J. Zool., Lond. (1989) 219, 201-208

Population dynamics of the narwhal Monodon rnonoceros: an initial assessment (Odontoceti: Monodontidae)

MICHAEL KINGSLEY Arctic Management Research, Department of Fisheries and Oceans, 501 University Crescent,

Winnipeg, Manitoba R3T 2N6 Canada

(Accepted 13 December 1988)

(With 1 figure in the text)

A simplified population equation for Monodon Pnonoceros shows that current estimates of the values of the life history variables are inconsistent with the hypothesis of a stationary population. Instantaneous adult mortality must be less than 0,10/yr, not the published estimate of 12-13%/yr, for accepted values of the other variables to be consistent with stationarity. For sustainable harvest, permissible exploitation rates are no larger than 3-4%/yr, and instantaneous natural mortality must then be well below 10%/yr. Present uncertainty in the values of survival rates, both of adults and young, contributes twice as much to uncertainty in population growth rate as does uncertainty in reproductive rates.

Contents Page

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Population dynamic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Estimates of vital rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Adult mortality and population growth rate. . . . . . . . . . . . . . . . . . . . 204 Sensitivity of population growth rate to changes in vital rates . . . . . . . . . . . . 206 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Introduction

The narwhal, Monodon monoceros, is a medium-sized odontocete, the males averaging 4.7 m in length and 1600 kg in weight, the females 4 m and 900 kg (Mansfield, Smith & Beck, 1975). It was hunted in the eastern Canadian Arctic by native people and by commercial whalers (Mitchell & Reeves, 198 l), but is now hunted only by natives. They hunt at the fast-ice edge and at leads, and in open water from small boats (Finley, Davis & Silverman, 1980; Kemper, 1980). Recent international concern for the species has prompted action against trade in narwhal tusks: the European Economic Community (Regulation 3626/82) imposed an embargo on their import from Canada, and the Federal Republic of Germany proposed to list narwhal in Appendix I of the Convention on International Trade in Endangered Species at the 1985 meeting of signatories in Buenos Aires. The information on population size and population dynamics of the species is imperfect and this impedes management (Davis, Finley & Richardson, 1980). Information on

Present address: Institute Maurice Lamontagne, 850, Route de la Mer, Mont-Joli, Que. Canada G5H 324

0952-8369/89/010201+08 $03'00 20 1

0 1989 The Zoological Society of London

202 MICHAEL KINGSLEY

population size is reviewed by Smith et al. (1985). In this article, I examine population dynamics, reviewing available estimates of vital rates, and examining the implications of possible errors in them on the population growth rates. The vital rates of some other species, notably the closely- related beluga (Delphinupterus leucas), have been used for comparison because the information on vital rates of narwhal is not definitive.

Population dynamic equation

I have used an analytical method here rather than a computer simulation, as the data available will not support complex modelling. My approach has been to use a general model equation, fit parameters based on available data, and examine the effects of small variations in parameter values on population growth rates. Analytical methods allow general statements about relationships between variables and ‘focus one’s attention on gaps in the data’ (Eberhardt, 1985). The standard Lotka (1 907) population equation is:

00

1 I, * m, * exp( --r * x) dx = 1 0

where 1, is survival to age x; m, is instantaneous reproductive rate at age x; and r is the instantaneous rate of population increase (Pielou, 1969). This equation requires rates of reproduction at, and survival to, every age. An approximately equivalent birth-pulse model may be derived by these substitutions:

I, = K-exp( - s - x )

where s is the constant adult instantaneous mortality (/year) and Kis subadult survival relative to adult (Eberhardt & Siniff, 1977);

m X = R / 6 t , a < x < a + 6 t , a + I < x < a + l S 6 t , . . . in, = 0 elsewhere

where R is the probability that a calf will be female; a is the age (years) at first reproduction; and l i s the calving interval (years). The resulting equation:

(I + nl + f i r 00

1 lim f ( K * R/6t) - exp( - ( r + s)x) dx = 1 n = O B t - r O a + n I

models a birth-pulse every I years starting at age a, and integrates and sums to:

K - R = exp(a-(r +s))-( 1 - exp( - l . ( r + 3))) ( 3 ) Reproduction is seasonal, therefore individual whales have ages at first reproduction a which

are approximately integer, and individual values of reproductive interval I are also integer. However, the population means of these variables, which are the effective statistics in the dynamics of populations, are non-integer, so continuous ranges are considered for them. All uncertainty of absolute mortalities is incorporated in s; while K, and ( r+s ) , could in theory be estimated from stable age distributions. If l . ( r + s ) is small, equation (3) simplifies further to:

