An Improved Temporal Formulation of Pupal Transpiration in Glossina
aa r X i v : . [ q - b i o . O T ] M a y An Improved Temporal Formulation of PupalTranspiration in
Glossina
S. J. Childs
Department of Mathematics and Applied Mathematics, University of the Free State,P.O. Box 339, Bloemfontein, 9300, South Africa.tel: +27 51 4013386, email: [email protected]
Mathematical Biosciences, 262: 214–229, 2015
Abstract
The temporal aspect of a model of pupal dehydration is improved upon. The observeddependence of pupal transpiration on time is attributed to an alternation between two,essential modes, for which the deposition of a thin, pupal skin inside the puparium andits subsequent demise are thought to be responsible. For each mode of transpiration, theresults of the Bursell (1958) investigation into pupal dehydration are used as a rudimen-tary data set. These data are generalised to all temperatures and humidities by invoking theproperty of multiplicative separability. The problem, then, is that as the temperature varieswith time, so does the metabolism and the developmental stages to which the model datapertain, must necessarily warp. The puparial-duration formula of Phelps and Burrows(1969) and Hargrove (2004) is exploited to facilitate a mapping between the constant-temperature time domain of the data and that of some, more general case at hand. Theresulting, Glossina morsitans model is extrapolated to other species using their relativesurface areas, their relative protected and unprotected transpiration rates and their differ-ent fourth instar excretions (drawing, to a lesser extent, from the data of Buxton and Lewis,1934). In this way the problem of pupal dehydration is formulated as a series of integralsand the consequent survival can be predicted. The discovery of a distinct definition forhygrophilic species, within the formulation, prompts the investigation of the hypotheticaleffect of a two-day heat wave on pupae. This leads to the conclusion that the classificationof species as hygrophilic, mesophilic and xerophilic is largely true only in so much as theirthird and fourth instars are and, possibly, the hours shortly before eclosion.
Keywords: pupal water loss; transpiration; dehydration; pupal mortality; tsetse;
Glossina .1 Childs S.J.
Early mortality is considered to be the most significant, by far, in any model of tsetse populationdynamics (Hargrove, 1990 and 2004) and the vastly different dynamics of pupal and teneraltranspiration afford each the status of a topic in its own right. While both are crucial in decidingthe viability of any tsetse population, the literature suggests pupal dehydration to be the mostchallenging aspect of modelling early mortality. The consequences of pupal dehydration are inno way limited to pupal mortality and the prospects of eclosion alone. Transpiration continuesafter eclosion up until the moment the teneral fly has its first meal. The ultimate effect ofcumulative water loss on a given cohort is therefore likely to be best assessed in terms ofthe proportion of original larvae which have sufficient reserves to achieve their first feed astenerals (the hypothesis of Childs, 2014). Combined dehydration and fat loss are thought toculminate in significant early mortality and, while rates of teneral dehydration are generallyseveral times higher than those characterising the pupal stage (Bursell, 1958 and 1959), thepupal rates prevail many times longer. Water loss during the pupal phase can therefore decidethe fate of the teneral and the significance of pupal dehydration only becomes clear in thecontext of the model for teneral dehydration (Childs, 2014). Pupal stage mortality is of stillgreater relevance in the context of control measures, since the pupal stage is neither susceptibleto targets, nor aerial spraying.This research is particularly concerned with modelling the temporal dependence of pupal tran-spiration rates and the variation in the metabolic time-table to which those rates pertain. Thevariation of the metabolic rate is ultimately temperature dependent, consequently, so are thetemporal domains to which each mode of transpiration applies. The formula of Phelps andBurrows, 1969 (modified by Hargrove, 2004) assumes a significant role in resolving this de-pendence in the absence of any more detailed information. It facilitates the formulation ofboth a mapping and its derivative. This research is otherwise based on the investigations ofBursell (1958) and, to a lesser extent, the data in another authoritative work, Buxton and Lewis(1934). One of the problems with the Bursell (1958) investigation is that many of the data arenot of much use in the format in which they were presented. Many of the data are for steadyhumidities at 24.7 ◦ C , a problem that is overcome by re-interpreting the data to be a functionof calculated, accumulated water loss, or vice versa.The transpiration-rate data for Glossina morsitans may be considered to fall into four, essentialcategories: Temperature-dependent data, humidity-dependent data, history-dependent data andtime-dependent data. One observes a certain amount of corroboration between points on therespective curves. Some of this corroboration is demanded, for example, where the data setsintersect, however, in other instances it comes as a pleasant surprise. The time-dependent dataare a case in point. Time-dependent transpiration would appear to be nothing more than analternation between two basic rates. These two essential modes of transpiration are thought toarise as a result of the protection afforded by the deposition of a thin, relatively impermeable,pupal skin inside the puparium and its subsequent slow, then finally precipitous, demise. Itshould, nonetheless, be emphasized that this research is neither concerned, nor reliant on anyparticular biological explanation for the different rates. Two different rates are simply observedto exist. mproved Temporal Formulation of Pupal Transpiration in
Glossina
G. morsitans -basedmodel to the rest of the
Glossina genus and linking total water loss to observed pupal emer-gence. In the latter instance, the challenge could be described more specifically as utilizingthe dependence of survival on humidity, at 24 ◦ C , when only the total water loss is known. Asimilar problem pertains to the historical conditioning undergone prior to and at the beginningof the sensu strictu pupal stage. The final formulation, hence the solution to the problem, ispredicated on six important assumptions. Two others are taken for granted, in addition to thoseexplicitly stated and explored. The first is that the Bursell (1958) investigation is comprehen-sive, to the extent that it encapsulates all salient aspects of pupal water loss. The second is thatthere is no transpiration at dewpoint. The problem is then reduced to a series of integrals whichcan be performed on the original, constant-temperature, time-domain of Bursell (1958). Al-though these integrals are extremely simple, they are both numerous and voluminous and thereare issues pertaining to differentiability and continuity. Since a high degree of accuracy fromthe data, itself, is not expected and the model is not intractably large, expedience takes prece-dence over taste and the midpoint rule is the preferred integration technique. A least squares fit,Newton’s method and a half interval search are the only other numerical techniques employed.The aims and broader applications of this research, in order of priority, are:1. The completion of the most challenging compartment of an early mortality model.2. Pupal habitat assessment.3. A better comprehension of the tsetse pupa’s biology (particularly the Bursell, 1958, en-deavour).The main causes of early mortality could be summed up as dehydration, fat loss, predation andparasitism. Most of the experimental work needed for a model of early stage mortality haslong been complete, notwithstanding that Bursell (1959) would appear to have a small amountof data outstanding insofar as teneral water loss’ dependence on temperature is concerned andBursell (1960) and Phelps (1973) lack a few data points pertaining to teneral fat consumption’sdependence on activity. Rogers and Randolph (1990) present a limited amount of data linking Childs S.J. predation and parasitism to the density at pupal sites, an observation corroborated by Du Toit(1954).So far as habitat assessment is concerned, this author is by no means the first to postulatethe existence of localised and highly confined sites in which some species larviposit. Du Toit(1954) concluded that such sites existed and attributed the successful extirpation of
Glossinapallidipes from KwaZulu-Natal to focusing their efforts on the Mkhuze region. Although theexistence of the pupal sites are not specifically attributed to soil humidity, there can be no doubtas to the motive for the Bursell (1958 and 1959) investigations and Glasgow (1963) identifiedriver terraces as being of key importance in the control of tsetse. Rogers and Robinson (2004)found that cold cloud duration was far and away the most frequently occurring variable in theirtop five for determining the distribution of both the fusca and palpalis groups, using satelliteimagery. Normalized difference vegetation index (NDVI) ranked second by a significant mar-gin in those two groups and only just beat cold cloud duration for the morsitans group. It isnot too great a stretch of the imagination to entertain the possibility that cold cloud durationand NDVI translate directly into soil humidity, as might elevation in the context of river basins,vleis and low-lying, coastal areas, through which rivers typically meander before terminatingin estuaries. Rainfall was also found to be even more relevant when it came to abundance, asopposed to distribution.Habitat assessment and tsetse biology are intricately entwined. That the third and fourth in-star larva should be more vulnerable to dehydration than at the sensu strictu pupal stage isalready apparent from the Bursell (1958) data. Appearances can, however, be deceptive andjust how vulnerable the organism is, is something which should not be considered in terms ofindividual stages in isolation. Certainly there are surprises in store so far as the mesophilic andxerophilic species are concerned. Making a distinction between the hygrophilic, mesophilicand xerophilic species on the basis of their response to a hypothetical heat wave proves to bean interesting exercise. The implications of this research for habitat assessment only becomeproperly apparent in the context of the teneral model of Childs (2014). It could partly ex-plain why South Africa’s sympatric,
Glossina austeni - Glossina brevipalpis population persiststo this day.
