OCTOBER 1984 

ESTIMATING DOLPHIN JUVENILE 

SURVIVAL RATES FROM THE 

PROPORTION OF CALVES NURSING 

Tom Polacheck 

NOAA-TM-NMFS-SWFC-5 1 

U.S. DEPARTMENT OF COMMERCE 

National Oceanic and Atmospheric Administration 

National Marine Fisheries Service 

Southwest Fisheries Center 
NOAA Technical Memorandum NMFS 

The National Oceanic and AtmosphericAdministration(NOAA), organized in 1970, hasevolved intoan 

agency which establishes national policies and manages and conserves our oceanic. coastal, and atniospheric resources An organizational elenlent within NOAA, the Office of Fisheries is responsible for 

fisheries policy and the direction of the National Marine Fisheries Service (NMFS) 

In addition to its formal publications, the NMFS uses the NOAATechnical Memorandum series to issue 

informal scientific and technical publications when complete formal review and editorial processing are 

not appropriate or feasible Docunlents within this series, however, reflect sound professional work and 

may be referenced in the formal scientific and technical literature 
NOAA Technical Memorandum NMFS 

This TM series is used for documentation and timelv communication 01 Dreliminary results. imterim reLYom. or 

Specla1 purpose information. and have not received complete formal revtew. editorial control. or delailed ediling 

OCTOBER 1984 

ESTIMATING DOLPHIN JUVENILE 

SURVIVAL RATES FROM THE 

PROPORTION OF CALVES NURSING 

Tom Polacheck 

South Pacific Commission 

B.P. D5 

Noumea Cedex 

New Caledonia 

NOAA-TM-NMFS-SWFC-5 1 

U.S. DEPARTMENT OF COMMERCE 

Malcolm Baldrige. Secretary 

National Oceanic and Atmospheric Administration 

Anthony J. Calio. Deputy Administrator 

National Marine Fisheries Service 

William G. Gordon. Assistant Administrator for Fisheries 
CONTENTS 

Page 

INTRODUCTION .......................................................... 1 

METHODS. .............................................................. 2 

RESULTS AND DISCUSSIONS........ ....................................... 4 

LITERATURE CITED ...................................................... 7 

APPENDIX.. ............................................................ 13 

i 
LIST OF TABLES 

Tab1 e Page 

1 

2 

Figures 

1 

2 

3 

Estimates of the values of the parameters used 

in the analysis of survival rates ......................... 8 

The range of estimates for the average annual 

survival rate from birth to maturity based on 

Method I1 estimate of the length of lactation 

when Sw was varied between 0.70 and 0.98. ................. 9 

LIST OF FIGURES 

Page 

Estimated annual survival rate of non-nursing calves 

(SM) plotted as a function of the estimated survival 

rate of nursing calves (S 1 for a range of values for 

the net rate of increase yf) for the northern off-shore 

spotted stock. Figure la is for Method I estimate of 

the length of lactation and Figure lb is for Method I1 ....... 10 

Estimated annual survival rate of non-nursi ng cal ves 

(SMl plotted as a function of the estimated survival 

rate of nursing calves (S,) for a range of values for 

the net annual rate of increase (A) for the northern 

whitebelly spinner stock. Figure 2a is for Method I 

estimate of the length of lactation and Figure 2b is 

Method 11. Note that in Figure 2b no curve has been 

plotted for a net rate of increase equal to 1.05 

because no solutions exist in which both of the 

estimated survival rates are less than one................... 11 

Estimated annual survi Val rate of non-nursi ng calves 

(SM) plotted as a function of the estimated survival 

rate of nursing calves (S,) for the eastern spinner 

stock for a range of values for the annual net rate 

of increase (A) for Method I1 estimate of the length of lactation .................................................. 12 

ii 
ESTIMATING DOLPHIN JUVENILE SURVIVAL 

RATES FROM THE PROPORTION OF CALVES NURSING 

Tom Pol acheck* 

Southwest Fisheries Center 

National 

La 

Marine Fisheries Service, NOAA 

Jol 1 a , Cal i forni a 92038 

and 

Department of Bi 01 ogy 

University of Oregon 

Eugene, Oregon 97403 

INTRODUCTION 

Information on the age structure of a population is often used to 

estimate survival by comparing the number of individuals in successive age 

classes. Techniques for determining the age of individuals in a population 

may not be available or may be prohibitively costly or time consuming. In 

contrast, cl assi f icati on of i ndi vi dual s into reproduction categories is often 

