Present address: Malorie B. Gélinas, Département des Sciences Biologiques, Université de Montréal, C.P. 6128, Succ. Centre-Ville Montréal, H3C 3J7, Canada.
Derek A. Roff, Department of Biology, University of California, Riverside, CA 92521, USA. Tel.: 909 787 2437; fax: 909 787 4286; e-mail: email@example.com
Abstract In the wing dimorphic sand cricket, Gryllus firmus, there is a pronounced trade-off between flight capability and fecundity. This trade-off is found both between morphs and within the macropterous morph, in which fecundity is negatively correlated with the mass of the principle flight muscles, the dorso-longitudinal muscles (DLM). In this paper, we examine how this trade-off is affected by a reduction in food and its genetic basis. We find that the relative fitness of the two wing morphs is not changed although both fecundity and DLM mass are decreased. A quantitative genetic analysis shows that the trade-off function is genetically variable but that most of the variation occurs in the intercept rather than the slope of the function. Analysis further indicates a very high genetic correlation between environments (food ration) supporting the hypothesis of a strong functional constraint between reproduction and flight capability.
A basic tenent of life history theory is that because resources are limited there are trade-offs between pairs or suites of traits (Stearns, 1976, 1992; Roff, 2001). This hypothesis is supported by a large body of empirical evidence (see reviews in Reznick, 1985; Stearns, 1992; Roff, 2001). A second general component of life history theory is that the environment is heterogeneous in both space and time. The optimal combination of trait means is unlikely to be the same under all conditions (Via & Lande, 1985; Stearns & Koella, 1986). Consequently, organisms should be phenotypically plastic, altering their life histories according to circumstance. The mathematical relationship between the environmental cue and trait value or between covarying traits is termed as a norm of reaction (Stearns & Koella, 1986). In this paper we examine experimentally the effect of variation in the amount of food – one of the most fundamental resources – on fecundity – one of the most important life history traits – in the sand cricket, Gryllus firmus. In addition, we examine how this reaction norm interacts with the trade-off between fecundity and an ecologically important trait, flight capability. We investigate both the phenotypic and genetic components of the reaction norm/trade-off function.
Laboratory studies have shown that fecundity decreases with food quantity in a wide range of taxa: copepods (Ebert et al., 1993a), polychaetes (Qian, 1994), gastropods (Cheung & Lam, 1999), sea-urchins (Thompson, 1982), insects (Grill et al., 1997; Blanckenhorn, 1998), spiders (Kessler, 1971), fish (Wootten, 1973) and reptiles (Seigel & Ford, 2001). Although this effect of food might appear to be intuitively obvious there are counter examples which demonstrate that selection can favour alternative responses. For example, Kessler (1971) studied four species of spiders in the genus Pardoso and found that whereas two species showed the ‘expected’ decline in fecundity with a 25% reduction in ration, the other two species kept their reproductive output constant by utilizing body reserves. It was only when food was reduced severely that all four species showed a reduction in fecundity.
The optimal response in terms of maximizing fitness depends upon the consequences of maintaining present fecundity. In the spider example, two species give away body components for present fecundity, which could reduce their future survival and/or future fecundity. The response to a reduced ration thus depends not only upon possible unavoidable effects but also upon the allocation ‘decisions’ of the organism. Such patterns must be genetically variable to evolve. Statistically, this variation is indicated by a significant genotype by environmental interaction. Genetic variation for phenotypic plasticity in response to variation in food quantity or quality has been observed in the copepod, Daphnia (Glazier, 1992; Ebert et al., 1993a,b), the insects Harmonia axyridis (Grill et al., 1997) and Scathophagus stercoraria (Blanckenhorn 1998) and the snake Thamnophis marcianus (Seigel & Ford, 2001).
