Pupal durations as a function of temperature

(i) T = 16–32°C: The data are selected from Phelps & Burrows (Reference Phelps and Burrows1969a) and Phelps (Reference Phelps1973) who measured, directly, the mean pupal duration, I _P(T), for G. m. morsitans incubated at constant temperatures between T = 16 and 32°C. The data were well described by the function:

$$I_{\rm P}\,(T)\, = \,(1\, + \,{\rm exp}\,(a + bT))\,/\,k$$

where a, b and k are all constants (Phelps & Burrows, Reference Phelps and Burrows1969a). Here we incorporate the constant in the denominator into the other constants, using the equivalent form:

(1)$$I_{\rm P}\,(T)\, = \,A\, + \,B\,{\rm exp}\,{\rm (}C(T-T_1){\rm )}$$

with the constants A, B and C suitably redefined. T ₁ is an arbitrary constant, which does not affect the goodness of fit, but simply ensures that the value of B falls in a convenient range: we take T ₁ = 16°C. The best fits to the data obtained with this model are shown in fig. 1a, using the parameters shown in table SM1. All tables, providing full details of each analysis, are available in the Supplementary Material: parameter estimates, with 95% confidence intervals, are presented in the legends of each figure. Observed and predicted values differed by a maximum of 0.04 days for all temperatures, except for males kept at 16°C, and even here the difference was only 0.24 days (0.24% of the observed mean duration of 101.8 days). The original raw data are not available so it is not possible to assign confidence intervals to the mean values – but the extremely good fit to the data in fig. 1a suggests that the variation in the data must have been very small.

Fig. 1. Pupal duration in male and female G. m. morsitans as a function of the constant temperature at which the pupae were maintained in the laboratory. (a) Data for temperatures in the range 16–32°C, where pupal durations were measured directly. Predicted values calculated using equation (1). Parameter estimates: Males A = 19.1 (17.4–20.8), B = 84.0 (81.7–86.4), C = −0.24 (−0.26 to −0.22). Females A = 17.9 (16.8–19.1), B = 82.3 (80.7–84.0), C = −0.25 (−0.27 to −0.24). Further details in table SM1. (b) Data as in (a) but with additional data for pupae where the development rates at 8, 10, 12 and 14°C have been inferred by maintaining pupae at constant temperatures for 24 h periods, the periods alternating between the test temperature and 25°C. Predicted values calculated using equation (7). Parameter estimates: Males B = 107.4 (75.1–139.8), C = −0.15 (−0.20 to −0.11). Females B = 101.3 (96.1–106.4), C = −0.17 (−0.18 to −0.16). Further details in table SM3. At 16°C the predicted values using the two equations are sufficiently similar that the difference between them cannot easily be separated visually.

Males and females are treated separately because, although there is overlap, for each parameter in table SM1, between the 95% confidence intervals for males and females, the observed pupal durations for males are higher at each of the 12 temperatures tested. This would happen by chance with probability P = 0.00024. The sexual dimorphism in the pupal period is thus a real effect, although the differences in pupal durations are small. The absolute differences between the sexes decrease with temperature over the range T = 17–31°C (fig. 2a), though the difference expressed as a percentage of the male pupal duration increases linearly with temperature over this range (fig. 2b). At 16 and 32°C, however, both the absolute and percentage differences lie below the general trend in the data. This may be related to the fact that, at these low and high limits, mortalities were much higher (52.2 and 45.8%) than at intermediate temperatures.

Fig. 2. Observed mean pupal durations for G. m. morsitans incubated at constant temperature (T): differences between males and females. (a) Absolute differences. (b) Differences as a percentage of the duration in males. A power function in (a), and linear function in (b), fitted to the data for T = 17–31°C indicate the decrease in the absolute value of the sex difference, and the increase in the percentage difference, with increasing temperature. The values at T = 16 and 32°C have been excluded from the fitting exercise.

