Age Estimation of the Tasmanian Bettong (Bettongia gaimardi) (Marsupialia : Potoroidae)

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-- - Aust. Wildl. Res., 1989, 16, 251-61 Age Estimation of the Tasmanian Bettong (Bettongia gaimardi) (Marsupialia : Potoroidae) R. W. Rose Zoology Department, University of Tasmania, G.P.O. Box 252C, Hobart, Tas. 7001. Abstract This paper presents data on the increase in size from birth until maturity of the body parameters of head, foot and tail length as well as weight of the Tasmanian bettong, Bettongia gaimardi, a small macropodoid marsupial. Additional data are presented on changing body characteristics, particularly molar eruption and wear. The data presented allow estimation of the age of bettongs from birth to over three years. During pouch life, a quadratic equation based on head length was the most useful estimator of age; after pouch vacation, molar eruption sequences should be used. Introduction Studies on growth are often undertaken for the purpose of estimating the age of individuals. Accurate information on this topic is necessary in order to obtain information on life history traits. Numerous studies on the growth of macropodoid pouch young are available, e.g. Potorous tridactylus (Guiler 1960); Setonix brachyurus (Shield and Woolley 1961); Macropus rufus (Sharman et al. 1964); M. giganteus, M. robustus, M. rufogriseus (Kirkpatrick 1965; Ealey 1967; Poole et al. 1982a); M. parma (Maynes 1972); M. eugenii (Murphy and Smith 1970); M. fuliginosus (Poole 1976; Poole et al. 19823); Aepyprymnus rufescens (Johnson 1978); Petrogale penicillata (Johnson 1979); P. xanthopus (Poole et al. 1985); Thylogale billardierii (Rose and McCartney 1982); and Peradorcas concinna (Nelson and Goldstone 1986). The Tasmanian bettong, Bettongia gaimardi, is one of two potoroids (rat-kangaroos) inhabiting the island of Tasmania. Aspects of the reproductive biology of this species have been described in Rose (1978, 1986a, 1987a, 1987b). After a relatively short gestation period of 21 days the newborn bettong climbs to the pouch and attaches itself to one of the four teats. After giving birth the mother enters post-partum oestrus, mates and produces a fertilised egg that remains quiescent (embryonic diapause) within her uterus. The pouch young remains in the pouch for 105 days, at which time it vacates the pouch and continues to suck milk from outside for an additional 8-9 weeks. Vacation of the pouch almost always coincides with the birth of another young as a result of the resumption of the previously diapausing blastocyst. Subsequently, two teats may operate concurrently. The bettongs' diet consists predominantly of underground fungi (Rose 1982, 19863). On this diet it is able to breed continuously in the wild. Though locally common, the bettong is vulnerable to many environmental pressures, particularly forestry operations (Rose 1986b). Hence it is important that accurate methods are available so that the age structure of bettong populations can be monitored; until the present study, this has not been possible. 03 10-7833/89/03025 1$03.00

