Relatively simple, precise methods to analyze temperature transients in ectotherms

Journal of Thermal Biology 26 (2001) 121–132

Relatively simple, precise methods to analyze temperature transients in ectotherms Margaret A. Voss*, F. Reed Hainsworth Department of Biology, Syracuse University, Syracuse, NY 13244-1270, USA Received 8 February 2000; accepted 20 June 2000

Abstract Relatively complex core-shell models have been used to precisely characterize times and temperatures for ectotherms. There is a simpler method using a second-order analysis of heat flux. We derive the method from an equivalent mechanical system, correct some previously published inaccuracies, and show how to use the method by analyzing thermal transients for House Wren eggs under natural conditions. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Core-shell models; Incubation; Second-order models; Thermal transients

1. Introduction As animals heat and cool, the product of their total resistance to heat transfer, R (8C W ÿ 1) and heat capacitance, C(J 8C ÿ 1) varies. The product has the dimension time and would be the time constant for a negative exponential process. Variations in RC are clear from many analyses showing ln(TB ÿ TB1 ) over time where the slope, the rate constant, is not constant, particularly near zero time (Bartholomew and Lasiewski, 1965; Fraser and Grigg, 1984; Gonzalez Gonzalez and Porcell, 1985; Rismiller and Heldmaier, 1985; Turner, 1987a). Commonly used first-order negative exponential models for temperature change in animals, where TBt ¼ ðTB0 ÿ TB1 Þeÿkt þ TB1; assume no change in RC through use of a single rate constant. Nevertheless, there is considerable evidence that RC varies, at least briefly, when any object changes from being cooled to being heated and vice versa. Even dead organisms and eggs that have not yet developed circulatory systems show variation in the slope for ln(TB ÿ TB1 Þ versus *Corresponding author. Department of Biology, State University of New York College at Potsdam, 44 Pierrepont Avenue, Potsdam, NY 13676-2294, USA. Tel.: +1-315-2672261; Fax: +1-315-267-3170. E-mail address: [email protected] (M.A. Voss).

time. Many physiologists have made this type of measurement as a ‘‘control’’ for ectotherm heating and cooling experiments (Spray and May, 1972; Rice and Bradshaw, 1980; Fraser and Grigg, 1984; Daniels et al., 1987; Tazawa et al., 1988). Experiments by Turner (1994a, 1994b, 1997) also show the importance of any change in temperature for variation in RC. As objects are heated to cool, or cooled to heat, the accompanied variation in RC will violate the first-order assumption for some time in any experiment involving cooling or heating. What apparently is not understood is how RC variation can be quantified simply and easily (Turner, 1987a). The variation produces second-order negative exponential dynamics, and there are well-developed analytical methods to find the behaviors of secondorder systems (e.g., Ford, 1955). Many instead have constructed complex core-shell models to explore consequences for RC variation (Porter and Gates, 1969; Tracy, 1976, 1982; Spotila et al., 1992; O’Connor, 1999; see the appendix). These models were originally designed to predict thermal equilibria for animals over time (Porter and Gates, 1969). The animal properties influencing rates of heat flux included metabolic rate, evaporative loss rate, and resistances and capacitances of various shells (e.g., fat, skin, fur, feathers) about a core. Because these models include variations in RC for

0306-4565/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 6 - 4 5 6 5 ( 0 0 ) 0 0 0 3 2 - 2

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Nomenclature A

B

C f ðtÞ k R t TB TB0

the first coefficient required for the general solution of a second-order differential equation the second coefficient required for the general solution of a second-order differential equation an object’s heat capacitance (J8 C ÿ 1) the forcing function governing a mechanical or thermal transient the rate constant associated with heating or cooling (min ÿ 1) the total resistance to heat transfer from the center of an object (8C W ÿ 1) time (min) body temperature at an animal’s core (8C) body temperature at an animals’s core (8C) at time zero

shells, they can be used to predict times to heat and cool (see O’Connor, 1999). Nevertheless, this is cumbersome and complicated from the many parameters required to specify heat flux through the shells and with an environment (see the appendix). Although core-shell models show how variation in RC can be important, they may not be necessary if there are sufficient data for transients that meet the requirements for a simpler second-order analysis. Because of their complexity, coreshell models may have been an impediment to analysis of times to cool and heat in ectotherms, an area of considerable interest in thermal biology (e.g., Bartholomew and Lasiewski, 1965; Spray and May, 1972; May, 1976; Bakken, 1976; Rice and Bradshaw, 1980; Webb and King, 1983; Fraser and Grigg, 1984; Turner and Tracy, 1985; Rismiller and Heldmaier, 1985; Daniels et al., 1987; Pages et al., 1991; Carrascal et al., 1992; Holland et al., 1992; Brill et al., 1994; Belliure et al., 1996; Kruuk et al., 1997). Thus, the purpose of this paper is to explain the basis and requirements for the simpler second-order analysis, to give the methods for analysis, and to show how to use them with data on temperature transients for House Wren eggs under natural conditions.