(ii) T < 16°C: For temperatures <16°C, it was not possible to measure pupal duration directly, because all pupae kept at these low temperatures died within the puparial case. Phelps & Burrows (Reference Phelps and Burrows1969a) did find, however, that if pupae were incubated at some low test temperature (T = τ in the region 8–14°C) and at 25°C, for alternating 24 h periods, survival probabilities were in excess of 78%. They were then able to infer the rates, r(τ), of development that would have occurred at a given low test temperature (T = τ), on the assumption that the rate of development over the whole pupal period was the mean of the rates for the times spent at τ and at 25°C, respectively. We follow these workers in treating I _P(T) and r(T) as reciprocals of each other so that, with temperatures alternated over equal periods, we have:

$$I_{\rm P}\,({\rm \tau}, {\rm} 25)\, = \,1/[(r({\rm \tau} )\, + \,r\,(25))\,/\,2]\, = \,2\,/\,(r({\rm \tau} )\, + \,r\,(25))$$

or, equivalently,

(2)$$r\,({\rm \tau} )\, = \,(2/I_{\rm P}\,({\rm \tau}, \,25))\,-r\,(25)$$

and we can then estimate the pupal duration appropriate for temperature τ from I(τ) = 1/r(τ). The implicit, and as yet untested, assumption is that this procedure would work if the low temperatures were alternated with some temperature other than 25°C, or if the periods of alternating temperature were not 25°C – assuming only that the chosen temperatures and periods of exposure were such that they did not cause severely elevated pupal mortality.

Phelps & Burrows (Reference Phelps and Burrows1969a) tested the procedure summarized in equation (2) by keeping pupae at alternating temperatures of 20 and 25°C, for 24 h at each temperature. At these two temperatures, the observed pupal durations were 47.9 and 26.4 days, respectively, and rates of development, therefore, took values of r(20) = 1/47.9 = 0.02087 days⁻¹ and r(25) = 1/26.4 = 0.03786 days⁻¹, respectively. The predicted pupal duration for the alternating regime is then:

$$\; I_{\rm P}\,(20,25)\, = \,2\,/\,(0.02087\, + \,0.03786)\, = \,34.1{\rm days}$$

which differed from the observed pupal duration of 34.4 days by <1%.

If this result holds more generally, for arbitrary temperature profiles, then we could estimate the proportion, d(t), of pupal development that has been completed by time t post-larviposition by evaluating the integral:

(3)$${\rm d}\,{\rm (}t{\rm )}={\rm } \mathop \int \limits_{\rm 0}^t r{\rm (}s{\rm )\;d}s$$

where r(s) is the instantaneous rate of development at time s. The function in equation (3) simply adds up these rates over each instant in time: the pupal duration is then given by the value of t for which d(t) = 1. In practice, we never have continuous measures of temperature, but one can approximate the integral by summing mean rates of development over a convenient time period, such as 1 day. If the mean temperature on day j is T(j) and the development completed on that day is r(T(j)) then the proportion of development completed by the end of i days post-larviposition is given by:

(4)$$D\,(i)\, = \,\mathop \sum \limits_{\,j = 1}^{\,j = i} r\,(T(\,j))$$

and the pupal duration is given by the value of i for which D(i) first exceeds 1.0. Again, the function in equation (4) sums rates over time – but, now, the rates are daily averages calculated from the mean daily temperature.

Pupal developmental rates as a function of temperature

As explained above, pupal durations cannot be measured directly at temperatures <16°C, but must be estimated instead as the inverse of the development rate at these low temperatures. Phelps & Burrows (Reference Phelps and Burrows1969a) modelled the relationship between the rate of pupal development, r(T), and temperature using the function:

(5)$$r\,(T)\, = \,k\,/\,(1\, + \,{\rm exp}\,{\rm (}a + bT{\rm )}$$

In the absence of the computer software required to fit the data, they used an iterative procedure due to Aitchison & Silvey (Reference Aitchison and Silvey1960): a slightly better fit (i.e., with a lower residual sums of squares) was achieved using Stata's non-linear fitting routine (fig. SI1, table SI1; Supplementary Materials). As will become apparent, however, using a single function to describe the data over the entire range of temperature leads to problems.