( r + s) .exp(a-(r + s)) - K - R/I= 0 (4)

POPULATION DYNAMICS OF M O N O D O N 203

R is assumed to be 0.5. The sensitivity of the rate of population increase to small relative changes in the vital rates is given by:

a.dr/da= - ~ ( r + s ) ~ / { l +a(r+s)}/yr ( 5 )

s.dr/ds= -s{ 1 +a(r+s))/{l +a(r+s)}/yr

I . dr/dI= - ( r + s)/{ 1 + a(r + s)}/yr

K.dr/dK=(r+s)/{ 1 +a(r+s)}/yr

Estimates of vital rates

Mortality may be estimated by analysis of age distributions, subject to assumptions on population growth (Eberhardt, 1985); by intensive mark-recapture, again subject to relevant assumptions; or by analysis of population trends, subject to information on reproductive rates (Polachek, 1984). There is no mark-recapture data for narwhal. Ageing has proved difficult and the number of growth layer groups laid down annually has proved controversial (Int. Whal. Comm., 1980; Mitchell, Kozicki & Reeves, 1987). There may be multiple layering in early life (Hay, 1980). Hay (1984) indicated, with reservations, that one layer was laid down per year in embedded teeth and mandibular bone to maxima of 30 layers in females and 50 in males. Bada, Mitchell & Kemper (1983) estimated 0.7-04 layers/year in the tusk. Chapman-Robson (1960) analysis of the distribution of layer counts in Hay’s samples (1984: figs 18 and 19) estimates adult mortality at 0*283/layer for females and O+107/layer for males, but the female estimate may be biased by limits on accumulation of layers in females (Hay, 1980). Adult mortality was given as 0.094-0.128/yr for narwhal (Braham, 1984) and as 0.12/yr (Hay, 1984: table 17). (Maximum?) longevity for narwhal is suggested to be 50 yr (Braham, 1984), which would imply (minimum?) adult mortality of only about 2%/yr. Such an estimate would give narwhal twice the reproductive longevity of beluga, and depends on unreliable ageing methods (Braham, 1984). For beluga, published values for s included 0.09-0.16/yr (Braham, 1984), and 0*054/yr (Burns &Seaman, 1985: from analysis of an age distribution).

Sergeant ( I 962) estimated mortality for Globicephala melaena, another large odontocete, at about 0.045/yr; a revised value (Kasuya, Sergeant & Tanaka, 1988, table 4, column B, averaged mortality 7-43 yrs) was 0.05/yr. Globicephala melaena is larger, slower-growing, and slower- maturing than the narwhal, and should have lower mortality. From comparison with both beluga and Globicephala, a mortality of 0.02 for narwhal is too low. I consider here 0-05 to be a lower limit for narwhal, and 0.13/yr an upper limit. The relative error for these values, range/midpoint, is (0.13 - 0.05)/(0.13 + 0.05)/2) or 0-9.

Absolute subadult mortality is difficult to estimate (Eberhardt, 1985). It may vary much between taxa, depending particularly on maternal care, which in odontocetes is prolonged (Brodie, 1969). But rates of subadult survival relative to adult survival may be derived from information on population structure, if a population is stable, and population structure data is free of age-specific capture or observation biases. Burns & Seaman (1985: table 7) show mortality of prime adult beluga at about 5.4%/yr, and total mortality of subadults from 0 to 5 yr at 50%; this makes relative subadult survival equal to 0.5/0,9465 = 0.66. For Globicephala melaena, first-year mortality was estimated at 35%, and mortality in all other years at 4.5% (Sergeant, 1962), giving K=0.65/0.955 =0.68. In a simulation model of dolphin population dynamics, Reilly & Barlow (1986), with limited information, considered a wide range of values from 0.5 to 0.94. I here consider values from 0.5 to 0.9. The relative error is 0.57.

204 MICHAEL KINGSLEY

TABLE I Female year-class lengths (Hay, 1980) and the parameiers of aJitted

Gompertz growth curve

Parameters of Gompertz Age (Yr) Length (cm) growth curve

0 163 1 247 L=400 em 2 290 k=0.193/yr

Adult 400 I/L = 0.85 at a = 3.21 yr

I(a) = L exp (- exp ( - ek(a-A)))

3 332 A = -0.258 YC

Average age at sexual maturity (ASM) for female narwhal has been estimated at 5-8 yr (Braham, 1984) or 12 growth layers (Hay, 1984, fig. 36). Interspecific comparisons predicted ASM at 6.4 yr (Ohsumi, 198 1 , cited in Hay, 1984). I fitted a Gompertz growth curve (Laird, 1966) (Table I) to lengths of female subadult year-classes (Hay, 1980, table 11). If the length at sexual maturity in marine mammals is about 85% of final length (Laws, 1956), the female ASM is predicted at about 3.2 yr and, with a 14-month gestation (Hay, 1984), the lowest possible age at first production is 5 Yr.