2. Methods from an equivalent mechanical system A mechanically equivalent system can be used to explain how transitions from heating to cooling, and vice versa, operate to change the rate constants during cooling and heating. We use mechanics to avoid a coreshell structure so methods will be as simple as possible. The specific equivalence for cooling following heating is

TBt TB1

Tbp TE0 TEt TE1

Te

x on

body temperature at an animal’s core (8C) at time t the asymptotic temperature at the core of a body allowed to equilibrate with its environment (8C) brood patch temperature (8C) the core temperature of an egg at time 0 (8C) the core temperature of an egg at time t (8C) the asymptotic temperature of an egg at equilibrium with its environmment (8C); Tbp during heating. the operative environmental temperature (8C), or TB1 when heat production minus evaporative heat loss=0 the damping ratio of an oscillating system (dimensionless) the natural frequency of an oscillating system (time ÿ 1)

induced damped oscillation of the center of a mass attached to a spring with the other end of the spring rigidly attached. This is a common example in texts on differential equations under vibrating or oscillating systems (e.g., Ford, 1955). The mass is set into motion by first applying a force, f(t) to lengthen the spring, and then releasing that force. By the first law of thermodynamics the additional applied force times distance (=work) is equivalent to heat added to a system. When the heat is ‘‘released,’’ how will the system respond during cooling? The answer depends on the forces operating within and on the system. If there were no friction, Newton’s second law of motion as mðd2 y=dt2 Þ ¼ ÿhy þ f ðtÞ would apply, where y is distance, t is time, and m and h are proportionality constants. But a spring with the mass moving in air is resisted in its motion by a force (friction) proportional by n to its rate of movement, so d2 y dy ¼ ÿhy ÿ n þ f ðtÞ: ð1Þ dt2 dt When the system is not influenced by any other forces, and when f ðtÞ ¼ 0, Eq. (1) can be written

m

d2 y dy þ b2 y ¼ 0, þ 2a ð2Þ dt2 dt where b2 is h=m and 2 a is n=m: This form of a secondorder, linear differential equation is connected with a large number of mechanical vibrations, electrical oscillations, etc. (Ford, 1955). What are the equivalents for heat flow from the center of a simple object such as an egg with no circulatory system? Loss of heat (y) proceeds with proportionality constant k, and the rate of heat loss from the center, dy=dt is opposed with proportionality constant j by gain

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of heat from any non-uniform distribution of heat within the object (the frictional equivalent), so

The solution of Eq. (2) is found from solving the algebraic auxiliary equation

d2 y dy ¼ ÿky ÿ j þ f ðtÞ, ð3Þ dt2 dt and, as in (2) above, if f ðtÞ ¼ 0, 2a is j=c and b2 is k=c,

m2 þ 2am þ b2 ¼ 0

d2 y dy þ b2 y ¼ 0: þ 2a dt2 dt A most important point is that a mechanical system or its heat equivalent will be influenced by the ‘‘forcing’’ function f ðtÞ. This is what is used to add heat to a system. When f ðtÞ varies with time, as it must, it is necessary to set the second-order equation equal to f ðtÞ

If the roots are imaginary, or a5b, the specific solutions contain trigonometric functions that characterize the oscillation of a system that is said to be ‘‘under damped’’. If the roots are real and equal, or a ¼ b, the system is said to be ‘‘critically damped’’. The roots are real and unequal when a > b, as for the case we are interested in, and the system is then said to be ‘‘over damped’’. When this is true the general solution for Eq. (2) is pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 2 2 2 y ¼ Aeÿðaþ a ÿb Þt þ Beÿðaÿ a ÿb Þt: ð6Þ

c

d2 y dy þ 2a ð4Þ þ b2 y ¼ f ðtÞ: dt2 dt When the forcing function is a constant, so first and second derivatives=0, then the solution for Eq. (2) plus a constant (the asymptotic temperature TB1 ) will completely describe the dynamics. The simplest representation for f ðtÞ ¼ constant is an instantaneous application of a constant source of heat over time that is removed instantaneously. This has been a common method used by biologists when an ectotherm was quickly moved from a constant lowtemperature environment to a constant high-temperature environment and back (e.g., Bartholomew and Tucker, 1963; Turner, 1987a). This could also represent the way in which a brood patch functions during egg heating and cooling or when an ectotherm quickly moves from sun to shade and back. Turner’s (1994a,b) experiments with an artificial brood path applied to chicken eggs have evaluated the more complex case for f ðtÞ 6¼ constant. In some of these experiments, for example, f ðtÞ was a sine function. The equipment dictated that f ðtÞ 6¼ constant occurred during cooling as well as heating. This was because the brood path remained on the egg during cooling, which does not normally occur. Nevertheless, from these experiments, and the behavior of a large number of equivalent mechanical and electrical systems, it is well known that complex phenomena (such as impedance, resonance, and beats) occur when f ðtÞ 6¼ constant but is a periodic function (Ford, 1955; Turner, 1997). It is still possible to use a relatively simple analysis when f ðtÞ is a step function. Then an appropriate time during f ðtÞ can be selected when the diminishing influence of its brief transient is effectively nil. Because any f ðtÞ produces unequal heat distribution (Turner, 1994a,b; 1997), or variation in RC, the effect will be part of the second-order analysis through the 2aðdy=dtÞ in Eq. (2), which represents the retarded rate of heat loss or gain at the center. We thus proceed to solve Eq. (2) for such a situation and then add a constant for the complete solution.

