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Techniques for the Study of Primate Population Ecology (1981)

Chapter: PRIMATE POPULATION ANALYSIS

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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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Suggested Citation:"PRIMATE POPULATION ANALYSIS." National Research Council. 1981. Techniques for the Study of Primate Population Ecology. Washington, DC: The National Academies Press. doi: 10.17226/18646.
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7 Primate Population Analysis Population analysis is a technique for describing quantitatively the (1) size and structure of a population in terms of the numbers of animals of different age and sex and (2) changes in population size and structure due to births, deaths, emigration, and aging. Estimates or measures of such numerical attributes of a popu- lation are referred to as vital statistics. They include the birth- rate, death rate, population growth rate, and others. Vital statis- tics describe the dynamics of a population, or its demography. The vital statistics of a population are a function of the genetic makeup of a species and of environmental influences. Therefore, they are useful for making comparisons of species, or of popula- tions of the same species in different environments, or of the same population at different times. Knowledge of the vital statistics of a species, especially as they change under different environmental conditions, is fundamental to understanding its strategy for liv- ing. Apart from the value of such data for theoretical considera- tions concerning the evolution of behavioral-ecological phenom- ena, knowledge of a population's demographic behavior is essen- 135

136 TECHNIQUES IN PRIMATE POPULATION ECOLOGY tial in the applied fields of wildlife management and conserva- tion. DATA REQUIREMENTS The raw data required for population analysis are animals, in suf- ficient numbers, of each age and sex. In studies of many species, especially game animals, "kill records" or trapping records have been used as raw data. For primates, such data are rare. Instead, free-living animals are usually censused by an observer who clas- sifies them according to established aging or sexing criteria. The data in population analysis must meet two requirements that are often difficult to fulfill. First, the data must represent a true sample of the population from which it was drawn. The sec- ond requirement is that the census data must be accurately classi- fied by age and sex. Errors in estimates of vital statistics that can arise from faulty age classification are discussed in Chapter 5. Changes in a population can be monitored by making an an- nual census of animals of all ages by sex and by making a series of annual censuses to find newborn infants. BIRTHS AND FECUNDITY The purpose of censusing for births is to obtain information on the distribution of births in time, the sex ratio at birth, and the rates of birth and fecundity. BIRTH SEASONS Theoretically, births can be distributed in time according to either of two models. In a birth pulse model, all births in the pop- ulation for the entire year occur on the same day and on the same date every year. In a birth flow model, births occur at the same rate every day throughout the year. No population fits either model exactly, but any population will resemble one more than the other. Modern human populations closely fit the birth flow model, whereas most animal populations reproduce seasonally

Primate Population Analysis 137 and come closer to the birth pulse model, which is illustrated later in this chapter. The determination of the mean or median birth day and the standard deviation of the birthdates should re- place the imprecise descriptions of birth seasons based on the overall spread and peak periods of births, because these statistics allow more accurate division of the population into age classes. Caughley (1977) suggests that populations whose season of birth has a standard deviation of less than 30 days can be described by the birth pulse model. To define the season of births one must sample a population for the presence of newborn infants during a period of at least 1 yr. Mortality among newborn infants can be high. Therefore, to reduce the error of not counting infants that were born and died in the interval between successive censuses, one should make the census interval as short as possible. Since pregnant females are conspicuous in some primates, the censusing effort can be fo- cused on specific individuals until all adult females in a troop have recently given birth or can be seen nursing infants. In a birth census it is important to detect all births and esti- mate birthdates as accurately as possible. This will depend on the interval between censuses and the accuracy of age estimates based on infant morphology. When a birthdate is uncertain it should be estimated within absolute confidence limits, that is, a range of dates between which the birth is certain to have occurred. The frequency of births throughout a year or study period is tabulated by dividing the year or study period into equal periods. The period should not be less than either the interval between birth censuses, or the birthdate with the largest confidence limits as defined above. This prevents the tabulation from appearing to be more precise than it actually is. For most primates a period of 14 or 28 days is a convenient interval for analysis. If the birth season is brief and if the census data are very accurate, 7-day periods may be more appropriate. The method of analysis as outlined by Caughley (1977, pp. 72-74) is given in Table 7-1 and is illustrated with data from 150 births that occurred in the population of toque monkeys, Macaca sinica, at Polonnaruwa, Sri Lanka, between 1968 and 1972 (W. P. J. Dittus, unpublished). The same data are plotted as a histo- gram in Figure 7-1.