Mean ASM for beluga was stated to be 5 yr (Brodie, 1971), and 5.3 yr (Burns & Seaman, 1985), implying first production no earlier than 6.2-6.5 yr.

For Globicephala melaena, ASM was 6-8 yr (Sergeant 1962, confirmed by Kasuya et al., 1988), 7 yr (Martin, Reynolds & Richardson, 1987), and 9-10 yr (Martin & Desportes, 1987). I fitted a growth curve by eye to the plot of female age and length data (Kasuya et al., 1988, fig. 4) and found that it confirmed Laws’s (1956) approximate rule.

Age at first pregnancy for narwhal was given as 10-12 yr (Braham, 1984), i.e. 4-5 yr after maturity. A suggested range for the population mean age at first production, a, is 7 to 13 yr; the relative error is 0.6.

Reproductive interval can be estimated from the proportions of females in early pregnancy, in late pregnancy, and lactating but not pregnant. The narwhal and the beluga gestate for about 14 months, and lactate for about 20, for a reproductive interval of 3 yr (Brodie, 1971; Hay, 1984; Burns & Seaman, 1985). Females may sometimes bear young only 2 yr after a previous birth (Hay (1984) estimated a 20% probability of fertile ovulation 9 months after bearing) or rest a year between pregnancies. The range of pregnancy rates (Hay, 1984) was 0.3-0.38, indicating reproductive intervals in the range 2.6-3.3 yr. Myrick et al. (1986) indicate that there is large variation in ovulation rates within odontocete populations, but these average out. The best estimate for mean productive interval is 3 yr, with range 2.5-3.5 yr. The relative error is 0.24.

Adult mortality and population growth rate

The relationships of the vital rates for stationary populations are shown in Fig. 1. The largest value for instantaneous adult mortality which could be compatible with a stationary population is about O.l/yr, and so high a value could only be reconciled with a stationary population if subadult survival and the reproductive parameters were at extreme values. For a stationary population with adult mortality higher than O.l/yr, the limits on the other vital rates would have to be extended.

POPULATION DYNAMICS OF M O N O D O N

101

h

% a 9- a

(D

K 0 0

- Y

.- c

$ 8- a P 2 c

;5 c : 7- 8

6-

205

/ I I I I

0.5 0.6 0.7 0.8 0.9 1 .o Relative subadult survival (K)

1=3.5 _ _ _ _ 1 ~ 3 . 0 - k 2 . 5 _ - FIG. 1. Relationship of age at first reproduction (a), instantaneous adult mortality (s), relative subadult survival ( K ) and

reproductive interval ( I ) for stationary populations. (Graphs for non-stationary populations can be obtained by adjusting adult mortality, percentage point for percentage point.)

Moderate values of the other parameters (0.6< K<O.75; 7 < a < 11 yr; I = 3 yr) limit s for stationarity to 0-055-0.073/yr. Using estimates a = 10 yr, K = 0.7, I= 3 yr and R = 0.5, eq’n (3) gives (r+s)=0.066.

The life expectancy at maturity is given by the reciprocal of the average adult instantaneous mortality; mortality values of 0.1 to 0.128/yr (Braham, 1984) yield life expectancies of only 8-10 yr. Sergeant (1973) suggested a life history for female beluga in which reproductive senility starts to occur at 23 yr; the same may be true of narwhal (Hay, 1984). If a typical mammalian U-shaped qx curve obtains, with very high survival of prime adults from maturity (at 6 yr) to reproductive senility and rapidly rising mortality at 23 yr, female reproductive life expectancy is somewhat less than 17 yr, i.e. reproductive mortality is about 0*059/yr.

Instantaneous adult mortality must therefore be lower than the 0-1-0.13/yr suggested by Braham (1984). It is evident that O.l/yr should be taken as a maximum value for total adult mortality. This reduces the range considered below to 0.05-0.1 O/yr, and the relative error becomes 0.67.

This slowly reproducing cetacean species cannot maintain stationary populations unless adult mortality is lower than values so far published for it. Sergeant (1981) calculated that monodontid whales could be sustainably exploited at a rate of 5%/yr without decline. The analysis above indicates that this value is too high, unless natural mortality is presumed to be completely replaced by exploitation, and that conservative harvests should be 3%-4%/yr.