Phelps & Burrows (Reference Phelps and Burrows1969a) found that their fit to the data was poor for T outside the range 20–30°C. We found that the measured rates for T = 16–32°C were very well fitted by equation (5) but, when data for T < 16°C were also included, the fit was not as good. At temperatures <16°C, the rate of development was, however, well described by a simple exponential function of temperature:

(6)$$r\,(T)\, = \,K_1\,{\rm exp}\,{\rm (}K_2T{\rm )}$$

where K ₁ and K ₂ are constants. The fit to the data of this composite function is nearly perfect and provides a good practical means of calculating the rate of pupal development at any temperature between 8 and 32°C (fig. 3): the parameters required for these fits are provided in table SM2. For the equivalent relationship between pupal duration and temperatures <16°C, we fitted the data using:

(7)$$I_{\rm P}\,(T)\, = \,B\,{\rm exp}\,(C(T-T_1))$$

Parameter values for the best fits for T < 16°C are shown in table SM3 and the combined fits for data across the whole temperature range T = 8–32°C are shown in fig. 1b. As with the development rates, using a single function to model the pupal duration over the whole temperature range led to unacceptable errors in the predicted duration at higher temperatures in the range (Supplementary Materials, fig. SI2, table SI2).

Fig. 3. Development rates for pupae of G. m. morsitans incubated in the laboratory at constant temperatures (T). Data for T ≥ 16°C fitted using equation (5). Parameter estimates: Males k = 0.053 (0.051–0.055), a = 5.3 (4.9–5.7), b = −0.24 (−0.26 to −0.22). Females, k = 0.057 (0.054–0.060), a = 5.5 (5.0–5.9), b = −0.25 (−0.27 to −0.22). Further details in table SM2A. Data for T < 16°C fitted using equation (6). Parameter estimates: Males K ₁ = 0.00064 (−0.00006 to 0.0013), K ₂ = 0.18 (0.09–0.26). Females K ₁ = 0.00066 (0.00050–0.00081), K ₂ = 0.17 (0.15–0.19). Further details in table SM2B. At 16°C the predicted values using the two equations are sufficiently similar that the difference between them cannot easily be separated visually.

Fat consumption by tsetse pupae maintained at constant temperature in the laboratory

Having maintained tsetse pupae at constant temperatures in the laboratory until they emerged, thereby estimating pupal duration and mortality, Phelps (Reference Phelps1973) measured the amount of fat in the newly emerged teneral flies. He then used linear regression to estimate the ‘size-specific’ fat content adjusted using the fat content for a fly of fat-free (or residual) dry weight (RDW) 4 mg – this being close to the mean RDW for flies in his study. With knowledge of the fat content of a newly deposited pupa of this RDW, it was then possible to estimate the total amount of fat used during pupal development. The daily rate of fat consumption was calculated by dividing this total by the pupal duration.

Phelps (Reference Phelps1973) modelled increases in the logarithms of the daily fat consumption as a two-part linear function, with a discontinuity in the slope of the function somewhere between 20 and 22°C (fig. 4a). Such a discontinuity is not biologically plausible and, indeed, the untransformed fat consumption data are well fitted by a linear model with no evidence of a discontinuity (Hargrove, Reference Hargrove, Maudlin, Holmes and Miles2004). The latter model is, however, not a good one because it predicts negative rates of fat consumption for sufficiently low temperatures. Fitting a logistic function of the form:

(8)$$F\,(T)\, = \,a\,/\,(1 + \,{\rm exp}\,{\rm (}-b{\rm (}T-c{\rm ))})$$

where F(T) is the daily rate of fat consumption at temperature T and a, b and c are constants, addresses the above problems and provides a good fit to the data for male and female pupae (fig. 4b; table SM4). Notice that, for large values of T in equation (8), the exponential term goes rapidly to zero and the rate of fat consumption converges to a value of a. Conversely, as T declines below values of about 20°C, the exponential term – and thus the denominator – becomes large and the rate of fat consumption converges to zero.