Wing dimorphism in insects is an excellent model for the study of the costs and consequences of different allocation patterns in response to food limitation. Wing dimorphic insects are characterized by two morphs, one that lacks or has fore-shortened wings and cannot fly and one that has fully developed wings and can typically fly at some point in the adult stage. The cost of producing wings per se is probably insignificant but there is undoubtedly a substantial energetic cost to the production and maintenance of the musculature required for flight. Consequently, we expect the volant morph to show a reduction in those life history traits that might compete for the resources allocated to flight muscle development and maintenance. In females a likely candidate is fecundity and in males the success of obtaining mates. There is considerable evidence that macropterous (long-winged and flight-capable) females generally show a delay in maturation and a reduced cumulative fecundity (Roff & Fairbairn, 1991). Similarly, macropterous males have a reduced probability of obtaining mates (Crnokrak & Roff, 1995; Fairbairn & Preziosi, 1996). Studies in several cricket species, most notably the sand cricket G. firmus, have shown a negative correlation between flight muscle weight and fecundity both between wing morphs and within the macropterous morph (Roff, 1989; Zera et al., 1997; Tanaka & Suzuki, 1998; Roff et al., 2002). Male crickets attract females by calling and in G. Firmus, the relative lack of success of macropterous males in attracting a female can be attributed to the reduced time spent in calling (Crnokrak & Roff, 1995), which can itself be attributed to the size and composition of the flight musculature (Crnokrak & Roff, 1998a).
The experimental analyses of the trade-off between wing morphology and reproductive traits were generally carried out giving the individuals unrestricted access to food, which suggests that the trade-off involves some internal limitation of resource input. The present experiment was designed to investigate how the two morphs of G. firmus respond to reduced food and to assess the quantitative genetic basis of this response. We focus upon changes in fecundity and the trade-off relationship between fecundity and flight capability, as assessed by the mass of the main flight muscles, the dorso-longitudinal muscles (DLM).
Materials and methods
Species description and husbandry
Gryllus firmus (Orthoptera; Gryllidae) is a relatively large cricket (0.7 g) found in sandy areas along the south-eastern USA as far north as Connecticut ( Alexander, 1968 ; Harrison, 1985 ). The crickets used in the present experiment were descended from approximately 100 individuals collected in Gainesville, Florida in 1998. This stock was maintained in the laboratory as a breeding group of around 150 adults for about five generations prior to the present experiment. All individuals were raised in cages (4 l buckets) at a constant temperature of 28 °C and a 15 : 9 h light : dark photoperiod. Crickets were fed an unrestricted amount ( ad libitum ) of crushed Purina rabbit chow and water was provided by a soaked cheese cloth wick connected to a water reservoir in a second bucket into which the first bucket was suspended. Adult pairs were housed in the same cages with an earth dish for oviposition. Offspring from the crosses were placed in rearing buckets at a density of 40 nymphs per bucket and then placed in an incubator set at the same temperature and photoperiod as described earlier.
Because fecundity in G. firmus shows some dominant variance (Roff, 1998, 2002; Roff et al., 1997), we used a half-sib mating scheme to estimate the genetic parameters. Individuals of the parental generation were randomly chosen, each male being placed with three different females for a minimum of 3 days. Each female was placed in a separate cage with her own earth dish. After 10–15 days the earth dish was examined for the presence of eggs. If no eggs were found, the male was allowed to mate with a fourth female. If eggs were found, the female was removed from the cage and the cage containing the earth dish was returned to an incubator. Newly hatched nymphs were collected over a 3-day period and placed in a cage containing a maximum of 40 nymphs. If, after 3 days, there were fewer than 40 nymphs, the cage was discarded. Three cages of nymphs were obtained per female. Once these nymphs reached adulthood, the females were used for the experiment, and the males were discarded. After the first nymphs reached the penultimate instar, the cages were examined daily for adult females. Virgin females were placed for 7 days either in environment 1 (high food treatment) or 2 (low food treatment), after which time they were preserved for later dissection (see below). The final data set consisted of 1056 offspring distributed among 20 sires with three dams per sire.
Treatments and measurements
As described above, females were placed in one of the two food environments for a period of 7 days. In environment 1, females were fed ad libitum (high food), whereas in environment 2, the level of food was restricted (low food) to 0.55 g for the 7 day period. A preliminary experiment showed that macropterous (LW) females fed ad libitum during 7 days had an average total ovary weight of 0.1205 ± 0.0942 (SE) (n = 13), whereas females fed with 0.55 g/7 days had an average ovary weight of 0.0601 ± 0.0494 (n = 14). All females were maintained individually to eliminate competition between females. Each 7-day-old female was preserved in Bouin's fluid, which was replaced with 70% ethanol several days prior to dissection (Bouin's fluid solidifies all the organs, which facilitated the dissection).