The same transpiration dependences on temperature, humidity and historical water loss areused as in Childs (2009). A brief summary of their derivation follows. Bursell (1958) ob-tained one set of water loss data for variable temperature, in dry air, and another for variablehumidity, at 24.7 ± ◦ C . Yet a third set of data points can be inferred by reason. One ex-pects no transpiration at dewpoint, regardless of the temperature. The existence of separate,temperature-dependent and humidity-dependent data sets lends itself favourably to the assump-tion of multiplicative separability. mproved Temporal Formulation of Pupal Transpiration in Glossina SSUMPTION The transpiration rate is a multiplicatively separable function of humidityand temperature. Put succinctly, if dkdt is the transpiration rate, then there exist two functions φ and θ , dependent exclusively on humidity and temperature respectively, so that dkdt ( h, T ) = φ ( h ) θ ( T ) , in which h denotes humidity and T , the temperature. How reasonable is this assumption? Certainly it is consistent with, and replicates, the third,inferred set of data points entertained above. For the ‘H’ of data which exists across thehumidity-temperature domain, one reasonably expects rates to be bounded by the inferred,wet-end and known, dry-end data, furthermore, to be close to monotonic. The multiplicativelyseparable result, below, is consistent with the simplest, such surface. The perceived wisdomis that the domain of interest lies mainly between 16 ◦ C and 32 ◦ C , (although atmospherictemperatures hotter than that certainly do occur in G. morsitans country). This means thatthe detailed, 24.7 ◦ C data are only being extrapolated over ± ◦ C , based on the known, dry-end data and consistent with the inferred, wet-end data. One would also expect any unusual,capricious behaviour, or even failure in the waterproofing, to manifest itself in dry air. Thedry-air, temperature-dependent data set is, fortunately, reasonably complete and suggestive ofbehaviour which is simple, smooth and monotonic (pure exponential, in this case). It shouldalso be pointed out that, even in the event that circumstances were more favourable and datain the ideal format of a grid were being interpolated, one would still ultimately be ignorantof the behaviour between grid points, with the possibility of an unpredictable value for some,unique combination of humidity and temperature. Thus, in the very likely event that water lossrates are not perfectly multiplicatively separable, multiplicative separability should not be abad substitute.The advantage in assuming transpiration to be a multiplicatively separable function is that theproblem of its dependence on temperature and humidity is immediately reduced to a simpleexercise in curve fitting, which yields φ ( h ) = 100 − h and θ ( T ) = e . T − . + 0 . initial-pupal-masses per hour, for the third and fourth instars (Childs, 2009). Sensu strictupupal-stage transpiration is not quite as straightforward. Transpiration rates are determined bythe temperature and humidity which prevailed during the third and fourth instars as well as atthe beginning of the sensu strictu pupal stage itself. A second assumption followed by minormanipulation is required in order to resolve the apparent dependence on historical water loss.A SSUMPTION The transpiration rate, conditioned by a given historical water loss, is thesame as the transpiration rate conditioned by a historically steady humidity, at 24.7 ◦ C , whichproduced an equivalent total water loss. Once this assumption has been used to quantify acclimation in terms of a historical water lossvariable, the multiplicative separability assumption (Assumption 1) once again facilitates a
Childs S.J. reduction of the problem to an exercise in curve fitting, which, with minor manipulation, yields φ ( w, h ) = c + c h + c w + c wh + c h + c w θ (24 . and θ ( T ) = e . T − . initial-pupal-masses per hour, in which w is the total historical water loss and the c i are theconstants of the fit (Childs, 2009 and 2009a). The first formula above describes the droughthardening reported in Bursell, 1958 (in Fig. 1, below). Transpiration Rate / Initial Pupal Masses H -1 h i s t o r i c a l w a t e r l o ss / pupa l m a ss e s Figure 1: The hourly transpiration rate as a function of humidity and water lost during the first8/30 of the puparial duration (for
G. morsitans ) at 24.7 ◦ C .Recent, qualitative data in Fig. 4 of Terblanche and Kleynhans (2009) now suggest that accli-mation may still be possible well after the onset of the sensu strictu pupal stage. Unfortunately,not enough is presently known to model a late acclimation properly. There is no data sug-gesting how the acclimation period is composed, what and when the onset of its effect is, orwhether it is lasting and permanent. Existing late-acclimation data were also never intendedfor the quantitative purposes of a model. Problems in this regard range from obvious inaccura-cies in the presentation, sample-dependent units, unknown sample sizes, under-weight pupae,unknown pupal history, indeterminate pupal age and the beginnings of exponential growth inthe transpiration rate at around the time of measurement. The absorption of water vapour fromthe atmosphere is imagined to be superficial (buffering by the puparium) and it is not known if mproved Temporal Formulation of Pupal Transpiration in Glossina
7a reverse process contributes to the measured transpiration values.
Glossina palpalis transpi-ration rates which are lower than those for
G. morsitans are also surprising when comparingthe habitats of these species, as well as other, related data. For these and other reasons a crudelate acclimation, not commensurate with the rest of the model, was investigated and reportedon separately. Although, the omission of late acclimation renders the model slightly deficientin detail, it seems to make little difference to the results and it stands to reason that the later theacclimation is, the more diminished is the benefit to the organism.
Time-dependent transpiration would appear to be little more than an alternation between anunprotected rate and the more protected rate associated with the sensu strictu pupal-stage. Thisobservation is not only supported by the temporal data (Fig. 1 of Bursell, 1958), it is alsocorroborated by the different temperature-dependent (Fig. 8 of Bursell, 1958) and humidity-dependent (Figs. 2, 3, 5 and 6 of Bursell, 1958) relationships which prevail prior to, thenduring the sensu strictu pupal stage. In both the aforementioned data sets, a clear distinctionexists between data which pertain to the stages prior to the commencement of the sensu strictupupal stage and those which pertain subsequent to its commencement.What is less trivial is the manner in which the Bursell (1958) time-domain must necessarilywarp with fluctuations in temperature. The assumption that parts of the puparial duration varywith temperature, in the same way as the whole, is inevitable in the absence of any moredetailed information on the individual stages of the pupa’s development. The constituent thirdand fourth instars, the sensu strictu pupal stage and the pharate adult stage are all assumed towarp in unison with the puparial duration, predicted by the formula of Phelps and Burrows(1969) and Hargrove (2004). A uniform dependence on an overall metabolism for all partsof the puparial duration might be considered adequate justification for such an assumption.Before the model can be extended to this more general time frame, however, the temporaldependence on the original, constant-temperature time domain of Bursell (1958) first needs tobe established.