252 R. W. Rose Tyndale-Biscoe (1968) studied the growth of six pouch young of B. lesueur and provided details of qualitative changes and tooth eruption. The brush-tailed bettong, B. penicillata, has been studied by Sampson (1971) and Christensen (1980), but in both of these studies there is a paucity of data on specimens with known ages. There has been only one previous study of the growth of B. gaimardi (Lukschanderl and Lukschanderl 1969), in which information is limited to the development of a single hand-reared pouch young (150 g) of unknown age that had been rejected by its mother. The present study gives for the first time details of the growth and development of captive B. gaimardi of known age from birth to sexual maturity, and it allows the estimation of ages from birth to at least the fourth year of life. Concurrent studies (Rose 19873; Taylor and Rose 1987) have examined and compared the growth of field and laboratory populations of B. gaimardi and concluded that the field young were in poorer condition. Methods and Materials Measurements Data were obtained at varying intervals from 23 pouch young born in captivity whose dates of birth were known to within 24 hours. Weekly measurements and observations were made on ten of these pouch young and used to construct growth equations. The data from male and female pouch young were grouped once it was established that there were no significant differences between the sexes (Rose 1985). After final pouch emergence, observations on the young were less regular but generally continued until maturity. The measurements obtained were the length of the head, right foot and tail; body weight was also measured. Measurements were obtained following the procedure described by Sharman et al. (1964). Initially, five young were measured and weighed shortly after birth, but thereafter young were not removed from the teat until week 5 except for experimental purposes. Lengths were measured to the nearest millimetre with vernier calipers, and weights were measured to 'within 1 g with a field pan balance. Small pouch young were weighed with greater precision (to within 0.001 g) in the laboratory. Tooth Eruption Data on the eruption of molar teeth and the replacement of teeth on the right side were obtained regularly from 10 living animals of known age. Additional data were obtained from the cleaned skulls of 20 bettongs. In live animals, the number and degree of eruption of molar teeth were observed with a battery-powered otoscope while the jaws were held apart with loops of nylon cord. The criteria used to denote the stages of molar eruption were based on Sharman et al. (1964). The stages of eruption were given decimal notations accumulating in fifths, as described by Newsome et al. (1977) for the agile wallaby (M. agilis). This method has the advantage over the step/jump scale of the older notation of providing an almost continuous scale (up to five) for statistical analysis (Table 1). Table 1. Notation used to designate degree of molar eruption Notation Position of anterior loph Position of posterior loph -0 Below jaw Below jaw.2 Through jaw, below gum Through jaw, below gum.4 Through gum Below gum.6 Partly erupted Just through gum a 8 Fully erupted Completing eruption Roman numerals were used to indicate fully erupted teeth; thus MI11 denotes that the third molar was fully erupted, while MIII.4 denotes that the third molar was fully erupted and the anterior loph of the fourth molar had broken through the gum. Age of Appearance of Certain Body Characteristics Information was obtained from observations on both the live pouch young and 36 preserved young obtained and stored in formalin during the study. Most observations were made with the unaided eye, although a lox lens was used to determine the sex of small preserved young.

Estimating Bettong Ages 253 Table 2. Weekly measurements of head, foot and tail lengths and body weights of B. gaimardi young Values are given as a mean + s.d., with the range in parentheses

254 R. W. Rose Table 2 (Continued) Age Number of Head Foot Tail Weight (days) animals (mm) (mm) (mm) (g) Calculations and Statistical Treatment Regressions for the parameters measured during pouch life were obtained, and the equations of best fit are presented. The accuracy of these regression equations has been tested by using them to estimate the ages of individuals that were not involved in the construction of the equations. Equations are presented for the regression of the natural logarithm of age against molar eruption following the procedure of Dudzinski et al. (1977). Student t-tests were used to test for significant differences in size between the sexes and between wild and captive animals. Results Pouch Young Bettong young shortly after birth appear similar to other kangaroo neonates. The forelimbs possess deciduous claws that are shed 2-3 weeks later and are replaced by permanent ones. A small tail, the free end of which extends onto the ventral surface of the body between the hind limbs, obscures the pouch anlage or scrota1 region and makes the determination of sex difficult in live neonates. The weight of five neonates was 0-307+-0.015 g (mean+s.d.). The lightest of the five (0.298 g) was unattached in the pouch and it is likely that it had not yet attached to a teat. The mean mother-to-young weight ratio was 5542 : 1. Growth The mean weight and the mean lengths of the head, foot and tail over weekly intervals are presented in Table 2. Figs 1 and 2 illustrate these mean values together with their standard deviations. The regression equations of best fit are presented in Table 3 for the 15-week pouch life. The confidence limits of consecutive weekly samples for the estimates of foot and tail lengths start to overlap after the termination of pouch life (Rose 1985). This indicates that although measurements of these parameters are likely to be useful for the estimation of age during pouch occupancy, the accuracy of estimation will decline after this period. It is apparent that each of the measured parameters increases with increasing age and that the growth curves flatten out as maturity approaches. Most animals reach maturity by 272 days (Rose 1987a).

Estimating Bettong Ages Fig. 1. Increase in mean head length and weight (+s.d.) from birth to maturity for B. gaimardi. 120 100 80 n E E u 60.)lr 0 2 40 20 0 Fig. 2. Increase in mean foot and tail lengths (2s.d.) from birth to maturity for B. gaimardi.