The mass of the two ovaries (hereafter simply referred to as ovary mass) and that of the main flight muscles, the DLMs, were immediately weighed after dissection from the body. When kept as virgins, females of G. firmus produce but do not lay their eggs. Egg number is proportional to ovary mass (Roff, 1994) and the number of eggs produced by virgins in the first 7 days after the final moult is not significantly different from the total egg production (eggs laid + eggs in ovaries) of mated females during the same time interval (Roff, unpublished data).
Phenotypic changes in fecundity: the relative fitness of macropterous (LW) and micropterous (SW) females
As expected from the preliminary experiment, the mean weight of the ovaries of females given reduced food was approximately half than that of the ovaries from females fed ad libitum(Table 1). Also, as expected, micropterous (short-winged, designated SW) females had larger ovaries than LW females (Table 1). At high food the weight of the ovaries of SW females was 1.64 times as large as those of LW females, whereas under the low food regimen the ratio was slightly greater (1.79; Table 1). Based on the mean values, the SW females had a 9% increase in their relative fitness when under restricted food (1.79/1.64 = 1.09).
Table 1. Summary statistics (mean, x; standard error, SE; sample size, n) for Gryllus firmus females fed either ad libitum or on a reduced ration.
Weight of both ovaries
Weight of dorso-longitudinal muscles (DLM)
The null hypothesis for the effect of food reduction on the relative fecundities of the two morphs is SWA/ LWA = SWR/LWR, where SW and LW refer to ovary mass of the two morphs (SW = micropterous, LW = macropterous) and A, R to abundant and reduced rations. This hypothesis can be tested very simply by log transforming the ovary mass and using the model Log(Ovary masses) = a + bY + cX + dYX + error, where Y is a dummy variable representing food level, X is a dummy variable indicating wing morph and a, b, c, d are fitted coefficients. Under the null hypothesis d = 0, which is indicated in a two-way anova by a nonsignificant interaction term. The rationale for this can be seen by designating, without loss of generality, the dummy variables as X = 0 for SW and 1 for LW and Y = 0 for abundant food and Y = 1 for reduced food. Now the equations describing the four cases are Log(SWA) = a, Log(LWA) = a + c, Log(SWR) = a + b, Log(LWR) = a + b + d. Under abundant food, we have Log(SWA) − Log(LWA) = Log(SWA/LWA) = –c and similarly for restricted food the equation is Log(SWR/LWR) = –c − d. Thus, the ratio does not change only if d = 0. To ensure that these results were robust to the assumptions of anova, we further tested the hypothesis using randomization with F from the two-way anova as the test statistic (Manly, 1997).
Under the statistical model described above, the additive effects of wing morph and food level are highly significant (Wing morph: F1,1052 = 144.2, P < 0.001; Food: F1,1052 = 154.8, P < 0.001) but, as is visually evident from the plot (Fig. 1a), the interaction term is not significant (F1,1052 = 0.7, P = 0.199). The randomization test gives the same result (PMorph < 0.001, PFood < 0.001, PFood × Morph = 0.197, based on 1000 randomizations). Thus there is no significant statistical difference in the relative fitnesses of the two morphs at the two food levels.
Phenotypic analysis of the trade-off/reaction norm function
Because of the large differences between the DLM masses of the two wing morphs (Table 1), we used a log transformation to stabilize the variances. As expected from previous analyses (see Introduction), the present data show that SW females had very small DLMs (Table 1). Nevertheless, a two-way anova shows a significant main effect of both morph (F1,1052 = 393.4, P < 0.001) and food level (F1,1052 = 3.37, P < 0.001) but no interaction (F1,1052 = 1.09, P = 0.2976; Fig. 1b). Under reduced food, both wing morphs have smaller flight muscles (Table 1; Fig. 1b).
Log(ovary mass) is significantly correlated with Log(DLM), food level and wing morph (Table 2). The relationship between Log(ovary mass) and Log(DLM) depends upon wing morph but no other interaction terms are significant (Table 2). When the wing morphs are considered separately we find in the LW morph significant additive effects of Log(DLM) and food level (P < 0.001 in both cases) but no significant interaction (P = 0.641). Only food level is significant in SW females [food, P = 0.012; Log(DLM), P = 0.146; interaction, P = 0.107]. These results indicate that whereas in the LW females there is a marked negative relationship between ovary mass and DLM mass in which the intercept but not the slope is modulated by the level of food, in the SW females the trade-off is statistically insignificant (Fig. 2).