206 MICHAEL KINGSLEY

Sensitivity of population growth rate to changes in vital rates

How much small changes or errors in the separate life history variables affect the estimate of population growth rate can be analysed by the differentials of equations (5) through (8). The relative sensitivities of the growth rate to ASM, adult survival, reproductive interval and relative subadult survival are:

a.dr/da : s*dr/ds : 1*dr/dI: K - dr/dK = - a * (r + s>’ : - s{ 1 + a(r + s)] : - ( r + s) : ( r + s) (9)

If r is expressed as ks, these sensitivities are in the ratios:

a .s( l+k):a .s+l / ( l+k): l : -1 (10) The effects of changes in a and s relative to each other depend on the rate of population growth. If the population is declining (k < 0 and 1/ ( 1 + k ) becoming large) adult mortality dominates, and early reproduction is of little benefit. If it is stationary at carrying capacity ( k = 0 ) , these ratios become:

a.s:a.s+ 1 : 1 : - 1 (11)

and s-dr/ds is still largest. In a growing population, with small a, small s and k >> 1 , reproductive age becomes more significant. This suggests that reproductive age should tend to be labile and s stable against changes in population growth rate.

Considering central values for the ranges given above:

a= 10 yr s = 0.075/yr 1 = 3 yr

K= 0.7 for r = 0.0098/yr;

equations (5) through (8) give:

a-drlda = - 0.0257/yr s.dr/ds= -0*075/yr I-dr/dI= -0.0395/yr

K.dr/dK= @0395/yr.

The largest value, s-dr/ds= -0.075, shows that the population growth rate is most sensitive to small relative changes in the adult mortality. It is only about 112 as sensitive to changes in Iand K , and only 1/3 as sensitive to change in a.

The relative error (range/midpoint) is 0.6 for a; 0.9 for s; 0.24 for 4 and 0.57 for K. These are in the ratios:

2.5 : 3.75 : 1 : 2 4

The other parameters have two or three times the relative error of the reproductive interval. s has the highest, but if its upper limit is reduced to 0.1 /yr, its relative error becomes 067, and the ratios above become:

2.5 : 2.8 : 1 : 2.4

POPULATION DYNAMICS O F M O N O D O N 207

The approximate contribution of each error to the uncertainty in growth rate is the product of the relative error and the sensitivity of the population growth rate. The error in a has effect:

0.026.0.6 =0.015 yr-'; ins (reduced) 0.075 -0.67 = 0.050 yr-I; in Z 0.040 -0.24 = 0.0 10 yr- I ;

and in K 0.040-0.57=0.023 yr-I.

Zis precisely known, and a has little effect on population growth rate, so the current imprecision of these variables is not of great significance. Buts is imprecisely known, and affects r most strongly; it is the parameter of population dynamics for which current uncertainty is the most significant in affecting prediction of population growth rates.

Summary

The effects of error in the life history variables on predictions of population dynamics of narwhal are a paradigm of the central continuing problem of studying and managing the dynamics of wild populations: that survival rates, especially of adults, are critically important, but difficult to estimate with the necessary precision, while reproductive parameters, which are much easier to study and to estimate, have less effect on population trends.

REFERENCES

Bdda, J. L., Mitchell, E. & Kemper, B. (1983). Aspartic acid racemization in narwhal teeth. Nature, Lond. 303 418-420. Braham, H. W. (1984). Review of reproduction in the white whale, Delphinapterus leucas, narwhal, Monodon monoceros,

and Irrawaddy dolphin, Orcaella brevirostris, with comments on stock assessment. Rep. int. Whal. Commn Spec. Issue 6 81-89.

Brodie, P. F. (1969). Duration of lactation in Cetacea: an indicator of required learning? Am. Midl. Nut. 82: 312-314. Brodie, P. F. (1971). A reconsideration of aspects of growth, reproduction and behaviour of the white whale

(Delphinapterus leucas) with reference to the Cumberland Sound, Baffin Island, population. J . Fish. Res. Bd Can. 2 8

Burns, J. J . & Seaman, G . A. (1985). Investigations of beluga whales in coastal waters of western and northern Alaska. I I . Biology and ecology. Final rep. to NOAA OCSEAP under contract NA 81 RAC 0049 by Alaska Dept. Fish and Game, Fairbanks.

1309-1318.