ð5Þ

for its 2 roots. The roots are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÿa  a2 ÿ b2 :

A mechanical example is the weight attached to a spring moving in a viscous medium such as molasses. Since a and b are constants, Eq. (6) can be written y ¼ Aeÿk1 t þ Beÿk2 t :

ð7Þ

The next steps are to find the specific values for A and B from this general solution and the constant. Turner (1987a) gave a complete specific solution, but there was a sign error. We point this out here with a detailed description of the solution. In Eq. (7) k1 and k2 are the rate constants governing two temperature transients, and A and B are the coefficients required for a second-order solution. As Turner (1987a) pointed out, the complete solution for heating and cooling when f ðtÞ ¼ constant is to let y represent a proportionate change in temperature difference relative to time zero, or y¼

ðTB ÿ TB1 Þt ; ðTB ÿ TB1 Þ0

ð8Þ

where TB1 is the constant. Thus, Eqs. (7) and (8) yield y¼

ðTB ÿ TB1 Þt ¼ Aeÿk1 t þ Beÿk2 t : ðTB ÿ TB1 Þ0

ð9Þ

Since this is the complete general solution, values for the coefficients now can be found by evaluating Eq. (9) at t ¼ 0 when y ¼ 1 so A þ B ¼ 1 and when dy=dt ¼ 0 so 2Ak1 ÿ Bk2 ¼ 0. These conditions are satisfied by the following specific values for the coefficients: A¼ÿ

k2 ; k1 ÿ k 2

and

B¼

k1 : k 1 ÿ k2

Thus, the specific solution for Eq. (9) is y¼

ðTB ÿ TB1 Þt ðTB ÿ TB1 Þ0

¼ÿ

k2 k1 eÿk1 t þ eÿk2 t : k 1 ÿ k2 k1 ÿ k 2

ð10Þ

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This equation is similar to Eq. (15a) in Turner (1987a) with the exceptions that Eq. (15a) was written using time constants, and the second coefficient in Turner’s equation (15a) should carry a negative sign. This is crucial because this produces retardation in the rate of cooling and heating. Eq. (10) can be used to precisely find temperatures over time if the asymptotes, TB1 , are known so k1 and k2 can be found analytically.

3. Finding asymptotes There are two independent ways to find TB1 ’s for ectotherms. The first is the construction and use of Te thermometers as described in Bakken et al. (1985) and Bakken (1992). When metabolic heat production equals evaporative heat loss, measured Te over time can be used in Eq. (10) for TB1 [Bakken et al., 1985, Eq. (5)]. However, it is not always possible to use a Te thermometer, such as when an environment is not accessible. In some situations metabolic heat production for ectotherms may exceed evaporative heat loss (Walsburg and Wolf, 1996). When Te thermometers are unfeasible, or insufficient, a more data intensive method of curve fitting by least-squares regression can be used to find TB1 from recorded thermal transients (Riggs, 1963; Bakken, 1976). The method can be used in situations when a step change in TB1 is used or suspected. Since this method is best understood by example, we shall return to a specific description in Section 6.1 after we complete the general methods for analysis.

4. Finding rate constants Values for k1 and k2 are found by using ‘‘correct’’ TB1 values. General methods for both first- and secondorder analyses of negative exponential functions are found in Riggs (1963), and specifically for temperature transients in Turner (1987a). Second-order changes in temperature at the center of an object will be governed by [k1 =ðk1 ÿ k2 ފeÿk2 t after an initial transient. As a result, a linear regression near the end of cooling or heating graphed as ln ½ðTBt ÿ TB1 Þ=ðTB0 ÿ TB1 ފ versus time will have a slope of ÿk2 if f ðtÞ ¼constant (Fig. 1; also see Fig. 1 in Turner, 1987a). The antilog of this function’s intercept, B ¼ k1 =ðk1 ÿ k2 Þ, is then solved for k1 ¼ Bk2 =ðB ÿ 1Þ to find k1 from B and k2 . The information required to predict temperatures and times precisely using the second-order model (Eq. (10)) consists of a correct value for TB1 and the two accompanying rate constants. Although this method is data intensive, it is simpler than use of core-shell models, which are derived from numerous spatially explicit considerations for variation in RC (Porter and Gates,

Fig. 1. A first-order model of a transient, by definition, has an intercept for y=0. The rate constants for the second-order model can be found using regression analysis. A best fit line, determined by least-squares regression, was used to find k2 ðk2 ¼ ÿslope). When k2 was known, k1 was found from k1 ¼ Bk2 =ðB ÿ 1Þ. Data from the start and end of an apparent transient (shaded areas) were excluded to find ln B and k2 to minimize errors (see text). Note for cooling or heating eggs, TE is substituted in y for TB .