easier. If some of the reproductive states are of relatively constant 

duration and data exist on the time spent in these states, classification 

according to these states can provide a sorting of individuals into disjoint 

age groupings and, thus, provide information on the age structure of the 

population. The accuracy of determining the actual age structure of a 

popul ati on from reproductive cl assi f ications depends upon the vari abil i ty of 

the duration of the various categories being used. Biases may exist in such 

classification if reproduction is seasonal or if sampling is non-uniform. 

However, in the absence of other data, the use of reproductive-state 

classification may be the only source of information on survival rates. 

Anytime a sample can be divided into disjoint age groupings, these age 

groupings have a potential to provide information on survival rates. As with 

any survival estimate based on the age structure of a population, it is 

necessary to assume a stable age distribution, a net rate of increase, and 

some form of functional relationship for the survival rate within an age 

interval. (Note, the net rate of increase is usually assumed equal to zero 

while the relationship for the survival rate within an interval is assumed 

constant. 1 

In Polacheck (1984) , I discuss how the reproductive classification into 

immature and mature females, along with an estimate of the fecundity rate and 

the age of maturity, can be used to provide estimates of survival rates during 

these two stages, and I apply this method to data from samples of dolphins 

* Present address: South Pacific Commission, B.P.DS, Noumea Cedex, New 

Cal edoni a 
2 

killed incidentally in tuna purse seine fishery in the eastern tropical 

Pacific. In the present paper, general equations are presented for estimating the survival rates in two successive age groupings in which the number of age classes in each group differs. These equations are then applied to the 

reproductive data on dolphins in the eastern tropical Pacific to provide information on survival rates for nursing and non-nursing immature females. 

METHODS 

Two equations were used to estimate the survival rates; one is based on a 

continuous model of population growth, and the other on a discrete model. For 

a continuous model of population growth, the ratio, Rc, of the number of 

individuals in two successive age groupings which span different lengths of 

time can be expressed as: 

where w = the duration of the first stage 

m = the duration of the second stage 

qw = the instantaneous mortality rate during w 

qm = the instantaneous mortality rate during m 

r = the instantaneous net rate of increase 

A derivation of this expression is provided in the appendix. A similar 

expression for a discrete model of population growth is 

i 

S 

W W 

T: - 

i =O i 

h 

R= 

d 

where w and m are as above and 

m m 

sw 

- 

W 

T 

i 

s 

W 

x A 

Sw = the survival rate per unit time during w 

Sm = the survival rate per unit time during m 

x = the finite rate of increase per unit time. 
3 

Derivation of equation 2 follows from expressing the stable age vector for a 

Leslie matrix as successive products of Si/h, where Si is the survival rate at 

age i (e.g. Pielou, 1977). 

Given a current rate of increase and the ratio of the number in two 

successive age groupings, these expressions can be used to provide an estimate 

of the survival rate during the second interval as a function of the rate 

during the first. They do not provide a single estimate of the survival rate 

between ages w and m (as a normal age distribution would) because of the need 

to account for the individuals in age group w that will not survive to 

age m. However, in particular situations, as will be shown with the data on 

dolphin populations, given values for w, m, and R, the combination of 

meaningful solutions for the survival rates during those periods may be 

1 imi ted. 

Data are available on the number of immature females and the number of 

females lactating from the samples available of dolphins from the eastern 

tropical Pacific for the northern off-shore spotted stock of Stenella 

attenuata, and the eastern spinner, and northern whitebelly spinner stocks of 

- S. longirostris (Henderson et al., 1979 1 and Perrin and Henderson, 19792 and 

Perrin, et al., 1976). Estimated values for these three stocks of the length 

of lactation (i.e. the age of weaning), the age of maturity, and the 

proportion of immature females which are nursing are given in Table 1 and are 

the values used in the analysis below. This information can be used to 

examine the survival rates of nursing and non-nursing calves. The estimate of 

the proportion of immature females that are nursing was derived as follows: 

If one assumes a random sample, the number of lactating females provides an 

estimate of the number of nursing calves in a sample as dolphins almost always 

have a single birth. By assuming a 50:50 sex ratio for nursing calves, the 

ratio of one half of the number of lactating females to the number of immature 

females provides an estimate of the proportion of immature females that are 

nursing. Note that the ratio of nursing to non-nursing calves has been 

expressed in terms of the proportion of immature calves which are nursing so 

that the results can be easily compared with an earlier sensitivity analysis 

of the reproductive data (Polacheck, 1983). 