Table 2. Analysis of variance of the relationship between Log(ovary mass), Log(DLM), food level and wing morph in Gryllus firmus.
Log(DLM) × morph
Log(DLM) × food
Morph × food
Log(DLM) × morph × food
Quantitative genetic analysis: a reaction norm approach
One approach to the quantitative genetic analysis of the trade-off function is to regard the coefficients in the function as traits and estimate their heritabilities and the genetic correlations between each pair. In the present case this is difficult because we have only one value per individual. We can nevertheless test for a genetic basis in the trade-off function in the following way. The phenotypic analysis suggests that the trade-off function differs between morphs and within the LW morph differs only in the intercept. A test of the genetic basis for the phenotypic additive model in the LW females is given by the statistical model, Log(ovary mass) = constant + sire + dam(sire) + food + Log(DLM), where for simplicity the coefficients have been omitted. In this model all terms were significant (sire: F19,40 = 2.49, P = 0.007; dam(sire): F40,794 = 1.60, P = 0.012; food: F1,794 = 422, P < 0.00001; Log(DLM): F1,794 = 206, P < 0.00001). Food level and Log(DLM) account for 40% of the variance whereas the parentage terms account for a further 9%. These results show that there is genetic variation for the reaction norm at least with respect to the intercept coefficient.
Quantitative genetic analysis: a character state approach
In the character state perspective traits in different environments show phenotypic plasticity and genetically covary as a consequence of a genetic correlation between environments (Falconer, 1952; Via et al., 1995). We tested for phenotypic plasticity using the mixed model anova approach enunciated by Fry (1992). There were insufficient SW females for a separate analysis and so we used only the LW females in this analysis. Food level was entered as a fixed effect whereas sire and dam were entered as random effects. There are two ways of calculating the sire F-value: F‘Overall’ sire = MSsire/MSdam(sire) and F‘G × E’ sire = MSsire/(MSdam(sire) + MSsire × food − MSdam(sire) × food), where ‘MS’ designates mean square. The F‘Overall’ sire test for additive genetic effects averaged across environments whereas F‘G × E’ sire tests for the existence of an additive genetic by environmental covariance between environments (Shaw & Fry, in press). We estimated the genetic correlation using the covariance estimated from the mixed model anova and the variances from separate nested anovas (Fry, 1992). Heritabilities were also estimated from the separate nested anovas. Standard errors were estimated using the delete-one jackknife (Fox et al., 1999). The estimates from the anova and jackknife estimates (means) were not substantially different. However, the standard errrors from the jackknife were substantially higher than expected from the anova analysis. This appears to be the result of the occasional occurrence of outliers in the pseudovalues.
In agreement with the analyses of the preceding sections, for Log(ovary mass) there is a highly significant effect attributable to food (Table 3). There are also significant effects attributable to dam(sire) and ‘overall’ sire. These results indicate significant additive genetic variance averaged across the two environments. A significant ‘G × E’ Sire term shows that there is a significant ‘G × E’ interaction between environments. The lack of significant interactions between food level and sire or dam components (P > 0.4 in both cases; Table 3) suggests that the genetic correlation between environments is close or equal to +1. This is supported by the two estimates, which are 1.05 (SE = 0.20) for the sire and 1.01 (0.42) for the dam.
Table 3. Mixed model analysis of variance of Log(ovary mass) as a function of sire, dam and food level.
† Fdam(sire)×food = MS dam(sire)×food /MS error , Fsire×food = MS sire×food /MS dam(sire)×food .
Ffood = MS food /MS sire×food , Fdam(sire) = MS dam(sire) /MS error
F'overall′sire = MS sire /MS dam(sire) .
FG×E′sire = MS sire /(MS dam(sire) + MS sire×food − MS dam(sire)×food ).
Sire(G × E)
Sire × food
Dam(sire) × food
The analysis of Log(DLM mass) differs from that for Log(ovary mass) in that neither the ‘overall’ nor ‘G × E’ sire terms are significant (Table 4). A highly significant dam(sire) term suggests the presence of nonadditive (maternal or nonadditive genetic) effects. The estimated genetic correlations between environments are 1.09 (0.48) for the sire and 1.17 (0.27) for the dam.
Table 4. Mixed model analysis of variance of Log(DLM mass) as a function of sire, dam and food level.