1969; Spotila et al., 1973; Turner, 1987b; see the appendix). Two derived parameters often used to describe the second-order model represented by Eq. (4) are the damping ratio, x (dimesionless) and the natural frequency, on (time ÿ 1) (Turner, 1987a). These are parts of the coefficients in Eq. (4) as a is on x and b is on . Thus, the damping ratio and natural frequency can be expressed entirely ffiffiffi terms of rate constants, with pffiffiffiffiffiffiin x ¼ ðk1 þ k2 Þ=2 k1 k2 and on ¼ k2 . Note that these were derived from Eqs. (18) and (15b), respectively, in Turner (1987a), but Eq. (19), used by Turner to find on; is in error since it cannot be derived from his Eq. (15b) by algebra. Also note that this method for finding x and on applies only when f ðtÞ is a step function. When f ðtÞ is more complex, such as a sine function, there is a different specific solution for Eq. (4) (Turner, 1994a,b).

5. Errors from first-order models We compare temperatures and times during cooling found with a first-order model with those found with second-order model to evaluate the errors from ignoring the effects of unequal heat distribution or variation in RC. The rate constants for both examples were from analysis of data we recorded for a House Wren

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(Troglodytes aedon) egg cooling in a nest. Errors depend on TE0 relative to TE1 (see below). To illustrate the general nature of the errors we used TE0 ¼ 36:08C with TE1 ¼ 18:08C: The equations used were first-order: TEt ¼ ð18:0Þeÿ0:24t þ 18:0; based on TEt ¼ ðTE0 ÿ TE1 Þeÿkt þ TE1 ; second-order: TEt ¼ ð18:0Þ½ÿ0:24=ð2:42 ÿ 0:24ފeÿ2:42t þ ð18:0Þ  ½2:42=ð2:42 ÿ 0:24ފeÿ0:24t þ 18:0 based on Eq. (10) solved for TEt , or TEt ¼ ðTE0 ÿ TE1 Þ½ÿk2 =ðk1 ÿ k2 ފeÿk1 t þ ðTE0 ÿ TE1 Þ  ½k1 =ðk1 ÿ k2 ފeÿk2 t þ TE1 : Note that the first-order equation has the same rate constant as k2 for the second-order equation. This ignores the initial transient by using only the terminal linear change in ln½ðTEt ÿ TE1 Þ=ðTE0 ÿ TE1 ފ versus time. This is commonly done in first-order analyses (e.g., Robertson and Smith, 1981; Pages et al., 1991). If, on the other hand, both transients were represented by one first-order rate constant using all the data for a transient, it would be somewhat lower than k2 . In either case, errors will result in predictions of both temperature and time.

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approached TE1 ¼ 18:08C (Fig. 2). The ratio of second-order rate constants, k1 ðk2 Þÿ1 for k1 > k2 , determined the time-dependent maximum percent errors (Fig. 3). For TE0 ¼ 36:08C and TE1 ¼ 18:08C; (maximum % errors) ÿ 1=0.02027 [k1 ðk2 Þÿ1 Š þ 0:0309ðr2 ¼ 0:99996Þ can be used to find temperature maximum % error for k1 > k2 . The larger the first rate constant was relative to the second, the shorter was the initial transient and the lower was the maximum percent error. For these cooling conditions a ratio>48.0, which corresponds to a damping ratio of 3.54, would produce a maximum percent error51.0%. Of course, from use of a secondorder model, one need not accept even this error. Decreasing TE1 relative to TE0 increased the maximum percent error for any k1 ðk2 Þÿ1 . For example, with TE1 decreased from 18.0 to 0.08C a maximum % error of 9.88% occurred at 2.6 min (versus 4.18% at 1.3 min). During heating first-order values exceed second-order values and the maximum % errors follow a similar general function. However, the exact errors will also depend on TE0 and Tbp . How are these errors manifested in natural systems? Turner (1987a) found second-order time constants, for cooling eggs from 34 bird species with egg masses

5.1. Temperature percent errors We found temperature percent errors from 100 times the differences in temperatures between the second- and first-order values at the same times divided by secondorder values. Percent errors varied with time, so we varied time within 0.1 min in the above equations to find when a percent error was maximum. To generalize the results for TE0 ¼ 36:08C and TE1 =18.08C, we then asked how the maximum percent errors and the times when they occurred varied with the same first-order solutions but with larger or smaller ratios between the second-order rate constants, or k1 ðk2 Þÿ1 for k1 > k2 . This was done by keeping the second rate constant at 0.24 min ÿ 1 and varying the first rate constant from 0.2618 to 22.04 min–1. This method is similar to one described by Turner (1987a) where ratios of time constants were related to the damping ratio. The damping ratio and the errors both depend on the ratio of rate constants (see below and Eq. (18) in Turner, 1987a). For the example a maximum percent error of 4.18% occurred at 1.3 min. Prior to and following this time the error diminished to zero at t ¼ 0 min and as TE

Fig. 2. Rate constants determined from a cooling transient for an incubated egg were used to find temperatures with a firstorder model (solid circles) and a second-order model (open circles). See the text for the equations used for each model. A maximum percent error of 4.18% in temperatures for the firstorder model occurred at 1.3 min (*). This error is time dependent, as the error can be seen to diminish to zero t ¼ 0 min and as egg temperature approached TE1 . Errors for times to reach the same temperature are maximum near zero time and diminish with time cooling (see text for values).