Table 3. Regression equations of best fit for the parameters of head, foot and tail length (cm) and weight (g) during the 15-week pouch life of B. gaimardi Q, quadratic equation ( y = A + B x+ C x2); L, linear regression ( y = A + Bx); Ex, exponential regression ( y = ~ e(~~)); x, age (weeks); y, parameter (centimetres or grams); r, correlation coefficient; F, F-statistic Parameter Equation r F Head Q: y=0.7853+0-2265x+0.0081x2 0.9838 4839 L: y=0.3924+ 0.36~ 0.9056 6564 Ex: y = 1.0738~(0' '221 x) 0.9573 3573 R. W. Rose Foot Q: y=0.0992+0-1579x+0.0338x2 0.9808 3872 L: y= 1.8394+0.7378x 0.9504 2933 Ex: y = 0. 5667e(0'2058x) 0.9568 3390 Weight Q: y=48.9188+25.8393~+2.85162 0-9457 1227 L: y=130.6931+24.5407~ 0.7475 420 Ex: y=1.3281e(0'3821x) 0.9536 2921 SEX;di scernable naked eye EYES EXTERNAL EARS - POINTING FORWARD FREE POINTING BACKYARD OPEN ANTERIOR LATERAL L I P OPEN MOUTH END ONLY GROOVE DEVELOPS LATERALLY FULLY OPEN I SUCKING PIGMENT VIBRISSAE ALL PRESENT FULLY DEkELOPED HAIR NONE VOCALISATION LOCOMOTION POUCH LIFE AGE ( DAYS ) Fig. 3. Appearance of body characteristics during the pouch life of B. gaimardi. Developmental Stages of Pouch Life Fig. 3 summarises the appearance of various characters. The first appearance and variety of facial vibrissae was determined from the 36 preserved specimens. The first vibrissae to be observed are the mystacial and the supraorbital (Lyne 1959); these are apparent from about day 28, although their pre-emergent papillae could be distinguished at day 22. The genal vibrissae first appear at day 49, and submental, ulnar, carpal and interramal vibrissae were first seen in a preserved specimen 63 days old. Sexual Dimorphism Although the sample size was not large, Student t-tests showed that there were no significant differences between male and female head lengths or weights at any age from

Estimating Bettong Ages 257 birth to maturity, i.e. in these parameters there is no sexual dimorphism. In fact, apart from the obvious difference due to the presence of a scrotum or pouch, male and female bettongs are not easily distinguished. There are, however, small but significant differences (P<O.005) in the pes lengths of adult males and females. The pes length of males (12-1 5 0.3 cm) is larger than that of females (11.7 k0.3 cm). Tooth Eruption The dental formula of B. gaimardi is similar to that of other potoroids. The number of teeth present varies with age. During pouch life the first teeth to be observed are the lower incisors. By the time the young vacate the pouch, the deciduous premolar P2 is erupting with MI appearing shortly thereafter. No other molar teeth have pierced the gum at this stage. Molar stage Fig. 4. Plot of age (5s.d.) against stages of molar eruption in B. gaimardi. The arrow indicates eruption of premolar P3. Molar teeth erupt in the manner usual for other macropodids: the anterior cusp erupts prior to the posterior cusp. The age of first appearance of the various stages of molar eruption is presented in Fig. 4. It can be seen that each stage of molar eruption lasts proportionately longer than the previous one. Molar stage MIV is a relatively stable period that may last for three months or more before the anterior cusp of MV appears through the gum (stage MIV.4). Prior to this occurrence, however, the deciduous teeth P2 and MI are replaced by a single fluted permanent sectorial premolar, P3. This occurs after 12 months. It is apparent from examination of cleaned skulls that P3 replaces the deciduous teeth after MV has erupted through the jaw but before it has pierced the gum, i.e. at molar eruption stage MIV.2.