† Fdam(sire)×food = MS dam(sire)×food /MS error , Fsire × food = MS sire×food /MS dam(sire)×food .
Ffood = MS food /MS sire×food , Fdam(sire) = MS dam(sire) /MS error .
F'overall′sire = MS sire /MS dam(sire) .
F'G×E′sire = MS Sire /(MS dam(sire) + MS sire × food − MS dam(sire)×food ).
Sire(G × E)
Sire × Food
Dam(Sire) × Food
The SW females have small, insignificant flight muscles and, therefore, to find a trade-off between ovary masses and DLM masse could not be expected. Contrary to this expectation, a decline in DLM mass of the SW females measured between food levels was not observed, although there was no relationship within a food level (Figs 1b and 2). A genetic explanation for the observed decline is that it is a consequence of a genetic correlation between the two morphs. To test this we estimated the genetic correlation between morphs after correcting for food effects. For Log(DLM) the significant components are dam(sire) (F40,945 = 1.66, P = 007), ‘overall’ sire (F19,40 = 2.35, P = 0.011) and ‘G × E’ sire (F19,28 = 2.21, P = 0.027). The sire component gave an estimate for the genetic correlation of 0.68 (SE = 0.055): the dam component was not estimatable, because of a negative variance estimate.
Heritabilities and genetic correlations between Log(ovary mass) and Log(DLM) were estimated separately for each food level, for the LW females using the residuals from a one-way anova with food level as the independent factor, and for both morphs combined using the residuals from a two-way anova(Table 5). With one exception, the nested anovas indicate significant additive genetic variation in Log(ovary mass). However, the estimated standard errors of the heritabilities tend to be large. For Log(DLM) all dam effects are significant but none of the sire components is, which could indicate the presence of nonadditive genetic variance and/or a problem of low power. The phenotypic correlation between Log(ovary) and Log(DLM) masses is significant as is the sire estimate when the food levels are combined (Table 5). The sire genetic correlation estimate points to a correlation of −1, although the large standard error (0.33) cannot exclude a larger value. In contrast, the phenotypic correlation is clearly greater than −1.
Table 5. Estimates of the heritabilities (±SE) of ovary and DLM masses and the genetic ( rA ) and phenotypic ( rP ) correlations (±SE) between them.
a Heritabilities estimated separately for each food level (rows 1, 2), for the LW females using the residuals from a one-way anova with food level as the independent factor (row 3), and for both morphs combined using the residuals from a two-way anova (row 4). b Asterisks indicate the results from a nested analysis of variance. Significant effects ( * P < 0.05, ** P < 0.01, *** P < 0.001) indicate the presence of additive genetic variance (sire) or additive + nonadditive effects (Dam). Standard errors shown in parentheses were estimated using the jackknife. c Too many extreme outliers for reliable estimation of SE from Jackknife.
Both the macropterous and micropterous morphs had a reduced fecundity when given a restricted amount of food. As found in male G. firmus (Crnokrak & Roff, 1998b), the relative reduction in reproductive allocation was not statistically different between the two morphs and thus there was no change in relative fitness. The nonsignificant difference in the relative fitnesses was 9% in favour of the SW females. Macropterous (LW) females cannot or do not appear to be able to compensate more than the SW females by making use of energy allocated to flight muscle maintenance. That the LW females do reduce the allocation to their flight muscles is evident from the decrease in the mass of these muscles (Table 1). Surprisingly, we also observed a drop in the DLM mass of the SW females. As these muscles are nonfunctional in SW females this drop appears paradoxical. It may, however, be simply a nonadaptive response resulting from a genetic correlation between the morphs. The observed genetic correlation of 0.68 (±0.05) is consistent with this hypothesis.
Tanaka (1993 ) and Tanaka & Suzuki (1998 ) examined the effect of food limitation on fecundity in the wing dimorphic cricket Modicogryllus confirmatus by providing newly eclosed adults with a fixed amount of food. This species differs from the Gryllus species studied in that flight muscle histolysis only occurs after dealation (natural or artificial). Natural dealation has not to our knowledge been observed in Gryllus species, although artificial dealation does initiate flight muscle histolysis ( Roff, 1989 ). In micropterous M. confirmatus and macropterous females that had removed their wings, there was a positive correlation between fecundity and amount of food (approximately five eggs at zero food increasing to 35 at 16 mg of food). Macropterous females that retained their wings produced very few (<5) eggs and showed no relationship between fecundity and ration. The reduction in ration in this experiment was extreme leading to a greater than 10-fold reduction in fecundity (and a lifespan of <15 days). Under ad libitum rations, the fecundity of micropterous females is greater than either intact or dealated macropterous females ( Tanaka, 1993 ; Tanaka & Suzuki, 1998 ).