Fig. 4. Models for fat consumption by male and female pupae of G. m. morsitans maintained at a constant temperature in the laboratory. (a) Logarithm of fat vs. temperature with a discontinuity in the slope between 20 and 22°C (redrawn from Phelps, Reference Phelps1973). (b) All data fitted using a logistic function (equation 8). Parameter estimates: Males a = 0.14 (0.09–0.19), b = 0.15 (0.10–0.20), c = 25.9 (20.5–31.3). Females a = 0.16 (0.11–0.20), b = 0.14 (0.11–0.18), c = 27.4 (22.6–32.3). Further details in table SM4. R ² = 0.997 for males, 0.998 for females.

The estimates of daily fat consumption, F(T), and pupal duration, I _P(T), can then be used to estimate the total fat consumed during a pupal period for a given incubation temperature T (fig. 5a). Total fat consumed takes a minimum in the region of T = 25°C, increasing monotonically as temperatures increase or decrease from this optimum level.

Fig. 5. Fat metabolism during the pupal period of male and female G. m. morsitans maintained at a constant temperature in the laboratory. (a) Total fat consumed during the pupal period. Figures estimated using the functions for pupal duration (table SM1) and daily fat consumption (table SM4). (b) Fat remaining at adult emergence, for a teneral fly of 4 mg residual dry weight.

The predicted level of fat in the emerging teneral adult is calculated by subtracting the fat consumed from the estimated level of fat in a newly deposited pupa (fig. 5b). The amount of fat available to an emerging fly declines as temperatures move away from 25°C. In particular, as temperatures decline below 16°C, it is predicted that fat reserves may be exhausted before development is complete (fig. 5b). If that were to happen the pupa would then, of course, starve and the adult would not emerge. This suggests the possibility, to be addressed in the following sections, that total fat consumption during development might be used to predict pupal survival.

Pupal mortality

Tsetse pupal mortality can be due to biological factors such as predation (Rogers, Reference Rogers1974; Rogers & Randolph, Reference Rogers and Randolph1984) and parasitism (Buxton, Reference Buxton1955; Heaversedge, Reference Heaversedge1969). It can also be due to physical factors such as desiccation (Bursell, Reference Bursell1958), but we will be concerned here only with deaths due to the effects of temperature. These can take different forms: for sufficient extremes of temperature, death will obviously occur more or less instantaneously. Pupae can also be killed rapidly even at temperatures that could conceivably be found in nature: Potts (Reference Potts1933) found that a 30 min exposure to temperatures of 45–50°C killed all pupae of G. m. morsitans. Even at less extreme temperatures, commonly experienced in the hot seasons in areas where tsetse are abundant, such as the Zambezi Valley of Zimbabwe, Phelps & Burrows (Reference Phelps and Burrows1969c) found that virtually all pupae died if exposed repeatedly to temperatures of 38°C for 6 h a day. They concluded that the high temperature interfered directly with normal physiological development processes. As indicated previously, pupal mortality can also result from indirect effects of temperature, whereby fat reserves are exhausted before pupal development is complete. In the following sections, we investigate the extent to which mortality is due to this indirect effect of temperature, as opposed to the more direct effects of extreme temperatures.

Mortalities of tsetse pupae maintained at constant temperature in the laboratory

Phelps & Burrows (Reference Phelps and Burrows1969a) and Phelps (Reference Phelps1973) provide data on the survival of the pupae of G. m. morsitans maintained at a constant temperature in the laboratory (fig. 6a). There was a problem with the calculation of the daily mortality rates for pupae because, while Phelps knew the sex of the flies that emerged, he did not always know the sex of the pupae that died. He was thus unable to disaggregate mortality by sex. From the combined data of Phelps & Burrows (Reference Phelps and Burrows1969a) and Phelps (Reference Phelps1973), we can see that, in their combined experiments, a total of 496 male and 479 female adult flies emerged. Using the quoted mortality figures (pooled on sex), it is then possible to estimate, very roughly, that the total number of pupae incubated was 1212. If we assume that the sex ratio among the pupae at the start of the incubation was 0.5, we can estimate survival probabilities of 0.818 (95% CI 0.785–0.848) for males and 0.790 (95% CI 0.756–0.822) for females. The complete overlap of the 95% confidence intervals indicates that there is no statistically significant difference between the survival probabilities for males and females.