Data exist for the temporal dependence of the water-loss-rate between parturition and eclosion,at 0% r . h . and 24.7 ± ◦ C (Fig. 1 of Bursell, 1958). Two, essential modes of transpirationare discernable. The rates differ by an order of magnitude in dry air at room temperature. It isthe interpretation of this research that other, intermediate rates arise largely due to transitionsbetween these two, basic modes of transpiration. The two, basic rates are thought to be aconsequence of the protection, afforded by a thin, pupal skin and its subsequent slow, thenfinally precipitous, demise. It should nonetheless be emphasized that this research is neitherconcerned nor dependent on any particular biological mechanism. Two different rates, withtransitions between them, are simply observed to exist. Childs S.J.
The elevated rate, that which occurs in the absence of any waterproofing, is dkdt unprotected ( h, T ) = 24 100 − h
100 ( e . T − . + 0 . (1)initial-pupal-masses per day in which h denotes humidity and T , the temperature (Childs,2009). This rate prevails from parturition and persists for the duration of the third and fourthinstars, only to reappear again, shortly before eclosion. The vagaries of the time-dependenceduring the third and fourth instars were ignored, for want of better data, and only the subse-quent stages were modelled based on the Bursell (1958) Fig. 1 data. Transpiration plummetsat the end of the fourth instar as a result of the new-found protection afforded by the depositionof the thin, pupal skin inside the puparium. The rate which ultimately serves as the basis to thesensu strictu pupal stage was determined to be dkdt protected ( w, h, T ) = 24 ( c + c h + c w + c wh + c h + c w ) e . T − . initial-pupal-masses per day, in which w is the total historical water loss and the c i are the con-stants of the fit (Childs, 2009). In the brief period before acclimation is concluded, the same,linear humidity-dependence as for the third and fourth instars, − h , is temporarily assumedand used in conjunction with the protected-stage temperature-dependence, e . T − . .Institution of the protected stage rate is followed by a long, slow and slight rise in transpirationrates, from the initial minimum attained at the start of the sensu strictu pupal phase. One pos-sible cause is that the puparial exuviae lose some of their competence as they age, developingminute cracks etc. as the pupa develops inside. The transpiration rate finally makes a spectac-ular return to its initial levels, an event which coincides with the cracking of the pupal case,shortly before eclosion.Three, successive, weighted averages were used to model the mix of the two rates. The mixof the two rates for the days following the fourth instar was determined by taking the unpro-tected rate to be 0.001102571 initial pupal masses per hour and the initial pupal rate to be0.000135555 h − , in dry air at room temperature. Exponential decay was deemed to producea good model of the transition down to the protected rate (between day four and day eight,at room temperature), the unprotected rate then creeping in, linearly, over time (between dayeight and day 21, at room temperature). Exponential growth was used to model the transitionback to the unprotected rate (between day 21 and day 29, at room temperature). The functionsin Fig. 2 were consecutively used to model the relative mix of the two rates with the followingresults, in which ¯ t denotes time in the constant temperature time frame. mproved Temporal Formulation of Pupal Transpiration in Glossina unp r o t e c t ed c on t r i bu t i on time / 30ths of a puparial duration14.0727 e - 0.630097 - t - 0.09526320.00262629 - t - 0.02363666.08165 x 10 -7 e - t + 0.0443087 Figure 2: The functions used in the successive weighted averages are the degree to which theorganism is unprotected by the thin pupal skin inside the puparium or the proportion Eq. (1)contributes to the transpiration rate.
The Period ¯ t = (4 , During this stage there is an adjustment from the unprotected rate down to the protected ratedictated by exponential decay, so that the transpiration rate was deemed to be dkdt = h − (cid:16) . e − . t − . (cid:17)i dkdt protected + (cid:16) . e − . t − . (cid:17) dkdt unprotected . (2) The Period ¯ t = (8 , Transpiration during this phase is predominantly at the pupal-stage rate. That is, dkdt = [1 − (0 . t − . dkdt protected + (0 . t − . dkdt unprotected . (3)0 Childs S.J.
The Period ¯ t = (21 , Transpiration then begins its return to the unprotected rate during the pharate adult phase. Thatis, dkdt = h − (cid:16) . × − e . t + 0 . (cid:17)i dkdt protected + (cid:16) . × − e . t + 0 . (cid:17) dkdt unprotected . (4)All three rate formulae, together with Eq. 1, constitute a sequence of first order, ordinary differ-ential equations which one expects to be Lipshitz continuous over each of the stages identified(likely, even a contraction over the greater portion of the time domain). They are the waterloss rates which prevail for the constituent intervals identified on the constant-temperature timedomain of Bursell (1958). The circumstances one is confronted with in reality are usually not those for which the temper-ature is constant, let alone a constant 24.7 ◦ C . What if the temperature varies? The problem isthat as the temperature varies, so the metabolic process of the organism is either accelerated, ordeaccelerated. As the temperature varies, so the duration of the various developmental stages,established on the constant-temperature time interval, warp; stretching at low temperature andshrinking at high temperature.What the Bursell (1958) temporal data have so far been used to establish is the metabolic time-table for transpiration as a function of the constant-temperature time-frame or, more succinctly,a function dkdt ( t (¯ t ) , w, h, T ) dependent on ¯ t ∈ [0 , . This time-frame is rooted in a particularmetabolic rate, namely that associated with a constant temperature of 24.7 ◦ C , whereas whatis of interest is the total water loss associated with more general temperature conditions. Iftime-dependent transpiration rates really can be simplistically regarded as being dependent onthe stage of pupal development and the historical conditioning of the puparium alone (ignoringthe ultimate, underlying dependence of humidity and temperature on time for the present), thenthe solution to the problem lies in a mere change in the variable of integration; should this bepossible. That is, ˆ τ dkdt ( t, w, h, T ) dt = ˆ dkdt ( t (¯ t ) , w, h, T ) dtd ¯ t d ¯ t, in which τ is the puparial duration for the temperature history in question. Only the lack ofa knowledge of one function prevents the integral on the right from being evaluated. Only thelack of knowledge of t (¯ t ) prevents the performance of the integration in the ¯ t time-frame. Sinceits derivative is obviously required and since one’s only knowledge of temperature and humiditydata can be expected to be in terms of the actual time, t , a one-to-one, invertible mapping from mproved Temporal Formulation of Pupal Transpiration in Glossina t (¯ t ) , or its derivative, dtd ¯ t , either onecan be deduced from the other and the correct T ( t (¯ t )) and h ( t (¯ t )) data be retrieved.The formula for the puparial duration is the only, potentially exploitable information in thisregard and could be the key to what is sought. The puparial duration in days, τ , has been foundto vary according to the formula τ = 1 + e a + bT κ , (5)in which T is temperature (Phelps and Burrows, 1969). For females, κ = 0 . ± . , a = 5 . ± . and b = − . ± . (Hargrove, 2004). For males, κ = 0 . ± . , a = 5 . ± . and b = − . ± . (Hargrove, 2004). The puparial durations of all species,with the exception of G. brevipalpis , are thought to lie within 10% of the value predicted bythis formula (Parker, 2008).