As indicated in Table 1, the length of lactation has been estimated in 

two different ways (see Henderson, et al., 19791 for a complete description of 

both methods). Method I is actually an estimate of the average length of time 

all females nurse, which includes the females which stop nursing because of 

the death of their young. Method I1 is a direct estimate of the age of 

lHenderson, J., W.F. Perrin and R.B. Miller. 1979. Gross annual production 

in dol phi n popul ati ons ( Stenel 1 a spp) i n the eastern tropical Paci f i c , 1973- 

1978. NMFS-SWFC, SOPS/79/33.- 

*Perrin, W.F. and J.R. Henderson. 1979. Growth and reproductive rates in 

populations of spinner dolphins, (Stenella longirostris) with different 

histories of exploitation. NMFS-SWFC Admin. Rept. #LJ -- i9 29. 
4 

weaning based on an estimation of the age of the oldest individual still 

nursing. Method I underestimates the actual age of weaning ( DeMaster personal 

communication). The relationship between Method I estimates of the average 

length of lactation and the actual age of weaning can be expressed as 

PPP 

1-e -9 w w 

w, = (3) 

where wI = method I estimate of the length of lactation 

qw = the instantaneous cal f mortal i ty rate before weaning 

w = the actual length of lactation 

It has been suggested that qw can be estimated by substituting Method 11's 

estimate for the length of lactation for w in equation 3 (DeMaster, Personal 

communication). However, because of potential biases in both methods of 

estimating the length of lactation (Polacheck, 19841, separate analyses will 

be presented below for both Method 1's and Method 11's estimates of the length 

of the lactation period. Note that whenever results are presented for Method 

I estimates, the actual value for w used in solving either equation 1 or 2 is 

the value of w derived from solving equation 3 for w. In other words, the 

following expression for w was substituted into equations 1 or 2 

In (1 - wlqwl 

w= 

9, 

(4) 

When equation 4 was used in equation 2, qw was assumed to equal -In sw. 

RESULTS AND DISCUSSION 

Both equations 1 and 2 were solved for the survival rate of non-nursing 

calves using the parameter estimates in Table 1. Equation 2 was solved for a 

discrete time unit equal to one month. When converted to a common time scale, 

the survival rate estimate from either equation was essentially the same. 

Thus, only results from the continuous model (equation 1) will be presented. 

All rates presented below have been converted to a common finite time scale of 

one year. 

In Figures 1-3, the survival rate of non-nursing calves (Sm) has been 

plotted as a function of the survival rate of nursing calves (S ) for a range 

of values for the net rate of population growth for the lhree dolphin 

populations. The estimates of Sm are a decreasing function of Sw and the net 

rate of increase. for Method I 

estimates of the length of lactation than for Method I1 estimaTes due to the 

interaction between Sw and w for Method I (i.e. Equation 4). (Note in figure 

3 no results are shown for Method I estimates because no biologically possible 

Sm is a lot more sensitive to changes in S 
5 

sol utions exist for the combination of input parameters. ) 

When w is not a function of Sw but is held constant, as with Method I1 

estimates, the overall survival rate from birth to age m appears almost 

insensitive to any particular combination of values for S,,, and S which are 

used to solve equation 1. This insensitivity is illustrated in Talle 2 where 

the range of overall average annual survival rates has been tabulated for the 

three stocks using Method 11's estimate of the length of lactation when Sm is 

allowed to range from 0.70 to 0.98. The estimate of overall average annual 

survival rate varies by at most 0.019 for a given value of net rate of 

increase. This means that estimating survival rates by this method is useful 

because only the overall survival rate from birth to maturity and not the age 

specific rates within this period affect the overall dynamics of the 

population model. 