Fig. 6. Mortality among pupae of G. m. morsitans maintained at a constant temperature in the laboratory. (a) Raw data from Phelps & Burrows (Reference Phelps and Burrows1969a) and Phelps (Reference Phelps1973). The 95% confidence intervals calculated assuming proportions dying are binomially distributed. (b) Instantaneous mortality, ν(I _P(T)): units 1/I _P(T). Data fitted using the model given in equation (11). Parameter estimates: K ₀ = 0.059 (0.032–0.086), K ₁ = 0.68 (0.60–0.75), K ₂ = 1.12 (0.81–1.43), K ₃ = 0.56 (0.48–0.63), K ₄ = 1.31 (0.83–1.80). Further details in table SM5. (c) Percentage mortality m(I _P(T)): the fitted line is calculated from the function for ν(I _P(T)) in (a), using the relationship between ν(I _P(T)) and m(I _P(T)) given by equation (10).

The pooled mortality estimates suggest a model where mortality might increase exponentially, both at high and low temperatures. Before fitting such a model, however, it is necessary to transform the original data – since the percentage mortality is constrained between 0 and 100%, whereas the suggested exponential functions tend to infinity as temperatures increase or decrease to +/− infinity, respectively.

If we define m (I _P(T)) as the probability that a pupa, maintained in the laboratory at a constant temperature T ^°C, dies prior to emergence after I _P (T) days, then the probability, Φ (I _P(T)), that the pupa survives the whole pupal period until adult emergence is given by

Φ (I _P(T)) = 1 – m (I _P(T)). We now define v (I _P(T)) as the instantaneous mortality over the period I _P (T). The survival and the instantaneous mortality rates over this period are related by:

(9)$$\Phi \,(I_{\rm P}(T))\, = \,\exp \,(-v\,(I_{\rm P}\,(T)))$$

or, equivalently, taking logs of both sides in equation (9), and re-arranging,

(10)$$v\,(I_{\rm P}\,(T))\, = \,-\,{\rm ln}\,(\Phi (I_{\rm P}\,(T)))\, = \,-\,{\rm ln}\,({\rm 1}\,-\,m\,(I_{\rm P}\,(T)))$$

where v (I _P (T)) can take any value in (0, ∞) and has units 1/I _P(T) – i.e., the inverse of the pupal duration.

Figure 6b shows how v (I _P (T)) varied with temperature for the Phelps (Reference Phelps1973) data. The data were well fitted using the sum of two exponential functions, with an additional constant term:

(11)$$ \hskip+-12pt v(I_{\rm P}(T)) = K_0 \!+ \! K_1\,{\rm exp}{\,\rm (} \!\!-K_2(T \! - \! T_1){\rm )} + K_3\,{\rm exp}{\rm (}K_4(T-T_2){\rm )}$$

with the parameter values shown in table SM5. Using the values of K ₀–K ₄ we can then also evaluate m (I _P (T)), defined above as the probability that the pupa dies prior to emergence. As expected, m (I _P (T)) converges rapidly to unity as temperatures decrease, or increase, to the neighbourhoods of 16 or 32°C, respectively (fig. 6c).

Pupal mortality rates: instantaneous daily rates as a function of temperature

Figure 7 shows that the first exponential term in equation (11) only has any visible effect for T < 21°C, and the second exponential only for T > 28°C. Between these temperatures, the instantaneous mortality rate is effectively constant at K ₀ = 0.05–0.06, with units of the reciprocal of pupal duration (I _P(T)) – which, itself, also changes with the temperature (T). When T is constant over the whole pupal period, we can, alternatively, define μ (T) as the daily mortality rate and we have:

$$(I_{\rm P}(T))\, = \,{\rm \mu} \,{\rm (}T{\rm )}\,I_{\rm P}\,(T)$$

or

(12)$${\rm \mu }\,{\rm (}T{\rm )}\, = \,(I_{\rm P}(T))\,/\,I_{\rm P}\,(T)$$

Using equation (12) with equations (1) and (11) we have:

