Heat transfer. Addison-Wesley,
Reading, Massachusetts.
Zimmerman, L. C. and C. R. Tracy. 1989. Interactions between the environment and ectothermy
and herbivory in reptiles. Physiol. Zool. 62:374409.