G. brevipalpis takes a little longer. To expect parts of the pupa’sdevelopment to be affected by temperature in the same way as the whole, that there are no‘bottle-necks’ in the pupa’s development, seems a reasonable assumption in the absence of anyinformation to the contrary.A
SSUMPTION The duration of any fraction of the puparial duration varies with tempera-ture as the whole.
That the duration of all the stages in the pupa’s development are determined by the same setof endothermic reactions, known collectively as the metabolism, could be considered justifica-tion for such an assumption. At constant temperature, a claim that the increments are relatedaccording to the formula d ¯ t = 30 τ ( T ) dt would then certainly be correct. What about real-life scenarios in which the temperature varies?The relationship between the increments is readily extended to variable-temperature scenariosto yield dtd ¯ t = τ ( T ( t ))30 . (6)Once dtd ¯ t has been formulated, the actual time, t , corresponding to a point, ¯ t , within the intervalof integration can be recovered from ¯ t ( t ) = ˆ t d ¯ tdt ( t ′ ) dt ′ = ˆ t dtd ¯ t ( t ′ ) dt ′ , (7)in which t ′ is a dummy variable of integration. It is in this way that a mapping between therespective time domains is formulated to facilitate integration on the constant-temperature time2 Childs S.J. domain. It is in this way that pertinent temperature and humidity input-data can be retrieved tofacilitate integration on the constant-temperature time domain. Should there be no interpolationbetween discrete, daily temperature data, the integral in Eq. 7 naturally becomes a summation(usually terminating with a fraction of a day’s contribution). An exposition of this is providedin Section 5.2.
G. morsitans
Model to Other Species
Transpiration-related rates for a number of species were measured at both the unprotected andprotected stages, by Buxton and Lewis (1934) and Bursell (1958). The rates are quoted in unitsof mg h − cm − (mm Hg) − . Surface-area data for pupae in the wild are likewise available andall are tabulated in Table 6 of Bursell (1958).In terms of fluxes, if ρ denotes the density of a fluid, v denotes the net velocity of that fluidthrough the surface of the organism, n denotes the normal to that surface, P is a pressure and ds is an element of area, then the quantities supplied in Table 6 of Bursell (1958) are p = ρ ‚ v · n dsP ‚ ds and s species = ‹ ds, whereas dkdt = ρ ‹ v · n ds = p × s species × P, is the transpiration rate sought. One can alternatively regard transpiration through the integu-ments of each species to be governed simplistically by something like Darcy’s law, since perme-ability is pressure dependent. The transpiration rate per fly is again the relevant rate multipliedby the surface area and a pressure.Using either argument, a species conversion factor for unprotected-stage rates can be definedas δ unprotected ≡ p unprotected p morsitans unprotected × s species s morsitans , in which p morsitans unprotected and p unprotected are the unprotected-stage rates for G. morsitans and thespecies in question, respectively, and s morsitans and s species are the surface areas of G. morsitans and the species in question, respectively. mproved Temporal Formulation of Pupal Transpiration in
Glossina δ unprotected δ protected (for minima) δ protected (for maxima) morsitans austeni morsitans pallidipes submorsitans swynnertoni palpalis palpalis tachinoides fusca brevipalpis fuscipleuris longipennis δ protected ≡ p protected p morsitans protected × s species s morsitans , in which p morsitans protected and p protected are protected-stage transpiration rates for G. morsitans and the species in question, respectively. The same surface area is used for both the pupariumand the third instar larva, the justification being that the puparial exuviae render the pupariummarginally bigger while the larval surface is not as regular. Actual values of δ unprotected and δ protected , for ten different species, are tabulated in Table 1.Notice that the ratio of the unprotected to protected conversion factors in Table 1 is curiouslyclose to two for hygrophilic species, whereas it is around unity for both the mesophilic and xe-rophilic categories. (It could suggest these categories have a layer of some, or other, protectionwhich is twice as thick.) Both the exceptions to this rule, Glossina submorsitans and
Glossinalongipennis , occur in Sudan as well as Ethiopia and the aforementioned ratio is around 1.5in both. Notice, also, that conversion factors alternatively deduced from the transpirationalmaxima, then minima, during the sensu strictu pupal stage, are remarkably similar for a givenspecies (for all except
G. brevipalpis and
Glossina fuscipleuris ). This is very encouragingand immediately suggestive of a similar slope in the time-dependence of the transpiration rateacross all species. The suggestion is that the ratios in Table 1 are fixed over time,
G. brevipalpis and
G. fuscipleuris being the possible exceptions (a lower, maxima-based value in the case of
G. brevipalpis might be due to the slightly longer puparial duration). The conversion factors al-ternatively deduced from the transpirational maxima, then minima suggest no behavioural dif-ferences. With this observation comes the dawning realization that the strategies and metabolic4
Childs S.J. time-table, adopted by the different species could all be very similar and that these conversionfactors might be all that is necessary to enable both the unprotected and protected transpira-tion rates for another species to be calculated from
G. morsitans values. Such speculation isfurther reinforced by the known, third-instar transpiration rates for both
Glossina brevipalpis and
Glossina palpalis , although very little data are available for other species. Conversion ofthird-instar,
G. morsitans -model, transpiration rates to
G. brevipalpis and
G. palpalis values,on this basis yields errors of 6% and 10% respectively (Childs, 2009). Finally, qualitative data,recently brought to light in Terblanche and Kleynhans (2009), suggest that the transpirationrates for a number of species are indeed approximate multiples of the
G. morsitans rate, duringthe sensu strictu pupal stage. Such observations present a strong argument for asserting thatthe only differences between the species, so far as water loss is concerned, are permeability,surface area and a fourth-instar excretion.A
SSUMPTION The water management strategies of the majority of tsetse fly species differonly with respect to relative pupal surface area, relative unprotected and protected transpi-ration rates and the different amounts excreted during the fourth instar.
On the face of it, Assumption 4 is certainly the most tenuous. How valid is it? Does sucha simplistic approach work? No dependence with pronounced variation in temperature andhumidity is indicated, since the pressures cancel when using the original data on which Table 1is based. It is, nonetheless, of considerable comfort that the conversion of the
G. morsitans model to other species involves relative rates. Assuming continuity, one should at least be ableto have confidence in results that are within a few degrees of room temperature. The perceivedwisdom is that the region of interest lies mainly between 16 ◦ C and 32 ◦ C . Collecting together all prior derivation and assumptions gives rise to a series of governing equa-tions. Their application to any, specific temperature and humidity data set is then facilitated bya mapping of the time domain and its derivative alone.