Using the ratios in Table 2 as estimates of the overall survival rate 

from birth to age of maturity, the results appear not to be consistent with a 

hypothesis of density dependent changes in juvenile survival rates given 

current understanding of the level of exploitations for these three stocks 

(Smith, 19793 1. Thus, the 1 east expl oi ted stock, the northern whi tebel ly 

spinner, has the highest estimated survival rates; the most exploited stock, 

the eastern spinner, has intermediate estimates. 

These estimates, however, may be more of an indication of sampling biases 

and inconsistency in the data than of evidence for or against density 

dependent changes in juvenile survival rates. Thus, the results for the 

northern whi tebel ly spinner suggest that the overall juveni 1 e survival rates 

must be extremely high, approaching absolute biological limits, if the 

population is not declining. A similar conclusion was reached from an 

examination of the estimates of the sexually mature proportion and the 

pregnancy rate for thi s stock (Pol acheck, 19834 1. 

For the eastern spinner stock, Method I1 estimates of the length of 

lactation provide biologically plausible survival estimates. The survival 

estimates derived from Method I are not biologically possible unless the 

population was rapidly declining. The two methods of estimating the length of 

lactation are only partially independent as both are dependent upon the actual 

number of lactating females sampled. Thus, a rejection of the juvenile 

survival estimates based on Method I estimates of the length of lactation 

because they are biologically unreasonable brings into question the juvenile 

3Smith, T. (ed.1 1979. Report of the status of porpoise stocks workshop 

(August 27-31, 1979, La Jolla, California). NMFS-SWFC Admin. Report No. 

LJ-79-41. 

bPolacheck, T. 1983. Estimating current rates of increase and survival rates 

from reproductive data for dolphin populations in the eastern tropical Pacific 

NMFS-SWFC Admin. Report LJ-83-11. 25 pp. 
6 

survival estimates from Method 11. The difficulties with the estimates from 

Method I suggest a possible bias in sampling towards juveniles. Such a bias 

would affect both the estimate of the mature proportion and Method I1 

estimation of the length of lactation (Polacheck, 1983). The source of the 

problem in the estimates for Method I most likely is affecting the estimates 

of Method 11. Similar difficulties in obtaining juvenile survival rate 

estimates that were biologically possible in a population which was not 

rapidly declining were also encountered for this stock when analyzing the 

estimates of the sexually mature proportion and the pregnancy rates 

(Polacheck, 19834) which further suggests sampling problems. 

For the northern offshore spotted stocks, the survival estimates from 

both Methods I and I1 estimates of the length of lactation are consistent with 

a wide range of potential rates of increase and suggest that a reasonable 

estimate for the average overall juvenile survival rate for this stock may be 

in the range of 0.80-0.90. While this range of rates is somewhat lower than 

the range of juvenile survival rates for this stock suggested by an 

examination of the estimate of the sexually mature proportion and the 

pregnancy rate (Polacheck, 19834), the two sets of estimates do not appear 

highly inconsistent given the possible sampling variabilities in the input 

parameters. 

Previous papers have suggested that the average annual survival rate for 

all ages for these dolphin populations is -0.90 (NMFS, 19765; Smith, 197931. 

This rate was derived from the comparison of estimates of the gross annual 

rates of reproduction in different dolphin populations and the assumption that 

these popul ati ons are not decl i ni ng . However , thi s previous f i gure cannot be 

considered a direct estimate and provides no basis for evaluating the juvenile 

survival estimates in the present paper. 

5NMFS. 

i nvol ved in the eastern Paci f ic ye1 1 owfi n tuna f i shery . Report. LJ-76-29. 52 pp. 

1976. Report of the workshop on stock assessment of porpoises NMFS-SWFC Admi n . 
7 

LITERATURE CITED 

Mertz, 0. B. 1971. Life history phenomena in increasing and decreasing 

populations. In E. C. Pielou and W. E. Waters (eds.), Statistical 

Ecology 2: 361-3m. 

Perrin, W. F., 3. M. Coe and J. R. Zweifel. 1976. Growth and reproduction of 

the spotted porpoise, Stenella attenuata in the offshore eastern tropical 

Pacific. Fish. Bull. XU .. S 1 /4:229 269. 

Pennsylvania State Univ. Press , University Park , PA. 