258 R. W. Rose MV is usually the last tooth to erupt through the gum, and this occurs well into the second year of the bettong's life. One skull had a molar stage of MV.2, but the teeth were extremely worn. This animal had been tattooed as an adult 4 years previously and had been recaptured in the wild. It was therefore at least 5 years old. The presence of a fifth molar was not described for B. lesueur by Tyndale-Biscoe (1968) or for B. penicillata by Christensen (1980), but it is not unknown in other macropodids (Sharman et al. 1964). Molar eruption stage MIV.4 lasts for a considerable period (about 12 months) before the posterior cusp of the fourth molar erupts. This very small cusp is difficult to detect in the back of the jaw of living animals. The fourth molar does not erupt completely until the start of the fourth year (158 weeks). Table 4 illustrates the gradual wear of teeth with age. Class 1 animals are mature and are 1 year old. Class 6 animals are the oldest and are at least 5 years old. The age ranges of the intermediate classes are not known. Table 4. Class Classes of the Tasmanian bettong, B. gaimardi, based on molar wear Description 1 Lingual cusps with points rounded and no dentine showing 2 Dentine of at least one labial cusp joined to dentine crescent of lingual cusps 3 Dentine of lingual cusps joined and dentine of both labial cusps joined to lingual cusps 4 As above, but not concave 5 As above, but enamel indented between anterior and posterior cusps 6 Cusps completely obliterated and crown of tooth concave Age Estimation Estimates of the ages of bettongs can be obtained by comparing the head, foot or tail lengths or, less satisfactorily, the weight of an animal of unknown age with the mean values in Table 2. Estimates of age can also be obtained by substituting values in the equations of best fit in Table 3. It can be seen from this table that, based on the square of the correlation coefficient (r2), the quadratic equation gives a better fit than the linear one. The head lengths of an additional eight pouch young of known ages were measured regularly. The actual recorded ages of these young were compared with their estimated ages using both the linear and the quadratic equations (Rose 1985). Regressions of actual ages against the ages estimated from either the linear or the quadratic equations had correlation coefficients of 0.994 and 0.987 respectively. The gradients of both lines were very close to unity, although the standard error of the slope was slightly less for the linear equation. Two major points from this comparison are that (i) during pouch life, estimates by either equation were generally accurate to within one week (usually closer) of the actual age, and (ii) little, if any, increase in accuracy was obtained by using the quadratic equation. In the case of weight estimations, an exponential relationship gives a better fit. Age Estimation Based on Tooth Eruption The data in Fig. 4 relate the stage of molar eruption to the ages of bettongs from near the end of pouch life (105 days) until the age of 3 + years. Either simple interpolation or the use of the regression equation presented below can be used to estimate age. Once again, the quadratic expression is the equation of best fit: log, y = 5-3 - 0-55x+ 0. 17x2, where y is the age in days and x is molar eruption stage (molars noted as in Archer 1978 and scored in fifths after Dudzinski et al. 1977); r2 = 0.928, F=296. Discussion The new-born bettong is extremely small in comparison with its mother, but it fits the predicted value for the allometric relationship between neonate mass at birth and maternal mass (Tyndale-Biscoe 1973). Unlike most macropodids, bettongs show little evidence of

Estimating Bettong Ages 259 sexual dimorphism, the only adult characters that differ in size between the sexes are the foot length, as noted here, and the canine length (slightly longer in males, Jarman 1983). This finding is unusual since Darwinian theory suggests that most males are larger as a result in part of competition among males for mates. Thus, the more polygynous the species the larger the males will be relative to the females (Ralls 1978). However, Jarman (1983) points out that in three diverse mammalian families (Bovidae, Cervidae and Macropodidae), the smallest species tend to be homomorphic. Wood et al. (1981) state that, in general, fitted growth curves for individual animals will differ significantly from each other and that this variation will increase the width of the confidence limits. They propose a method for allowing for such variation in the estimation of age. However, it is unlikely that this method will gain widespread use by field-workers since the method for obtaining the confidence limits is hard to apply by non-mathematicians 'as it is both tedious and involves fairly difficult computing' (S. M. Carpenter, personal communication). Poole et al. (1982) have shown that mathematically fitted growth curves can be used to estimate accurately the age of macropodids. From the point of view of field workers, quadratic equations are cumbersome to solve, and it may be that this inconvenience outweighs any inaccuracies introduced by simpler linear regressions. Other workers (e.g. Maynes 1972) have concluded that greater accuracy can be obtained if the average of more than one morphological parameter is used to age marsupials. Shield and Woolley (1961) have stated that simple linear interpolations from growth curves give better age estimates than values obtained from substitution into formulae. Data from 19 different pouch young (chosen to provide a full range of ages) showed that in general interpolations between the values in Table 2 provided better age estimates than values substituted into linear regression equations (Rose 1985). On average, the technique of simple interpolation allows an age estimation that is within two days of the actual age, and only marginal improvement in accuracy is achieved by averaging age estimates from the three length parameters. The simplest method of age estimation in the field would involve similar interpolation based on the graph of mean head length against age. Rose (1985) and Taylor and Rose (1987) compared the morphology of animals measured in the wild with animals reared in captivity. Although there were significant differences in weight (captives are heavier) and adult foot length (field animals have longer feet), head and tail lengths were similar in both wild and captive populations. Taylor and Rose (1987) also found that estimates of age from pes length were 0.2 days less than those from head length in captive pouch young, and 0-8 days less in field young, although this difference was not statistically significant. However, for the last 3-4 weeks of pouch life, bettongs in captivity are significantly heavier than those in the field. It seems likely that the very fast rates of growth exhibited by bettong pouch young in captivity (Rose 1987b) may be sustained by the protein-enriched milk used (Smolenski and Rose 1988). The most useful methods for age estimation after pouch vacation are those based on dentition. Molar eruption does not proceed at a regular pace; each eruption takes substantially longer than its predecessor, so confidence limits increase and the accuracy of estimation decreases in older animals. It is also possible that in field animals molar eruption occurs at a different rate from that in captive animals, partly because of the different abrasiveness of food. Nevertheless, the database is more substantial than that available for any other potoroid and is more accurate since all animals used were of known age. Dudzinski et al. (1977) showed that in M. agilis a linear relationship existed between the natural log of a kangaroo's age and the stage of molar eruption. By way of contrast, this study has shown that the relationship in the bettong is curvilinear and is better expressed by a quadratic equation. Acknowledgments I wish to acknowledge the provision of permits from the Tasmanian National Parks and Wildlife Service and the help of Ron Mawbey in capturing and handling the bettongs. Dr R. Swain provided assistance throughout the study, and Professor D. M. Stoddart criticised the manuscript.