(13)$$\eqalign{& {\rm \mu} \,{\rm (}T{\rm )}\, = \,v\,(I_{\rm P}(T))\,/\,I_{\rm P}\,(T)\, = \,[K_0\, + \,K_1\,{\rm exp}\,(-K_2\,(T - T_1)\, \cr & \quad + \,K_3\,{\rm exp}\,{\rm (}K_4{\rm (}T-T_2{\rm ))}]\,/\,[A\, + \,B\,{\rm exp}\,{\rm (}C{\rm (}T-{\rm} 16{\rm ))}]} $$

There is no obvious way to simplify this equation. Instead we evaluate μ(T) for T in the range 16–32°C using equations (1) and (11) and the parameter values in tables SM1 and SM5. The form of the graph of the data (fig. 8) again suggests fitting the sum of two exponentials, using a function of the form:

(14)$$ \hskip+-12pt {\rm \mu} \,(T) = k_0\, + \,k_1\,{\rm exp}\,{\rm (}-k_2(T - T_1){\rm )}\, + \,k_3\,{\rm exp}\,{\rm (}k_4{\rm (}T-T_2{\rm ))}$$

with the parameter values shown in table SM6.

Fig. 7. Instantaneous mortality rates (units 1/I _P(T) – i.e., per pupal period) among pupae of G. m. morsitans maintained at a constant temperature in the laboratory. The fitted function is given in equation (11) with parameters specified in table SM5. The plot as in fig. 6b but showing the function decomposed into the two exponential functions and the constant term. Raw data from Phelps & Burrows (Reference Phelps and Burrows1969a) and Phelps (Reference Phelps1973).

Fig. 8. Instantaneous daily mortality rate for female G. m. morsitans pupae as a function of the constant temperature at which the pupae were maintained in the laboratory. Raw data from Phelps & Burrows (Reference Phelps and Burrows1969a) and Phelps (Reference Phelps1973). Data fitted using equation (14). Parameter estimates: k ₀ = 0.0020 (0.0011–0.0029), k ₁ = 0.0053 (0.0027–0.0080), k ₂ = 1.55 (−0.65 to 3.76), k ₃ = 0.030 (0.027–0.032), k ₄ = 1.22 (0.95–1.49). Further details in table SM6.

Modelling pupal mortality under conditions of varying temperature

We saw above that the time taken to complete pupal development, under conditions of varying temperature, could be estimated by summing over time the daily, temperature-dependent, rates of development (equation 4). It would be convenient if we could, similarly, estimate the probability of surviving all of the pupal development as the product of the temperature-dependent survival probabilities of surviving for each day. If this were true and if ϕ(T(j)) were the survival probability on day j of pupal development, and Φ(I _P) the survival over the whole pupal period, then we could calculate the survival over the whole pupal period from:

(15)$$(I_{\rm P})\, = \,\mathop \prod \limits_{\,j = 1}^{\,j = I_{\rm P}} \,(T(\,j))$$

Here the symbol Π means that the specified survival terms, ϕ(T(j)), in equation (15) should all be multiplied together – with the implication that all of the survival events are independent.

For temperatures in the range 20–30°C, this works reasonably well. For example, when pupae were kept for alternate days at 20 and 25°C, the pupal duration was 34 days and, from equation (14), the predicted probability of surviving 17 days at each temperature was 0.966 in each case, giving an overall predicted survival of 0.966 × 0.966 = 0.933, close to the observed value of 0.911.