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was the same at 0.011 min for a percent error of 89%. Time percent errors reached 31.6% at 1.3 min and 4.4% by 10 min for the example, when TE was within 1.48C of TE1 . Time percent errors decreased as k1 ðk2 Þÿ1 increased. For example, with k1 ðk2 Þÿ1 ¼ 1:0908 instead of 10.0838, the time percent error at 10 min was 48% instead of 4.4%. Even for high values of k1 ðk2 Þÿ1 , when cooling or heating occur away from TE1 , use of a second-order model is important to avoid appreciable errors when interest is in times to cool or heat. We shall show this in Section 6.3.

6. Example analyses of House Wren egg transients 6.1. Finding asymptotes Fig. 3. For TE0 ¼ 368C and TE1 =188C, the ratio of the second-order rate constants, k1 ðk2 Þÿ1 for k1 > k2 , determines the time-dependent temperature maximum percent errors. This is true for any k1 > k2 , not just for the example given in the text used to construct this figure. The lowest maximum percent errors will occur when the first rate constant is large relative to the second. For this example a ratio >48.0, which corresponds to a damping ratio of 3.54, produced a maximum percent error51.0%. The solid circles show the mean (center with *) and 95% confidence intervals for the sampling distribution of k1 ðk2 Þÿ1 for cooling House Wren eggs. Larger eggs would be located to the left (see text for errors based on analyses of cooling eggs of different sizes by Turner, 1987). The 95% confidence interval for cooling House Wren eggs was from 1.11 to 1.92%, but the average error is skewed toward higher values from the shape of this function.

To illustrate the analytical techniques, we used temperature data collected from House Wren eggs near Syracuse, NY, in the summer of 1998. A Te thermometer for cooling eggs was made by filling an emptied egg with 1% agar and implanting a glass-insulated thermocouple at its center. The thermometer was placed on natural nesting material inside a nestbox similar to the thermal environment in which clutches of naturally incubated eggs were found. A similarly implanted egg was placed in the nestbox and anchored at the center of the clutch of naturally incubated eggs to measure TE . Clutch size range from 4 to 8 eggs, with a mean of 6  0.96 (S.E., n ¼ 4). A Li-cor data logger recorded temperatures for Te and TE every 15 s. Fig. 4 shows a typical cooling and heating record. Note that eggs were maintained somewhat below Tbp (found as described below) after

ranging from 1.12 g (Bluegray Gnatcatcher) to 1516.16 g (Ostrich). From this analysis k1(k2) ÿ 1=3.259 (kg) ÿ 0.435 based on separate scaling power functions for the time constants (Turner, 1987a). As average egg mass increases, average maximum percent error would increase. Based on the regression method, eggs with masses of 1,10, 100, and 1000 g cooling from TE0 =36.08C with TE1 ¼ 18:08C would on average have respective temperature maximum percent errors of 0.7, 1.9, 4.7, and 10.3%. We shall find, however, that averages for k1 ðk2 Þÿ1 are not sufficient to characterize maximum % errors for a set of cooling or heating conditions when there is variation in k1 ðk2 Þÿ1 (see Section 6.3). 5.2. Time percent errors Percent errors for times to reach the same temperature were maximum close to zero time and diminished toward zero as temperatures approached asymptotes. For the example in Fig. 2 the second-order temperature was 35.958C at 0.1 min, and the first-order temperature

Fig. 4. A time and temperature record for a naturally incubated House Wren egg during a cooling and heating cycle showing TE lower than Tbp during heating and higher than TE1 during cooling. The closed circles show the actual recorded TE , while the stars show the predicted pattern if TE were to continue to increase towards Tbp (TE1 during heating).

M.A. Voss, F. R. Hainsworth / Journal of Thermal Biology 26 (2001) 121–132

heating, and cooling usually ended at a temperature greater than TE1 . TE1 for heating (or brood patch temperature, Tbp ) was estimated using the least-squares regression method of curve fitting (Fig. 5). This method was also used to validate the TE1 estimated by the Te egg thermometer during cooling. Fluctuations in TE1 and occasional TE reversals made it impossible to analyze all of the transients recorded under natural conditions (n=256). To meet the requirements for the analysis, a subset of cooling and heating transients with a minimum number of TE reversals or TE1 variations was analyzed (n=166, or 65% of the total data). These were chosen unsystematically from data on four incubating birds over 38 days. Following methods given in Bakken (1976) and Riggs (1963), a series of estimated values for TE1 in 0.18C increments were used with recorded egg temperatures in the expression   TEt ÿ TE1 y ¼ ln ð11Þ TE0 ÿ EE1 plotted over time (Fig. 5). All of the data for a transient were included in this regression. This differs from the analysis for rate constants where data early or late in the transient must be excluded to accurately find k2 (Fig. 1, shaded regions; see below). In this analysis data late in the transient, as TE approaches TE1 are particularly important in determining the correct TE1 . Estimated

Fig. 5. The asymptotic egg temperature (TE1 ) during heating or cooling can be found through a series of regressions using estimated values for TE1 in 0.108C increments. The estimated TE1 that produces a regression line with a minimum value for the residual sum of squares is the assumed correct value. This is shown with the open points in the figure. The upper solid points show the behavior of the function when TE1 is 108C too low, while the lower solid points are when TE1 is 108C too high. Example data are for a cooling transient for a House Wren egg under natural conditions near Syracuse, New York, 1998.