References R. W. Rose Archer, M. (1978). The nature of the molar-premolar boundary in marsupials and a reinterpretation of the homology of marsupial cheekteeth. Mem. Qld Mus. 18, 157-64. Christensen, P. (1980). The biology of Bettongia penicillata Gray, 1837 and Macropus eugenii Desmarest, 1834 in relation to fire. Forest Department of Western Australia Bulletin 91. Dudzinski, M. L., Newsome, A. E., Merchant, J. C., and Bolton, B. L. (1977). Comparing the two usual methods for ageing Macropodidae on tooth-classes in the agile wallaby. Aust. Wildl. Res. 4, 219-22. Ealey, E. H. M. (1967). Ecology of the euro, Macropus robustus (Gould) in north-western Australia. IV. Age and growth. CSIRO Wildl. Res. 12, 67-80. Guiler, E. R. (1960). The pouch young of the potoroo. J. Mammal. 41, 441-51. Jarman, P. (1983). Mating system and sexual dimorphism in large terrestrial, mammalian herbivores. Biol. Rev. 58, 485-520. Johnson, P. M. (1978). Reproduction in the rufous rat-kangaroo (Aepyprymnus rufescens) (Gray) in captivity with age estimation of pouch young. Qld J. Agric. Anim. Sci. 35, 69-72. Johnson, P. M. (1979). Reproduction in the plain rock wallaby Petrogale penicillata inornata in captivity with age estimation of the pouch young. Aust. Wildl. Res. 6, 1-4. Kirkpatrick, T. H. (1965). Studies of Macropodidae in Queensland. 2. Age determination in the grey kangaroo, the red kangaroo, the eastern wallaroo and the red-necked wallaby, with notes on dental abnormalities. Qld J. Agric. Anim. Sci. 22, 301-17. Lukschanderl, L., and LukschBnderl, K. (1969). Einige Beobachtungen zur Jugendentwicklung von Bettongia cuniculus. Zool. Garten 37, 117-26. Lyne, A. G. (1959). The systematic and adaptive significance of the vibrissae in the marsupials. Proc. Zool. Soc. Lond. 133, 79-133. Maynes, G. M. (1972). Age estimation in the parma wallaby Macropus parma Waterhouse. Aust. J. Zool. 20, 107-18. Murphy, C. R., and Smith, 3. R. (1970). Age determination of pouch young and juvenile Kangaroo Island wallabies. Trans. R. Soc. S. Aust. 94, 15-20. Nelson, J. E., and Goldstone, A. (1986). Reproduction in Peradorcas concinna (Marsupialia : Macropodidae). Aust. Wildl. Res. 13, 501-6. Newsome, A. E., Merchant, J. C., Bolton, B. L., and Dudzinski, M. L. (1977). Sexual dimorphism in molar progression and eruption in the agile wallaby. Aust. Wildl. Res. 4, 1-5. Poole, W. E. (1976). Breeding biology and current status of the grey kangaroo Macropus fuliginosus fuliginosus of Kangaroo Island, South Australia. Aust. J. Zool. 24, 169-87. Poole, W. E., Carpenter, S. M., and Wood, J. T. (1982a). Growth of grey kangaroos and the reliability of age determination from body measurements. I. The eastern grey kangaroo, Macropus giganteus. Aust. Wildl. Res. 9, 9-20. Poole, W. E., Carpenter, S. M., and Wood, J. T. (19823). Growth of grey kangaroos and the reliability of age determination from body measurements. 11. The western grey kangaroos, Macropus fuliginosus fuliginosus, M. f. melanops and M. f. ocydromus. Aust. Wildl. Res. 9, 203-12. Poole, W. E., Merchant, J. C., Carpenter, S. M., and Calaby, J. H. (1985). Reproduction, growth and age determination in the yellow-footed rock-wallaby, Petrogale xanthopus Gray, in captivity. Aust. Wildl. Res. 12, 127-36. Ralls, K. (1978). When bigger is best. New Scientist, 9 Feb., 360-2. Rose, R. W. (1978). Reproduction and evolution in female Macropodidae. Aust. Mamm. 2, 65-72. Rose, R. W. (1982). Tasmanian bettong Bettongia gaimardi: maintenance and breeding in captivity. In 'The Management of Australian Mammals in Captivity'. (Ed. D. D. Evans.) pp. 108-11. (The Zoological Board of Victoria: Melbourne.) Rose, R. W. (1985). The reproductive biology of the Tasmanian bettong, Bettongia gaimardi. Ph.D. Thesis, University of Tasmania, Hobart. Rose, R. W. (1986a). Control of pouch vacation in the Tasmanian bettong. Aust. J. Zool. 34, 485-91. Rose, R. W. (19863). The habitat, distribution and conservation status of the Tasmanian bettong (Bettongia gaimardi). Aust. Wildl. Res. 13, 1-7. Rose, R. W. (1987a). The reproductive biology of the Tasmanian bettong (Bettongia gaimardi: Macropodidae). J. Zool. (Lond.) 212, 59-67. Rose, R. W. (1987b). Reproductive energetics in two Tasmanian rat-kangaroos (Potoroinae : Marsupialia). Symp. Zool. Soc. Lond. 57, 149-65. Rose, R. W., and McCartney, D. 3. (1982). Growth of the red-bellied pademelon, Thylogale billardierii, and age estimation of pouch young. Aust. Wildl. Res. 9, 33-8.