The total water loss of the organism is formulated as a series of five integrals, based on adecomposition of the temporal domain and an excretion. Note that water losses are in units of
G. morsitans , initial-pupal-masses (31 mg ). mproved Temporal Formulation of Pupal Transpiration in Glossina The Period ¯ t = [0 , The water loss rate for the greater part of the third and fourth instars is at the unprotected rate.Generalising Eq. 1 to all species and integrating leads to k | = ˆ (cid:0) e . T − . + 0 . (cid:1) − h p unprotected p morsitans unprotected s species s morsitans dtd ¯ t d ¯ t. The Period ¯ t = (4 , During this period there is an adjustment from the unprotected rate down to the protected ratedictated by the exponential decay in Eq. 2. Generalising to all species and integrating leads tothe expression k | = ˆ (cid:20)h − (cid:16) . e − . t − . (cid:17)i e . T − . p protected p morsitans protected + (cid:16) . e − . t − . (cid:17) (cid:0) e . T − . + 0 . (cid:1) p unprotected p morsitans unprotected (cid:21) − h s species s morsitans dtd ¯ t d ¯ t, the overall dependence on humidity being the one which exists prior to the dependence onhistorical water loss. Excretion
If water loss is sufficiently low during the first ¯ t = 8 days of the puparial duration, cognizancemust be taken of excretion. In such scenarios the formula k | + k | = 0 . h
100 (0 . − . was implemented for members of the morsitans group. Values of 0.0585 and 0.06 of the pupalmass were used for G. brevipalpis and
G. palpalis , respectively, for want of any greater wisdom.The only fourth instar excretion data are those for
G. morsitans , G. palpalis and
G. brevipalpis .The
G. morsitans rate was proportionally adjusted for other members of the morsitans group.
The Period ¯ t = (8 , Transpiration during this phase is predominantly at the protected rate. There is also deemed tobe a small component of loss at unprotected rates, which increases linearly over time, according6
Childs S.J. to Eq. 3. Generalising to all species and integrating leads to k | = ˆ h [1 − (0 . t − . e . T − . (cid:0) c + c h + c w + c wh + c h + c w (cid:1) p protected p morsitans protected + (0 . t − . (cid:0) e . T − . + 0 . (cid:1) − h p unprotected p morsitans unprotected (cid:21) s species s morsitans dtd ¯ t d ¯ t. The Period ¯ t = (21 , Transpiration begins its return to unprotected rates during the pharate adult phase. Generalisingthe exponential growth in Eq. 4 to all species and integrating leads to k | = ˆ hh − (cid:16) . × − e . t + 0 . (cid:17)i e . T − . (cid:0) c + c h + c w + c wh + c h + c w (cid:1) p protected p morsitans protected + (cid:16) . × − e . t + 0 . (cid:17) (cid:0) e . T − . + 0 . (cid:1) − h p unprotected p morsitans unprotected (cid:21) s species s morsitans dtd ¯ t d ¯ t. The Period ¯ t = (29 , There is a return to unprotected rates shortly before eclosion and Eq. 1 once again becomes theapplicable integrand, that is k | = ˆ (cid:0) e . T − . + 0 . (cid:1) − h p unprotected p morsitans unprotected s species s morsitans dtd ¯ t d ¯ t. If discrete values are used in Eq. 7 (on page 11), instead of interpolating between the τ pre-dicted by daily temperature, the integral becomes a sum whose final term is usually somefraction of a day’s contribution, floor { t (¯ t ) } X i =1 dtd ¯ t ( i ) + t − floor { t } dtd ¯ t (floor { t } + 1) = ¯ t. The expression, itself, is suggestive of the method of solution. A progressively increasingnumber of days’ contributions are summed, until the contributing interval, itself, overtakes mproved Temporal Formulation of Pupal Transpiration in
Glossina
Issues of non-differentiability and discontinuity dictate that a low order integration rule beused. The preferred choice in Childs (2009) was Euler’s method. Euler’s method is, however,no longer appropriate as the time-dependence is now exponential (‘stiff’) over parts of thedomain. The midpoint rule is as distasteful from the point of view of its error. The local errorper step, of length ∆ t , is O(∆ t ) . Since the required number of steps is proportional to t , theglobal error is O(∆ t ) . This is indeed primitive. The real strength of the midpoint rule and otherlow order methods lies in their robustness at discontinuities and points of non-differentiability.The maximum, additional error introduced at such points is of a similar order to the method’sglobal error. The same cannot be said for the higher order methods. Using the midpoint rule,one has one problem to solve, whereas using one of the higher order methods entails solvingfive, separate problems; each confined to its own respective domain of Lipshitz continuityetc. The handicap of a poor error is easily overcome computationally by using a small steplength. When considering the original pupal material used, the known error in the data, thatan engineering-type accuracy is anticipated from the model and a host of other factors, twosignificant figures are more than what are sought. Since the problem is not intractably large,expedience takes precedence over taste and the more pedestrian midpoint rule is considered theappropriate choice. How does one translate cumulative water loss into survival? Buxton and Lewis (1934) andBursell (1958) collected pupal emergence data for a variety of species, over a range of humidi-ties at 24 ◦ C (all reported in Bursell, 1958). Two challenges arise in using this pupal emergencedata. The first is to establish a credible relationship between the proportion which eclose andthe humidity of the substrate, while the second is how to relate this survival to total water losswhen survival is only known as a function of humidity at 24 ◦ C . Of course, the former problemreduces to an exercise in curve-fitting, once a suitable function has been determined.What is the relationship between pupal emergence and humidity in Fig. 12 of Bursell (1958)?Assuming the usual intra-specific variation, the simplest point of departure is that some pupaewill be slightly bigger, have slightly bigger reserves and more competent integuments. Yetothers will be slightly smaller, have slightly smaller reserves and less competent integuments.This justifies the following interpretation of the Bursell (1958) Fig. 12 data.A SSUMPTION The relationship between pupal emergence and humidity is a Gaussiancurve, or a part thereof. Childs S.J.
The parameters in E ( h ) = a exp (cid:20) − ( h − b ) c (cid:21) , for each species, are provided in Table 2. Convincing fits to the data are obtained in this way,regardless of whether or not the underlying logic is correct (Fig. 3 and Table 2). e m e r gen c e / % relative humidity / % G. austeniG. brevipalpisG. longipennisG. morsitansG. pallidipesG. palpalisG. submorsitansG. swynnertoniG. tachinoides
Figure 3: Percentage emergence data modelled as a Gaussian curve (Childs, 2009) for a varietyof species. All are at 24 ◦ C , except G. tachinoides (30 ◦ C ). A straight line had to be fitted tothe only two data points for the single exception, G. longipennis
A similar challenge to that encountered for historically-conditioned transpiration, compoundsmatters when it comes to the survival to eclosion for each species. Humidities are steady,furthermore, the data were obtained at a constant 24 ◦ C .A SSUMPTION The pupal survival to eclosion, for a given water loss, is the same as thatfor the steady humidity, at 24 ◦ C , that produced an equivalent total water loss. In other words, it is assumed that survival to eclosion can be re-expressed in terms of totalwater loss. Do different histories in temperature and humidity, which produce the same waterloss, imply the same pupal emergence, or is the level of the organism’s water reserve at someparticular stage more relevant to the pupa’s survival to full term? Such dependence is beyondthe scope of this research, as well as that of Bursell (1958). In practice, it is far easier toconvert a total computed water loss to a corresponding steady-humidity-at-24 ◦ C , instead of mproved Temporal Formulation of Pupal Transpiration in Glossina ◦ C constitute a monotonic declinewith steady humidity, barring excretion. These are ideal circumstances for the implementationof a half interval search (the rate of convergence is not bad in this instance).group species a b c morsitans austeni .