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values for TE1 that were too low caused the function to swing upward and away from a straight line, while estimated values of TE1 that were too high caused the function to turn down and away from a straight line (Fig. 5). The ‘‘correct’’ value for TE1 , when it could be found during a transient, produced a linear regression with a minimum residual sum of squares. This method for finding TE1 is described here since the use of a Te thermometer is problematic under some conditions, and, whenever possible, the two methods should be used to verify if heat production minus evaporative heat loss is zero. The analysis methods require that TE1 for cooling and heating be constant and act as a step function in a change from one to the other. Failure to find TE1 with this assumption when f ðtÞ is unknown would indicate f ðtÞ 6¼ constant for prolonged times. Thus, this method is a way to test for the general nature of f ðtÞ. Since we could analyze 65% of transients under natural conditions, f ðtÞ 6¼ constant occurred for 35% of the transients. We recorded an average Te of 21.19  7.398C (X  S.D.) with the Te thermometer during 83 cooling transients. A power analysis indicated that 30 transients would be sufficient to statistically test the hypothesis that the measured Te was not different from the estimated TE1 . The least-squares regression technique of Bakken (1976) indicated the 30 transients chosen for analysis were relatively constant for TE1 . These transients produced an average TE1 of 24.45  6.81 (X  S.D.; n ¼ 30Þ. A two-tailed paired t-test was used to compare this estimated mean TE1 with the corresponding recorded mean Te of 23.958C. The two were not statistically different (p ¼ 0:0690; a ¼ 0:05; df ¼ 29; Table 1), so we could be reasonably confident in our measured values with the Te thermometer for TE1 . We then found Tbp using the regression method for 43 heating transients. The method indicated Tbp also was relatively constant during the egg transients. The average estimated Tbp values were compared to average brood patch temperatures measured by Baldwin and Kendeigh (1932), who found the average breast temperature of 15 incubating female House Wrens to be 41.88C. Analysis of Tbp from the transients produced average brood patch temperatures that were not statistically different from this (X  95% C.I.:41.54  0.288C). As expected, analysis across transients showed Tbp was less variable than TE1 during cooling. 6.2. Finding rate constants for House Wren egg transients When the technique to find rate constants is used with actual thermal transient data from natural environments, it is necessary to standardize the regression method by not using the data during the initial, or

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Table 1 A comparison of average TE1 estimated by regression methods and average values recorded using a Te thermometer for House Wren (Troglodytes aedon) eggs

Mean S.D. Range Confidence interval ( a =0.05) p value (Two-tailed, paired t-test)

TE1 from Te thermometer (8C)

Estimated TE1 from least-squares regression (8C)

n=30 23.95  6.70 16.33–34.10

n=30 24.45  6.81 15.63–33.70

 2.40 0.0690

 2.44

perhaps, the final periods of temperature change (shaded regions in Fig. 1). k2 is found first. To find k2 it is important not to use the initial data governed by k1 . If the initial data were used, it would influence the slope of the function, and thus k2 . Once k2 is found, the intercept is used to find k1 , which does not ignore the initial data (Fig. 1). During the late period, the rate for one process, such as cooling, could be influenced by a change to the other process, such as heating. Such a shift into another phase of a heating and cooling cycle would also skew the values for k1 and k2 if data from the inappropriate time periods were included in the determination of the rate constants. Excluding late data is important when they are obtained from natural situations where there may be

uncertainty about the exact times when heating or cooling start. A least-squares regression can then be used to fit all the consecutive data points between the excluded time periods to find the residual sum of squares. If there are slight variations in the data (‘‘noise’’), or uncertainty about a shift in TE1 , this process may need to be repeated several times by removing data points from the end of the line and working with the remaining consecutive points until an absolute minimum value for the residual sum of squares is found for four or more data points. If an appropriately representative regression can be fitted to the data, the negative slope of the resulting line can be taken as k2 . Once known, values for k1 and k2 can be used in Eq. (10) with any value of t. Using this method, we found cooling and heating rate constants for 83 House Wren egg transients (Table 2). Rate constants for all cooling transients were found using measured values for Te . Heating rate constants were found using the ‘‘correct’’ estimated Tbp when known (n ¼ 43) or an average Tbp of 41.88C when a precise Tbp was not known for that transient (n ¼ 40). Values for k1 and k2 within the same transient were used to find k1 ðk2 Þÿ1 . The k1 ðk2 Þÿ1 ratios were then averaged over all transients analyzed for cooling and heating (Table 2). The k1 ðk2 Þÿ1 ratios for cooling and heating were not statistically different (p=0.6350; a =0.05; df=82). This implies that heating and cooling were mechanistically similar to each other for the episodes for which we used the method. Both appeared to occur with abrupt, rapid step functions, which, of course, is a requirement for successful use of the methods.