-- 

- 

Pielou, E. C. 1977. Mathematical Ecology. 385 pp. John Wiley and Sons, New 

York. 

Polacheck, T. 1983. Possible effects of sampling biases in reproductive rate 

estimates for dolphins in the eastern tropical Pacific. NMFS-SWFC Tech. 

Memo. NOAA-TM-NMFS-SWFC-26. 21p. 

Polacheck, T. 1984. Reproductive estimates as a source of information on 

survival rates. In Rep. Int. Whal. Commn. (spec. issue 6): 181:185. 
a 

Table 1. Estimates of the values of the parameters 

used in the analysis of survival rates. 

Proportion of Age of Length of Lactation 

immature females maturity Method I Method 11 

nursing 1 (years) (months 1 4 (months ) 

Northern offshore spotted 

~ ~~~~ 

0.337 82 16.7 

Northern white- 0.261 63 16.0 

bel ly spinner 

Eastern 

spinner 

0.210 53 19.3 

18.7 

17.5 

10.3 

lcalculated from the pooled data for 1975-1979 as given in Henderson, et a1 . (1979) (see text footnote 1). Mature females whose reproductive state was 

undetermined were prorated to the various reproductive states according to the 

percentages in the various states. 

*Based on estimates in Perrin, et a1 . , 1976. 

3Based on estimates in Perrin and Henderson, 1979 (see text footnote 2). 

4Based on estimates in Henderson et a1 . , 1979 (see text footnote 1). 
9 

Table 2. The range of estimates for the average annual survival rate 

from birth to maturity based on Method I1 estimate of the 

length of lactation when $ was varied between 0.70 and 0.98. 

Stock 

Annual Net 

Rate of Increase 

Average Annual 

Survival Rate From 

Birth to Maturity 

Northern off- 0.90 

shore spotted 0.95 

1 .oo 

1.05 

Northern white- 0.90 

bel ly spinner 0.95 

1 .oo 

1.05 

Eastern spinner 0.90 

0.95 

1 .oo 

1.05 

0.721 - 0.738 

0.764 - 0.781 

0.807 - 0.825 

0.850 - 0.869 

0.871 - 0.873 

0.920 - 0.922 

0.968 - 0.971 

1.017 - 1.020 

0.810 - 0.815 

0.856 - 0.861 

0.902 - 0.907 

0.948 - 0.953 \ 
10 

1.00 

0.95 

0.90 

0.85 

E 

v) 

0.80 

0.75 

0.70 

0.65 

- 

- 

- 

- 

- 

- 

- 

- 

= 1.05 

x = 1.00 

= 0.95 

x = 0.90 

0.6OI ' I I I I I I 

0.70 0.75 0.80 0.85 0.90 0.95 1.00 

sw 

0.95 '-O°F 

0.90 

0.85 

E 

a 0.80 

0.75 

0.70 

= 1.05 

x = 1.00 

0.95 

x = 0.90 

0.65 k 

11 I I I I I I 

0.70 0.75 0.00 0.85 0.80 0.95 1.00 

sw 

Figure 1. Estimated annual survival rate of non-nursing calves (S ) plotted as 

a function of the estimated survival rate of nursing carves (S,) for 

a range of values for the net rate of increase (f) for the northern 

off-shore spotted stock. Figure la is for Method I estimate of the 

length of lactation and Figure lb is for Method 11. 
11 

1.05 1 

0.90 

0.85 

0.80 

E 

v) 

= 1.05 

x = 1.00 

= 0.95 

x = 0.90 

I I I I I I 

0.70 0.75 0.80 0.85 0.90 0.95 1.00 

sw 

E 

v) 0.90 

0.85 

x = 1.00 

= 0.95 

x = 0.90 

0.80 

I1 I I I I I I 

0.70 0.75 0.80 0.85 0.90 0.95 1.00 

SW 

Figure 2. Estimated annual survival rate of non-nursing calves (Sm) plotted 

as a function of the estimated survival rate of nursing calves 

(S,) for a range of values for the net annual rate of increase 

(A ) for the northern whitebelly spinner stock. Figure 2a is for 

Method I estimate of the length of lactation and Figure 2b is 

Method 11. Note that in Figure 2b no curve has been plotted for 

a net rate of increase equal to 1.05 because no solutions exist 

in which both of the estimated survival rates are less than one. 
12 

1 .oo 

0.95 

0.90 

E 

v) 