The results for macropterous M. confirmatus that retained their wings were the same as those for G. firmus in the present study. The results for dealated females demonstrate that with reduced food macropterous females, by giving up their flight capability, have the potential to attain the same fecundity as micropterous females. This behavioural option does not appear to be part of the behavioural repetoire of G. firmus, although artificial dealation does initiate DLM histolysis and increased fecundity (Roff, 1989).
As observed previously (Roff, 1994; Roff et al., 2002), there was a strong phenotypic trade-off between fecundity (as indexed by ovary mass) and flight capability (as indexed by DLM mass or histolysis). Selection on fecundity produced a significant correlated response in DLM mass (Stirling et al., 2001), indicating a genetic basis for this trade-off. This is supported in the present study by two lines of evidence: a significant family effect in the trade-off function and an estimated genetic correlation of −1.04 (±0.33; Table 5). Roff (1994) using family mean values estimated a genetic correlation of −0.51 (±0.09). The family mean estimate is biased towards the phenotypic values (Via, 1984; Roff & Preziosi, 1994), which in the present case is −0.40 (±0.04; Table 5). Hence the previous estimate is probably biased upwards. The very high genetic correlation is evidence for a strong functional connection between the two traits, which is consistent with our hypothesis of a severe allocation constraint.
Given that functional fecundity and the flight muscles are ‘competing’ for a common resource pool, we would also expect that the response to changes in food quantity would produce a change in allocation and this change would itself be highly constrained. This is demonstrated by the between-environment correlations of 1.05 (±0.20) for Log(ovary masses) and 1.09 (±0.48) for Log(DLM masses). Thus the ranking of fecundity breeding values for females given ad libitum food is predicted to be the same as that when food is restricted. This result is supported by the finding that in the trade-off function, only the additive terms are significant (hence families differ statistically in intercept only).
Flight in G. firmus appears to be primarily a mechanism for dispersal rather than mate finding and the adult life history follows the well known oogenesis flight syndrome (Dingle, 1996) in which the adult life is divided into a prereproductive dispersal stage followed by a nondispersive reproductive stage. The latter in G. firmus and other wing dimorphic crickets is characterized by histolysis of the flight muscles, thus freeing resources to be reallocated for reproduction. Macropterous female G. firmus have two options when faced with limited food: they could give up their flight and hence dispersal capability and commence reproduction or they could give up their reproduction. An important factor in such a decision would be the possibility of making up any lost fecundity. In Drosophila melanogaster, the time course of egg production appears to be relatively fixed ontogenetically (David et al., 1971) and hence present sacrifice of fecundity probably means a decrease in maximum lifetime fecundity. The situation for G. firmus is unclear. The age-specific fecundity curve in LW females begins below that of SW females but at older ages the curves cross and LW females are more productive, although they do not make up the eggs lost to flight capability (Roff, 1994). The differences in the two fecundity curves indicates ontogenetic differences in the physiological processes, the details of which remain to be elucidated.
Both G. firmus and M. confirmatus change their allocation patterns according to the amount of food received. This pattern is observed in many other, but all, animal species (see Introduction). At some point, this phenotypic plasticity is forced upon an animal: for example, in very small mammals such as shrews the demands on maintenance metabolism are so great that under limited food any allocation to reproduction will lead to an early demise, with the concomitant death of the offspring, if they have not yet achieved independence. Under this circumstance, the optimal response is clearly to abandon the offspring (or abort egg production) and attempt to survive until sufficient reserves have been accumulated that another breeding attempt can be made. There will almost always be some food ration at which allocation to anything other than maintenance reduces fitness. The issue to be resolved is what is the appropriate reaction norm between the extremes of unlimited food and that in which only maintenance can be supported.
We are very grateful for the critical comments and insightful advice of Dr D. Fairbairn. This work was supported by a grant to DAR from the Natural Sciences and Engineering Research Council of Canada.