Table 2 Mean cooling and heating rate constants (min–1) for House Wren (Troglodytes aedon) eggs filled with 1% agara k1 Mean S.D. Range Mean ratios for paired values of k1 and k2

Cooling rate constants k2

n=83 5.93  3.63 1.02–19.23

n=83 0.27  0.14 0.06–0.61

k1(k2) ÿ 1 Mean 33.62 S.D.  42.99 Range 4.24–260.72 Confidence interval  9.42 ( a =0.05) Means comparison for cooling k1(k2) ÿ 1 and heating k1(k2) ÿ 1 p=0.6350 a =0.05 n=83

k1

Heating rate constants k2

n=83 8.18  4.22 1.86–18.79

n=83 0.32  0.13 0.09–0.64

k1(k2) ÿ 1 31.30  24.56 4.97–122.46  5.83

df=82

a Mean values were based on data for individual rate constants pooled across multiple transients (n=83). Mean values for k1(k2) ÿ 1 were calculated from the pairing of rate constants within individual cooling or heating periods.

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6.3. Errors from first-order models When the mean ratios for k1 ðk2 Þÿ1 for House Wren egg transients (Table 2) were used to find maximum percent error 95% confidence intervals, values of 1.11 to 1.92% were found for cooling with TE0 ¼ 36:08C and TE1 =18.08C (Fig. 3, solid points). Nevertheless, k1 ðk2 Þÿ1 varied widely, ranging from 4.24 to 260.72 for cooling (Table 2). This variation is not surprising given the use of a natural system, and the k1 ðk2 Þÿ1 variation produced a skewed distribution of temperature maximum percent errors. For the same interval below and above a mean k1 ðk2 Þÿ1 , temperature maximum percent error is greater below the mean (Fig. 3). When all the individual k1 ðk2 Þÿ1 values for House Wren eggs were transformed to maximum % errors, the modal error for cooling was 2.9%. Thus, even when a mean maximum % error may seem small, use of a second-order model is still important when there is appreciable variation in k1 ðk2 Þÿ1 . A second-order model is even more important to avoid increasing error for times when temperature transients occur away from TE1 , such as during egg cooling (Fig. 4). Incubating birds seldom allow their eggs to cool below 25.08C (Haftorn, 1988). When cooling from 36.08C at TE1 ¼ 18:08C the first-order model reached 25.08C at 3.93 min, while for the secondorder model time was 4.36 min. The minimum error during cooling then would be 9.9%. Another way of interpreting this ‘‘error’’ is variation in RC prolongs time off the nest for the parent by at least 9.9% (0.43 min). Using average rate constants for heating (Table 2) the % error in time to heat from 25.0 to 36.08C was 3.8% (3.45 min for second-order versus 3.32 min for first order). It was less partly because heating occurs closer to Tbp . Variation in RC over a complete cycle for this example of cooling and heating would produce an extra 0.30 min during cooling after subtracting the extra time (0.13 min) needed to heat.

7. General discussion In spite of widespread interest in ectotherm heating and cooling, and some methods for precise analysis (Turner, 1987a), there is still confusion about the appropriate ways to analyze thermal transient data. For instance, authors of two recent papers justified the use of a linear, or zero order, model of heating in lizards because body temperatures were never recorded at or near TB1 (Carrascal et al., 1992; Belliure et al., 1996). However, the exponential nature of heating and cooling is not dependent on body temperature reaching TB1 , a condition that need not normally occur in nature (Fig. 4). Others use a somewhat more appropriate first-order negative exponential model (Spray and May,

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1972; May, 1976; Bell, 1980; Webb and King, 1983; Fraser and Grigg, 1984; Turner and Tracy, 1985; Gonzalez Gonzalez and Porcell, 1985; Rismiller and Heldmaier, 1985; Daniels et al., 1987; Holland et al., 1992; Marden et al., 1996; Diaz et al., 1996). Use of this model rests on the assumption that there is no variation in RC. This is unlikely, as even dead organisms show variation in RC. Thus, models other than second-order cannot precisely account for changes in temperature over time in animals. Although precise methods of second-order analysis exist, few have used them, either because they misunderstand the physical system, or they are not aware of the methods. We thus wished to clarify the description of the physical system and to show that the appropriate analysis need not be overly complex. If sufficiently detailed information about the thermal transients of an organism is available, and if the forcing function is appropriate, then it is possible to derive all of the second-order information required for a precise analysis. However, many studies designed to understand variation in ectotherm temperatures record body temperature over periods so long (e.g., from every 5 min to only once an hour) that the thermal transients could not possibly be obtained from the records (for examples see Hertz, 1992; Hertz et al., 1993; Diaz, 1994; Christian and Bedford, 1995; Christian and Weavers, 1996; Diaz, 1997). The sampling intervals used depend on the hypotheses being evaluated. For these studies the hypotheses are curiously independent of time used for functions influenced by temperature (such as foraging, mating, digestion, etc.). When time is important, sampling intervals should obviously depend upon the size and thermal properties that influence the time scales of cooling and heating for the animal in question. It may be necessary to measure temperature at intervals of 30 s or less for small organisms to quantify the transients and to test for the nature of the forcing function. Without appropriate information during the transient periods of heating and cooling, it is impossible to use the power of a second-order analysis to precisely test hypotheses proposed for why ectotherms heat and cool at different rates. These hypotheses include minimizing heating time in a cycle of cooling and heating (Buttemer and Dawson, 1993), keeping TB within a favorable range during cooling (Bartholomew and Lasiewski, 1965), and maximizing the proportion of time foraging in a cycle of heating and cooling (Hainsworth, 1995). The original rationale for the development of coreshell models was that a first-order model could not adequately describe the physical properties of heat flux for animals (Porter and Gates, 1969; Tracy, 1972). These models have evolved to n-dimensional complexity to try to account for important features such as blood flow changes (O’Connor, 1999; see the appendix). RC can vary from unequal heat distribution among various ‘‘compartments’’ or variation in R with