When, however, temperatures <16°C were alternated with 25°C, the approximation was poor. For example, when the pupae were kept at 14 and 25°C on alternate days, the pupal duration was 52 days: the predicted survival probability, from equation (15), of surviving 26 days was 0.949 at 25°C, but only 0.043 were predicted to survive this period at 14°C – giving an overall predicted survival probability of 0.949 × 0.043 = 0.041. In practice, however, when pupae were kept at these alternating temperatures, the probability of surviving to adulthood was up to 20 times the predicted value. Discrepancies were even larger for pupae kept at temperatures alternating between T = 25°C and T < 14°C: in all these cases, it is predicted that no pupae survive until emergence – whereas the observed survival probabilities at 12, 10 and 8°C ranged between 79 and 87%.

It thus appears that we cannot use equation (15) to calculate overall survival probability in real-world environments where temperatures are changing from day-to-day. Slightly better agreement with the observed survival probabilities resulted if we predicted survival using the mean temperature that pupae experienced. For example, flies kept at alternating temperatures of 20 and 25°C are assumed to be living at a mean temperature of 22.5°C, at which temperature the predicted survival is 93.5%, in good agreement with the observed value of 91.1%. When temperatures were alternated between 25°C and either 14, 12, 10 or 8°C, the mean temperatures were 19.5, 18.5, 17.5 and 16.5°C with predicted and observed survivals of 90 vs. 85%, 88 vs. 79%, 84 vs. 87% and 69 vs. 84%. While the agreement is not perfect, the use of a mean temperature should provide a good first approximation for situations where temperatures do not dip below 10°C.

Pupal mortality as a function of total fat consumption during development

Phelps & Burrows (Reference Phelps and Burrows1969a) found that pupae incubated at any constant temperature <16°C do not survive to adulthood – but that 79–87% of pupae incubated on alternate days at temperatures <16°C and at 25°C did survive. This is consistent with the idea that incubation at the low temperature does no direct harm to the pupa: the pupa simply completes a very small percentage of its development, and uses little fat. As temperatures fall, the daily rates of fat consumption decline at a quasi-linear rate (fig. 4b), but there is an exponential increase in the pupal period (fig. 1b). Consequently, while pupae can even survive 24 h at 0°C (Phelps & Burrows, Reference Phelps and Burrows1969c), the probability of surviving the much-extended pupal period at low temperatures, and emerging as an adult, is much reduced. Indeed, as is obvious from fig. 5b, pupae incubated at any temperature below 15°C exhaust their fat reserves before completing development. Notice, however, that high temperatures are not predicted to result in the exhaustion of fat reserves in the same fashion. Predicted levels of remnant fat are always in excess of 0.6 mg even at a constant 35°C (fig. 5b) when we know that tsetse pupae kept at this constant temperatures do not survive even a single day.

The amount of fat consumed on a given day of pupal life, at a given temperature, can be calculated from the logistic function (equation 8) using the parameter values from table SM4. The total amount of fat consumed during pupal life is then the sum of the daily consumption values – and these totals can be calculated both for the flies kept at constant, and at alternating, temperatures. There is a strong correlation between this estimated total fat consumption during pupal life and observed pupal mortality, for all flies incubated at temperatures ≤30°C (fig. 9).

Fig. 9. Fat consumption as a predictor of death in pupae of G. m. morsitans. Total fat consumption estimated using the functions for pupal duration (table SM2) and daily fat consumption (table SM4). Death rates calculated from figures in table 1 of Phelps (Reference Phelps1973). Function fitted to pupae incubated at constant temperatures T < 31°C: m = 2.49 exp(2.91 F). R ² = 0.87. Figures in boxes indicate the incubation temperatures for selected results.

For flies kept at constant temperatures of 31 or 32°C, however, mortality is much higher than expected given the fat consumption. These results are consistent with the idea that, as T exceeds 30°C, direct effects of temperature come into play that interfere with normal pupal development, generally resulting in pupal death before fat reserves have been exhausted. Phelps & Burrows (Reference Phelps and Burrows1969a) noted this effect: for example, when they kept flies at a constant temperature of 36°C, the small numbers of flies that did emerge exhibited puparial durations about 10 days longer than predicted values. This phenomenon has never been fully explained: Phelps & Burrows (Reference Phelps and Burrows1969c) suggested that high temperatures interfered directly with normal physiological development but it is not known how this interference might be mediated.