663 73 . . morsitans . . . pallidipes . . . submorsitans . . . swynnertoni . . . palpalis palpalis . . . tachinoides . . . fusca brevipalpis . . . Table 2: Parameters for the fit of a Gaussian curve to the Bursell (1958) and Buxton and Lewis(1934) pupal emergence data for a variety of species (Childs, 2009). All are at 24 ◦ C , except G. tachinoides (30 ◦ C ). Validating the model presents something of a challenge. The deficiency in data, synonymouswith the need for a model, is extreme in this case. Almost all available data have been incorpo-rated into the model. The veracity of the results is, to a certain extent, suggested by consistencywith the model itself. For example, transecting the Figs. 4–9 surfaces of emergence at 24 ◦ C should replicate the Gaussian curves in Fig. 3 and it does. Predicted pupal mortalities dueto water loss should also, logically, never exceed any pupal mortalities observed in the fieldfor similar humidity and temperature conditions. It is also relevant to Figs. 5 and 7 that On-derstepoort Veterinary Institute (O.V.I.) keep their G. austeni and
G. brevipalpis colonies at75% r . h . (De Beer, 2013).One set of data on which the model is not based is the measured initial water reserves for thevarious species (Table 3). In a world in which normal distributions are assumed in the absenceof any other information, the measured, critical water loss contour should correspond to thatof the 50%, emergence contour. The sensitivity of G. brevipalpis and
G. palpalis pupae todehydration makes these species arguably the most challenging tests, as well as of particularinterest to this research.
G. brevipalpis is, furthermore, a topic of intense interest in South0
Childs S.J.
G. morsitans
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. morsitans
Water Loss / mg 8.8 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. morsitans
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. morsitans
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C Figure 4:
G. morsitans pupal emergence (top left) and water loss (top right);
G. morsitans pupal emergence for a 35 ◦ C and 25% r . h . heat wave on the first two days after larviposition(bottom left) and on day fifteen and sixteen (bottom right). mproved Temporal Formulation of Pupal Transpiration in Glossina G. austeni
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. austeni
Water Loss / mg 5.3 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. austeni
Pupal Emergence / % 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. austeni
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C Figure 5:
G. austeni pupal emergence (top left) and water loss (top right);
G. austeni pupalemergence for a 35 ◦ C and 25% r . h . heat wave on the first two days after larviposition (bottomleft) and on day fifteen and sixteen (bottom right).2 Childs S.J.
G. palpalis
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. palpalis
Water Loss / mg 7.7 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. palpalis
Pupal Emergence / % 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. palpalis
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C Figure 6:
G. palpalis pupal emergence (top left) and water loss (top right);
G. palpalis pupalemergence for a 35 ◦ C and 25% r . h . heat wave on the first two days after larviposition (bottomleft) and on day fifteen and sixteen (bottom right). mproved Temporal Formulation of Pupal Transpiration in Glossina G. brevipalpis
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. brevipalpis
Water Loss / mg 18.7 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. brevipalpis
Pupal Emergence / % 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. brevipalpis
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C Figure 7:
G. brevipalpis pupal emergence (top left) and water loss (top right);
G. brevipalpis pupal emergence for a 35 ◦ C and 25% r . h . heat wave on the first two days after larviposition(bottom left) and on day fifteen and sixteen (bottom right).4 Childs S.J.
G. pallidipes
Pupal Emergence / % 86 68 50 32 14 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. pallidipes
Water Loss / mg 10.5 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. pallidipes
Pupal Emergence / % 86 68 50 32 14 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. pallidipes
Pupal Emergence / % 86 68 50 32 14 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C Figure 8:
G. pallidipes pupal emergence (top left) and water loss (top right);
G. pallidipes pupal emergence for a 35 ◦ C and 25% r . h . heat wave on the first two days after larviposition(bottom left) and on day fifteen and sixteen (bottom right). mproved Temporal Formulation of Pupal Transpiration in Glossina G. swynnertoni
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. swynnertoni
Water Loss / mg 8.5 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. swynnertoni
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C G. swynnertoni
Pupal Emergence / % 90 70 50 30 10 0 10 20 30 40 50 60 70 80 90 100relative humidity / % 10 15 20 25 30 35 40 t e m pe r a t u r e / o C Figure 9:
G. swynnertoni pupal emergence (top left) and water loss (top right);
G. swynnertoni pupal emergence for a 35 ◦ C and 25% r . h . heat wave on the first two days after larviposition(bottom left) and on day fifteen and sixteen (bottom right).6 Childs S.J.
Africa at the moment and
G. palpalis is known to be a major culprit in the spread of humantrypanosomiasis. Information on the fourth instar excretions is available for both species, aswell as for
G. morsitans . The critical water reserve is also known for three other species forwhich, it is hoped,
G. morsitans -proportionate, fourth instar excretions will suffice. The threespecies in question are
G. austeni , G. pallidipes and
Glossina swynnertoni . G. austeni G. brevipalpis G. morsitans G. pallidipes G. palpalis G. swynnertoni mg mg mg mg mg mg Table 3: Initial water reserves after Bursell (1958).A heat wave was also simulated and more especially, the timing thereof, was experimentedwith for reasons expounded in the conclusion and which have their origins in Section 4. Ahypothetical heat wave of 35 ◦ C and 25% r.h., lasting two days, was based on the kind ofatmospheric temperatures and humidities one might expect in G. morsitans habitat. The heatwave was modelled to alternatively coincide with the first two days after parturition and dayfifteen and sixteen of the puparial duration, these times usually occurring within the unprotectedand sensu strictu pupal stages, respectively. In what way atmospheric conditions relate to thosewithin the substrate of larviposition is not really known, however, the mesophilic and xerophilicresults may shed some light on the matter.The same axes for the presentation of the results are used throughout, to facilitate easy com-parison. The reason that results above 35 ◦ C are blanked out in relevant plots, is that heat waveexperiments above 35 ◦ C are obviously meaningless insofar as the “heat wave” still being aheat wave is concerned. Relevant results below 15 ◦ C are similarly blanked out, since dayfifteen and sixteen no longer occur within the sensu strictu pupal stage once the temperaturefalls to a little above 15 ◦ C . Although a hypothetical, late-stage acclimation was experimentedwith, the results were deemed not significant enough to include. This research gives rise to a series of integrals and an algorithm which predict pupal water lossand consequent survival, given the prevailing soil temperatures and humidities. High transpi-ration rates are generally a consequence of high temperatures and low humidities. They leadto a dehydration of the tsetse pupa which can be fatal. Although the diametrical opposite istrue of transpiration rates at low temperatures, metabolic processes are slowed, the puparialduration becomes too long and the cumulative effect of transpiration can be just as fatal. Thelow transpiration rates, associated with low temperatures, are more than compensated for bythe lengthening of the duration over which they prevail. High water losses, in the days imme-diately following larviposition, also trigger a more conservative mode of transpiration duringthe sensu strictu pupal stage (Bursell, 1958, and Childs, 2009 and 2009a). mproved Temporal Formulation of Pupal Transpiration in
Glossina
G. morsitans model is extrapolated to other species, is that the onlydifferences between species are their relative surface areas, the relative permeability of theirunprotected-stage and protected-stage integuments and their different fourth instar excretions.One might therefore suspect that all tsetse species share a common strategy to actively minimisewater loss for the majority of modern habitats and have hydrational mechanisms preventativeof dehydration. Despite the good correspondence in measured and calculated critical waterlosses, some caution may still be necessary at temperatures remote from 24.7 ◦ C . The relativepermeability of the species’ membranes could change at such extremes.The G. morsitans -based model seems to provide a reasonably reliable and concise definition ofhygrophilic species, by way of the factors which facilitate its extrapolation to other species (inTable 1). The ratio of the unprotected to protected conversion factors is curiously close to twofor all hygrophilic species, whereas it is around unity for the mesophilic and xerophilic species(it could suggest the latter categories have a layer of some, or other, protection which is twiceas thick, during the instars). Both exceptions to this rule,