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temperature (Robertson and Smith, 1981). Can a simple analytical method using a second-order model account for all net effects of this complexity at the core? Only attempts at analysis will tell. The answer may depend on both the frequency of sampling and the nature of the data during a transient. Complex transient data will require an increased number of samples for analysis. The reason is the method should fail when there is unpredicted variation in either body temperature (TB ) or the asymptote TB1 , or non-exponential variation in (TB ÿ TB1 ) during a transient. In the case of small objects when high-frequency sampling is possible, the method will work if TB1 remains constant for only a short time. The method may be more problematic for large animals because more time is required to find rate constants and asymptotes during which TB or TB1 may change in unpredicted ways. To begin to understand the ecological and evolutionary significance of ectotherm heating and cooling rates, it is necessary first to have a physically precise model that can predict times and temperatures with minimal error. It helps considerably if the precise model is simple to use. Fortunately, natural egg transients early in development comprised a system for which a simple second-order analysis could be used. We found considerable data during which the forcing function was a simple step function. The methods presented here can now be used to ask specific questions about the adaptive significance of heating and cooling rates for small naturally incubated eggs just after they are laid. For example, it is now possible to use the model to test predictions about the egg temperatures that would maximize the percent time a parent bird can devote to foraging under a variety of heating and cooling regimes. Only a few attempts have been made to evaluate the dynamics of trade-offs between temperature variation and other functions (Dreisig, 1985; Vispo and Bakken, 1993; Hainsworth, 1995). Unfortunately, because of assumptions or available data, all were based on firstorder models. Acknowledgements We thank Miguel Rodriguez-Girones and an anonymous reviewer for valuable comments on early versions of this manuscript.

(1999) considered a reptile to be a set of nested cylinders in a series of n concentric shells, each exchanging heat with adjoining shells by conduction and with blood via convection from circulation through the shell. The surface shell additionally exchanges heat with the environment through convection and radiation. For a shell between the center and surface, according to Eq. (A.1) in O’Connor (1999): Net heat gain=Inner shell conduction ÿ Outer shell conduction+Blood advection+Conduction through blood or Ms Cp

dTs ¼Csÿ1:s ðTsÿ1 ÿ Ts Þ ÿ Cs:sþ1 ðTs ÿ Tsþ1 Þ dt þ Qs Ms Cp ðTb ÿ Ts Þ þ Cv:s ðTb ÿ Ts Þ; ðA:1Þ

where Ms is the mass of shell layer s (kg), Cp the specific heat of shell or blood ( J kg ÿ 1 8C ÿ 1), Ts the temperature of shell s (8C), t the time (s), Csÿ1:s the thermal conductance between shells s ÿ 1 and s (W 8C ÿ 1), Tsÿ1 the temperature of next inner shell (8C), Cs:sþ1 the thermal conductance between shells s and s+1 (W 8C ÿ 1), Tsþ1 the temperature of next outer shell (8C), Qs the mass specific blood flow to layer s (kg kg ÿ 1 s ÿ 1), Tb the temperature of central blood (8C) and Cn:s the thermal conductance from shell s to blood (W 8C ÿ 1). For the core with no interior shell, a separate equation for Ms Cp ðdTs =dtÞ is written with only the final three terms on the right of Eq. (A.1) (Eq (A.2) in O’Connor, 1999). For the surface, the separate equation for Ms Cp ðdTs =dtÞ (Eq. (A.3) in O’Connor, 1999) replaces the second term on the right of Eq. (A.1) with ÿ HðTs ÿ Te Þ, where H is the conductance to the environment (W 8C ÿ 1). H is equal to A (hc þ sÞ; where A is the surface area (m2), hc is the convection coefficient (Wm ÿ 1 8C ÿ 1), and s is the linearized IR conductance (W m ÿ 2 8C ÿ 1). It can be seen from this brief summary that many parameters must be estimated to find how core temperatures will change with time. This type of model is very useful to explore how variables such as blood flow could influence rates of temperature change (O’Connor, 1999). Nevertheless, if the goal is to understand the net effects of heat exchanges over time for the core, and if a second-order analysis can be used, then it provides this information with precision and with much greater simplicity.

Appendix Here we provide a brief summary of a core-shell model for comparison with the analysis provided in the text. It is abstracted from O’Connor (1999), who provided the most recent form of a core-shells model to find times to heat and cool in reptiles. O’Connor

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