Chapman, D. G. & Robson, D. S. (1960). The analysis of a catch curve. Biometrics 16 354-368. Davis, R. A,, Finley, K. J. & Richardson, W. J. (1980). Thepresent status andfuture management ofArctic marine mammals

in Canada. Science Advisory Board of the Northwest Territories, Yellowknife, NWT. Eberhardt, L. L. (1985). Assessing the dynamics of wild populations. J . Wildl. Mgmt 4 9 997-1012. Eberhardt, L. L. & Siniff, D. B. (1977). Population dynamics and marine mammal management policies. J . Fish. Res. Bd

Finley, K. J., Davis, R. A. & Silverman, H. B. (1980). Aspects of the narwhal hunt in the eastern Canadian Arctic. Rep. int.

Hay, K. A. (1980). Age determination of the narwhal Monodon monoceros L. Rep. Int. Whal. Commn Spec. Issue 3: 119.-

Hay, K. A. (1984). The life history qfthe narwhnl (Monodon monoceros L . ) in the eastern Canadian Arctic. Unpubl. PhD

International Whaling Commission (1980). Report of the workshop. Rep. in/. Whal. Commn Spec. Issue 3 1-50. Kasuya, T., Sergeant, D. E. & Tanaka, K. (1988). Reexamination of life history parameters of long-finned pilot whales in

Kemper, J. B. (1980). History of use of narwhal and beluga by Inuit in the Canadian eastern Arctic including changes in

Can. 34: 183-190.

Whal. Commn 30: 459-464.

132.

thesis, McGill University, Montreal, PQ.

the Newfoundland waters. Sci. Rep. Whal. Res. Inst. 3 9 103-1 19.

hunting methods and regulations. Rep. int. Whal. Commn 30: 481-492.

208 MICHAEL KINGSLEY

Laird, A. K. (1966). Postnatal growth of birds and mammals. Growth 3 0 349-363. Laws, R. M. (1956). Growth and sexual maturity in aquatic mammals. Nature, Lond. 178 193-194. Lotka, A. J. (1907). Studies on the mode of growth of material aggregates. Am. J. Sci. 24: 199-216. Mansfield, A. W., Smith, T. G. & Beck, B. (1975). The narwhal, Monodon monoceros, in eastern Canadian waters. J . Fish.

Res. Bd Can. 3 2 1041-1046. Martin, A. R. & Desportes, G. (1987). Reproduction and foetal growth in the pilot whale Globicephaia melaena around the

Faroe Islands. Abstracts, Bienn. Conf. Biol. Mar. Mamm., Miami, Fia, December 1987 No. 7. SOC. Mar. Mamm. Martin, A. R., Reynolds, P. & Richardson, M. G. (1987). Aspects of the biology of Pilot whales (Globicephala melaena) in

recent mass strandings on the British coast. J . 2001.. Lond. 211: 11-23. Mitchell, E. D. & Reeves, R. R. (1981). Catch history and cumulative catch estimates ofinitial population size ofcetaceans

in the eastern Canadian Arctic. Rep. int. Whal. Commn 31: 645-682. Mitchell, E. D., Kozicki, V. M. & Reeves, R. R. (1987). Estimation of relative age from non-destructive examination of

erupted tusks in male narwhals (Monodon monoceros). Abstracts, Bienn. Conf. Biol. Mar. Mamm., Miami, Flu. December 1987 No. 7. SOC. Mar. Mamm.

Myrick, A. C. , Jr, Hohn, A. A,, Barlow, J. & Sloan, P. A. (1986). Reproductive biology of female spotted dolphins, Stenella attenuata, from the eastern tropical Pacific. Fish. Bull. U S . natn. ocean. atmos. Admn 8 4 247-259.

Pielou, E. C. (1969). An introduction to mathematical ecology. New York: Wiley-Interscience. Polacheck, T. (1984). Reproductive estimates as a source of information on survival rates. Rep. int. Whal. Commn Spec.

Reilly, S . B. & Barlow, J. (1986). Rates of increase in dolphin population size. Fish. Bull. U S . natn. ocean. atmos. Admn 84:

Sergeant, D. E. (1962). The biology of the pilot or pothead whale, Globicephala melaena (Traill) in Newfoundland waters.

Sergeant, D. E. (1973). Biology of white whales (Delphinaprerus leucas) in western Hudson Bay. J . Fish. Res. Bd Can. 3 0

Sergeant, D. E. (1981). On permissible exploitation rates of Monodontidae. Rep. int. Whai. Commn 31: 583-588. Smith, T. G., Hammill, M. O., Burrage, D. J. & Sleno, G. A. (1985). Distribution and abundance of belugas,

Delphinapterus leucas, and narwhals, Monodon monoteros, in the Canadian high Arctic. Can. J . Fish. aquat. Sci. 4 2

Issue 6 161-173.

527-533.

Bull. Fish. Res. Bd Can. No. 132: 1-84.

1065 -1090.

676-684.