G. submorsitans and
G. longipennis ,occur in Sudan as well as Ethiopia and the aforementioned ratio is remarkably similar, havinga value of around 1.5. A third denizen of these climes,
G. tachinoides , has an extraordinarilymesophilic pupa for a palpalis group fly and the 30 ◦ C exception in the Fig. 3 data can bemisleading. Of course, both G. tachinoides and
G. submorsitans also have habitat in Chad(Ford and Katondo, 1977) and it is such habitats that elicit interest in a hot, dry spell.There would therefore appear to be a certain merit in advancing the course of this enquiry byway of experimenting with a simulated, two-day heat wave, the intention being to further elu-cidate the classification of species as hygrophilic, mesophilic or xerophilic and, thereby, con-tribute novel, biological insight. The xerophilic and mesophilic categories are found not to beas distinct from one another in such a context as the hygrophilic category is. The classificationof tsetse into separate mesophilic and xerophilic entities could, possibly, even be consideredartificial, in the sense of the one merely being a more extreme adaption. The responses inFigs. 4, 8 and 9 are all qualitatively similar. One clear distinction to emerge between them andthe hygrophilic species is that the latter are far more vulnerable to dehydration during the earlystages not protected by the thin, pupal skin inside the puparium. If the onset of the heat waveis timed to coincide with the days immediately following larviposition, instead of during thesensu strictu pupal phase, the effect on hygrophilic species is profoundly detrimental. In thecase of
G. brevipalpis it is nothing short of catastrophic (Fig. 7). In contrast, the suggestionfor mesophilic and xerophilic species is that such a heat wave is only mildly detrimental and,even then, mostly effects the higher levels of survival. More than one mechanism is thought tobe responsible for the mesophilic and xerophilic response. One is the historical conditioningof the transpiration rate (already referred to), a mechanism by which hot, dry weather tempers8
Childs S.J. the puparium against later losses. Although sensu strictu pupal transpiration rates are an orderof magnitude lower than the earlier transpiration rates which condition them, any difference inthem prevails for a much longer duration and so the cumulative effect is potentially as, or more,damaging. Higher transpiration rates associated with a hot and dry spell are also compensatedfor, to a certain extent, by a metabolic quickening of the relevant part of the third and fourthinstars over which they prevail (although, conversely, under certain circumstances, the fastermetabolism allows less time for early acclimation, in this model). For hygrophilic species,unprotected transpiration rates are simply too high for either mechanism to make a difference.A hot, dry spell in the days immediately following larviposition is profoundly detrimental tohygrophilic species. Such is the strength of this conclusion that further experimentation withlate acclimation suggests the phenomenon to be as irrelevant to hygrophilic species as the sensustrictu pupal stage is, itself.A lack of any early conditioning of the puparium gives rise to one further, surprising featureof mesophilic and xerophilic species: A hot, dry spell is more detrimental when it coincideswith the sensu strictu pupal stage, than when it occurs immediately after larviposition e.g.Fig. 4. This certainly is counter-intuitive when the transpiration rates in Bursell (1958) Fig. 1are contemplated in isolation, as well as being in stark contrast to the hygrophilic responses(Figs. 5–7). Yet, however compelling it may be to propose an artefact, it would be difficultto refute the existence of the phenomenon in the case of the
G. morsitans calculation for dryair at room temperature. Such adaption begs the question of whether an heat wave could beexpected to manifest itself in a different way in the pupal environment, at different stages ofpupal development. Just how the atmospheric conditions associated with a heat wave manifestthemselves in soil, rot holes, compost etc. is unknown, however, one would certainly expectatmospheric conditions to prevail at the start of the third instar. Thereafter, one might wonderabout a departure from atmospheric conditions, the pupal environment having become moreinsulated. This might explain the phenomenon if, indeed, it is an adaption. Of course, a moreprosaic explanation might be that the mesophilic and xerophilic species have the capability tosurvive adversely hot and dry conditions. They respond by taking countermeasures against anyfurther, unnecessary losses, preparing themselves for the worst; something which may not thenmaterialise. For the hygrophilic species, however, the opposite is true. For the hygrophilicspecies, things are as expected: A hot, dry spell which occurs immediately after larvipositionis more detrimental than one coinciding with the sensu strictu pupal stage.On these grounds one might argue an apparent qualitatively-different response for hygrophilicspecies. For the hygrophilic species it can certainly be said that third and fourth instar waterlosses are extremely high. Not only can they be as much as ten times those of
G. morsitans ,perhaps more important is the fact that the ratio of the unprotected to protected conversionfactors (in Table 1) is double the same ratio for the mesophilic and xerophilic species. Atwhat point these early losses render the long duration of the sensu strictu pupal stage irrele-vant is not clear. Whether a milder heat wave would reproduce the mesophilic and xerophilicresponse in hygrophilic species, is not known. Whether a milder heat wave would induce thesame preference for an early, rather than later, exposure to hot and dry conditions is a questionnot answered in this research. One would not expect atmospheric conditions anywhere near asdry as those used for the hypothetical heat wave in
G. austeni habitat, although they certainly mproved Temporal Formulation of Pupal Transpiration in
Glossina
G. brevipalpis country. Despite this, the soil humidity and soil temperaturewithin the levees and river terraces of large drainage lines, which are so often the habitat of
G. brevipalpis , could differ altogether from those which characterise the atmosphere. For thatmatter, it is not known how atmospheric conditions manifest themselves in the pupal environ-ments of any of the other species, either. In many tsetse habitats the mean daily temperaturesseldom change by more than two degrees at a time, and a change of four degrees from one dayto the next may be considered extreme. The transition to and from the heat wave condition istherefore completely unrealistic for large parts of the domain investigated. Experimenting insuch a manner does, however, allow the conclusions to be generally stated and so makes for aninteresting study nonetheless. One might otherwise have been tempted to conclude that thereis a continuous progression from
G. pallidipes to G. austeni , that
G. austeni is simply a moreextreme adaption. It is not, if the experiments with the hypothetical heat wave can be regardedas relevant. If one considers this pupal work, in conjunction with the work pertaining to theteneral stage (Childs, 2014), it points to one inevitable conclusion: That the classification ofspecies as hygrophilic, mesophilic and xerophilic is largely true only in so much as their thirdand fourth instars are and, possibly, the hours shortly before eclosion. The fate of hygrophilicspecies is largely decided by the conditions which prevail during the third and fourth instarsand, possibly, the hours shortly before eclosion.An explanation for the ‘squiggle’ associated with low temperature in the lower, right plots ofFigs. 4–9 might be in order. At very low temperatures the puparial duration becomes so longthat an heat wave, generally timed to occur during the sensu strictu pupal phase, falls within thefourth instar, instead. As the temperature drops and the metabolism slows, the puparial durationlengthens. The steep transition zone between the unprotected and protected transpiration rates(evident in Fig. 2) begins to enter the fifteenth day, the day scheduled for the commencementof the intended, sensu strictu pupal-stage heat wave. The pupa then begins to incur the massivewater losses that a lack of waterproofing implies. At some point this rapid transition in ratesenters the heat wave just enough for the optimum water loss to be incurred. At some point thewater loss is just sufficient to produce a maximum conditioning for a minimum tax on reserves,thereafter it becomes more damaging; hence, the ‘squiggle’ in the results which occurs between15 ◦ C and 20 ◦ C . At a little above 15 ◦ C , the coincidence of the fourth instar and the heat wavebecomes complete. The heat wave no longer coincides with any part of the sensu strictu pupalstage, as intended, hence the omission of the results at temperatures any lower than 15 ◦ C.The real value of the Terblanche and Kleynhans (2009) experiment is that it suggests acclima-tion is still possible well after the onset of the sensu strictu pupal stage, as well as reinforcingthe assumption that the transpiration rate of many species is roughly a multiple of the