0.85 

0.80 

0.75 

x 

x 

x 

x 

= 1.05 

= 1.00 

= 0.95 

= 0.90 

I I I I I I I 

0.70 0.75 0.80 0.85 0.90 0.95 1.00 

sw 

Figure 3. Estimated annual survival rate of non-nursing calves (S,) plotted 

as a function of the estimated survival rate of nursing calves 

(S,) for the eastern spinner stock for a range of values for the 

annual net rate of increase ( A) for Method I1 estimate of the 

length of lactation. 
13 

APPEND1 X 

Derivation of an expression for the ratio of the number of 

individual s in two successive age groupings of unequal derivation 

for a continuous breeding population in a stable age distribution. 

From Mertz (19711, the probability density function of an individual being age 

x for a continuous breeding population with a stable age distribution is: 

c (x) = be'r Io() 

where 

q(y) = instantaneous death rate 

r 

b 

= instantaneous rate of increase for the population 

= asymptotic instantaneous birth rate per individual 

If q(y) is a constant between ages i and j and if qj is defined as 

qj = q(y) for i < y - < j 

then for x > i and x< - j 

Also note, that if q(y) is constant between ages i and j then for x>i and x<j - 

-r(x-i) -qj(x-i) c(x) = c(i) e e 

since c(i) = b e-ri I(i) 

Thus, if f equal s the lowest age of an individual in the first age grouping 

and (f+w) equals the ages of the oldest individual and qw equals the 

instantaneous mortal i ty rate during this period, the proportion of the 

population between ages f and f+w is 
14 

f +w f +w -r(x-f) -qw(w-f) 

c(x)dx = If c(w) e e dx If 

w -rx -qwx 

= c(w) Io e e dx 

By the exact same reasoning, the proportion of the population in the age 

grouping running from f+w to f+w+m is 

From expression 1, 

-rw -qww c(f+w) = c(f) 0 e 0 e (4) 

Thus by combining expression 2, 3 and 4 and then simplifying the ratio of the 

number of individual in the two successive age groupings can be expressed as 

f +w - (r+q,,, )w 

If c(x)dx = (r+qm) (1 - e 1 

- 

Rc - 

f+w+m -rw -q w -(r+qm)m 

) W If, e(x)dx (r+qw) e e (1-e 
RECENT TECHNICAL MEMORANDUMS 

Copies of this and other NOAA Technical Memorandums are available from the National 

Technical Information Service, 5285 Port Royal Road, Springfield, VA 22167. Paper copies vary in 

price. Microfishe copies cost $3.50. Recent issues of NOAA Technical memorandums from the 

NMFS Southwest Fisheries Center are listed below: 

NOAA TM-NMFS SWFC 41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

Diving patterns of the Hawaiian Monk Seal, Lisianski Island, 

1982. 

F. V. SCHLEXER 

(February 1984) 

D. J. ALCORN 

(April 1984) 

Atlas of airborne sea surface temperature observations in 

nearshore California waters, 1980-1 983. 

P.N. SUND 

(July 1984) 

Potential impact of deep seabed mining on the larvae of 

tunas and billfishes. 

W.M. MATSUMOTO 

(July 1984) 

Sampling commercial rockfish landings in California. 

A.R. SEN 

(July 1984) 

Histopathological manual for monitoring health of striped 

bass in relation to pollutant burdens. 

J.A. WHIPPLE, M. JUNG, R.B. MacFARLANE and R. FISCHER 

(August 1984) 

Hawaiian monk seal population research, Lisianski island, 

1982. 

H.S. STONE 

(August 1984) 

J. BARLOW and A.A HOHN 

(August 1984) 

Observations of the Hawaiian Monk Seal on Laysan Island 

from 1977 through 1980. 

B.W. JOHNSON and P.A. JOHNSON 

(October 1984) 

Hawaiian Monk Seal observations on French Frigate Shoals, 

1980. 

P.A. JOHNSON and B.W. JOHNSON 

(October 1984) 

The Hawaiian Monk Seal of Laysan Island: 1982 

Interpreting spotted dolphin age distributions