Estimating Bettong Ages 26 1 Sampson, J. C. (1971). The biology of Bettongia penicillata Gray, 1837. Ph.D. Thesis, University of Western Australia, Perth. Sharman, G. B., Frith, H. J., and Calaby, J. H. (1964). Growth of the pouch young, tooth eruption and age determination in the red kangaroo Megaleia rufa. CSIRO Wildl. Res. 9, 20-49. Shield, J. W., and Woolley, P. (1961). Age estimation by measurement of pouch young of the quokka (Setonix brachyurus). Aust. J. Zool. 9, 14-23. Smolenski, A. J., and Rose, R. W. (1988). Comparative lactation in two species of rat-kangaroo (Marsupialia). Cornp. Biochem. Physiol. 90A, 69-63. Taylor, R. J., and Rose, R. W. (1987). Comparison of growth of pouch young of Bettongia gaimardi in captivity and the wild. Aust. Wildl. Res. 14, 257-62. Tyndale-Biscoe, C. H. (1968). Reproduction and post-natal development in the marsupial Bettongia lesueur (Quoy and Gaimard). Aust. J. Zool. 16, 577-602. Tyndale-Biscoe, C. H. (1973). 'Life of Marsupials.' (Edward Arnold: London.) Wood, J. T., Carpenter, S. M., and Poole, W. E. (1981). Confidence intervals for ages of marsupials determined from body measurements. Aust. Wildl. Res. 8, 269-74. Manuscript received 2 November 1987; revised 3 June 1988; accepted 17 April 1989