G. morsi-tans rate. A late-acclimation effect, along the lines of the effect in Bursell (1958), was crudelyimplemented for the sensu strictu pupal stage. Experimenting with this added level of complex-ity revealed no obviously discernable differences in the results for hygrophylic species. For
G.morsitans , the high-temperature boundary for survival (the 50% and 30% emergence contours)moved to a position 3 ◦ C higher, into the minor, shallow and flat band, across the hot and drycorner of Fig. 4. The reason for the slight upward shift in the relevant contours is thought to bethat early acclimation can now be completed in the sensu strictu pupal stage. The only otherdifference caused by a late acclimation, was that the xerophilic and mesophilic preference for0 Childs S.J. the timing of a heat wave during the third and fourth instars, became less apparent.The model formulated is obviously intended for more ambitious purposes than the mere inter-pretation and visualization of data. It, nonetheless, proves to be an invaluable tool in the inter-pretation and visualization of the Bursell (1958) endeavour. A substantial body of literature isof the opinion that pupal mortality due to dehydration is either irrelevant, can be assumed con-stant, is linearly dependent on temperature, or is dependent on temperature alone. Dehydrationdoes tend to be more temperature-dependent in the mesophilic and xerophilic species (e.g. Fig.4), although, even then, that dependence is certainly not linear. Notice that even in the morsi-tans group, daily pupal mortality is neither linear, nor a function of temperature alone. Evenfor a hardy fly such as
G. morsitans , its prospects deteriorate rapidly once out of favourablehabitat. When it comes to a species such as
G. brevipalpis , however, there would be more meritin assuming dehydration-related survival to be entirely humidity-dependent (Fig. 7). When itcomes to hygrophilic species, that pupal mortality due to dehydration is both relevant and pal-pable is beyond contention. The effect of soil humidity is profound. Soil humidity defineshabitat. This is despite the fact that water loss and any consequent pupal mortality are also verydifferent things (water loss may culminate in and ultimately take its toll on the teneral). If itcould be said that there was a single, overriding fact that the Bursell (1958) investigation wasable to reveal, it is that the water reserve is a limiting factor in tsetse pupae. Dehydration is achallenge to pupae, if not a major threat and it is generally accepted that most of the
Glossina genus is not well adapted to arid environments (Glasgow, 1963). The
Glossina genus maywell derive from a common, tropical, rain-forest dwelling ancestor, adjusted to moist, warmclimates (Glasgow, 1963).Notice that this is not to claim that pupal mortality due to dehydration is always decidedlyhigher than any other causes of mortality, only that causes of higher mortality are likely to begeographically more uniform, random, or cyclical. There may also be additional mortality,only indirectly attributable to dehydration. A dearth of humid substrates might lead to morelocalised and concentrated larviposition, consequently, to increased predation and parasitism.The results in Figs. 4–9 point to the fact that pupal sites for some species are very much con-fined in the dry season, particularly in the case of South Africa’s two extant species,
G. austeni and
G. brevipalpis . These would be obvious places in which to concentrate control measuresand one immediate application of this research. Barriers of the type modelled in Childs (2010)might be far more efficacious if deployed in the immediate vicinity of pupal sites, rather thanfor the purposes of containment. While early stage mortality is considered to be the most sig-nificant, by far, in any model of tsetse population dynamics it is of even greater relevance whenin the context of control measures. This is since the pupal stage, alone, is neither susceptibleto targets, nor aerial spraying. In this regard, it is noteworthy that the eclosion rates in Childs(2011) should probably be closer to those in Childs (2013), if not higher, for a steady-state equi-librium and the projected outcomes can be adjusted by a very similar factor. The humiditiesand temperatures referred to in this research have generally been attributed to the pupal sub-strate, however, one might still wonder whether atmospheric conditions can be wholly ignored.Hygrophilic species’ apparent vulnerability during the third instar, coupled with the discoverythat atmospheric conditions may sometimes be profoundly different from those which charac-terise the pupal substrate (Childs, 2014) are cause for concern, although some
G. brevipalpis mproved Temporal Formulation of Pupal Transpiration in
Glossina
G. bre-vipalpis (Fig. 7) succeeds, as well as how the equally contradictory pupal and adult strategiesof
G. longipennis perfectly complement those of
G. brevipalpis . The pupal and adult strategiesof
G. austeni complement those of
G. brevipalpis to a lesser extent, although sufficiently wellto allow their frequent sympatric association.The results in Figs. 4–9 must ultimately be described as projections, nonetheless. They areno substitute for good, field data, were such data to exist. The historical conditioning of thepuparium, in all species, has been based proportionately on that of
G. morsitans , as has a
G. morsitans -proportionate excretion been used for the other species belonging to the mor-sitans group. It would now appear that the modelled acclimation is simplistic, even for
G.morsitans . Acclimation would, however, appear to be irrelevant in the case of hygrophilicspecies. Consider then, for instance, that if Bursell (1958) is mistaken insofar as it attributeswaterproofing to being the cause of the precipitous drop in transpiration rates concomitant withthe end of the fourth instar, if that drop was, instead, actually the result of a puparium, of finitewater content, drying and the timing thereof coincidental, such a mistake might introduce largeerrors to the calculation. Such pessimism should be tempered by the observation that the modelhas already been demonstrated sufficiently sound, to the extent that any contrary data can beused to effect improvements. For example, any data which would demonstrate Assumption 3 tobe problematic could be used to replace dtd ¯ t ( T ) with a new function, dtd ¯ t (¯ t, T ) , predicting how thedifferent parts of the puparial duration change with temperature. The same data which woulddemonstrate Assumption 4 to be problematic at the extrema of the temperature domain, couldsimilarly be used to replace δ with a new relative-to- G. morsitans -rate, δ ( T ) , one dependent ontemperature. Although the assumptions are simplistic, the model, at very least, elucidates thetrends and the suggestion is that simplicity is all that is needed for hygrophilic species. Theneed for a better understanding of acclimation and related experimental data is possibly alsoof importance. Lastly, Phelps (1973) substantiated the Bursell (1958) claim that the laboratorypupae in question were slightly inferior, showing that they only correspond to pupae in the wildduring the most unfavourable season. Consider, however, that if a model is able to be adaptedand successfully make predictions with respect to other species, how much more suitable mustit be for adaption to the phenotypic plasticity within the same species. Since pupal dehydrationwould appear to be the most challenging aspect of modelling early stage mortality, the progno-sis for this model would be one of greater significance than any problems arising from issuessuch as inferior quality pupae and differing puparial durations and the shortage of statisticallysignificant data can be corrected at some stage. It is with the remainder of the pupal reservesthat the newly-eclosed teneral fly either hops onto a hock, for example, at sunset (Vale et. al.1976) or more likely, waits through the night until dawn to feed; the topic of Childs (2014).2 Childs S.J.
The author is indebted to Neil Muller, as well as to Schalk Schoombie, Johan Meyer and GlenTaylor for hosting this research.
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