138 TECHNIQUES IN PRIMATE POPULATION ECOLOGY TABLE 7-1 Estimating Averages and Standard Deviations of Birth Season in a Population of Toque Monkeys at Polonnaruwa, Sri Lanka Period Limits (days) Period Code No. Births f Date x fx fc2 June 25-July 22 0-28 1 0 0 0 July 23-Aug 19 29-56 2 2 4 8 Aug 20-Sep 16 57-84 3 6 18 54 Sep 17-Oct 15 85-113 4 6 24 96 Oct 16-Nov 12 114-141 5 4 20 100 Nov 13-Dec 10 142-169 6 8 48 288 Dec 11-Jan 7 170-197 7 25 175 1,225 Jan 8-Feb 4 198-225 8 55 440 3,520 Feb 5-Mar 4 226-253 9 20 180 1,620 Mar 5-Apr 1 254-281 10 13 130 1,300 Apr 2-Apr 29 282-309 11 9 99 1,089 Apr 30-May 27 310-337 12 2 24 0.288 May 28-Jun 24 338-365 13 0 0 0 Totals 150 1,162 9,588 Source: W. P. J. Dittus, unpublished data, 1968-1972. The mean date of birth and its standard deviation are first cal- culated in terms of the period codes and then converted to their real values in days or dates. Mean date, Mn = 1,162 150 = 7.75 periods, where / = the number of births during a period, and x — the period code = 7.75 periods X 28 days per period = 217 days after midpoint of period 1 (June 25-July 22) = July 8 + 217 days = February 10. Grouping the data into intervals of constant width (each equal to one unit in coded form) results in a slight overestimate of the

Primate Population Analysis 139 variance. This is corrected by using Sheppard's correction for grouping (Snedecor and Cochran, 1967)—that is, by subtracting 1/12. Variance, s2 = 9,588 - (1.162)2/150 149 -Vi2 = 3.85 periods Most birth seasons are skewed in the positive direction—that is, the frequency of births rises steeply to a peak and then de- scends in a long tail to the right. In such asymetrical distributions 60 -i 50 - 40 - ~ 30 - 10 - Period code l 2 3 4 5 6 7 8 9 10 ll 12 13 Date Jly Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jly FIGURE 7-1 Distribution of births in toque monkeys at Polonnaruwa, Sri Lanka, from 1968 to 1972. The mean birthdate (Mn), the median birthdate (Md), and the standard deviation are shown on the bar above the histogram. Source: W. P. J. Dittus, unpublished data.

140 TECHNIQUES IN PRIMATE POPULATION ECOLOGY the median date of birth is a better indicator of the average birth day of all infants. The median birthdate is the day when half the infants have been born—in this case, the day when 150/2 = 75 infants have been born. The median birthdate is located in two steps. First, the median birth period is found by cumulatively summing the frequencies of births (/) until the median birth is located. In this example the median birth number, 75, falls within the median birth period from January 8 to February 4. Second, the median birthdate is calculated as follows (after Snedecor and Cochran, 1967): Standard deviation, s = Vvariance = V3.85 = 1.96 periods = 1.96 periods X 28 days per period = 55 days Standard error, s/VE/ = 55 days/Vl50 = 4.5 days g(C) Median, Md = L f where L = lower limit (in days) of the period in which the median lies = 198 days g = serial number of the median birth minus the cumula- tive frequency of births up to the upper limit of the period preceding the median period = 75 - 51 = 24 C = period interval = 28 days / = frequency of births in the period containing the median = 55. Thus, Md = 198 + 24(28)/55 = 210 days from the start of period 1 = June 25 + 210 days = January 21.

Primate Population Analysis 141 The mode of a birth season may be of interest in comparisons of birth periods or for comparisons with some environmental event. The modal period is the period with the highest frequency of births. In this example it is January 8 to February 4. An ap- proximate mode is calculated as follows (after Stanley, 1963): fa+fb where L = lower limit of modal period = 198 days /„ = frequency of births in the period following the modal = 20 births fb = frequency of births in the period preceding the model = 25 births C = period interval = 28 days. Thus, mode = 198 + 20(28)/20 + 25 = 210 days = 210 days from start of period 1 on June 25 = January 21. By chance the major mode = median in this sample from the toque monkey. Another method for obtaining birth season statistics involves probit analysis. This method is especially useful when regular census work is not possible, because it requires no constant sam- ple size or regular interval between samplings, nor does the be- ginning or end of the birth season have to be sampled. It is an in- direct method that relies on estimating the birth distribution from the cumulative proportion of births that have occurred up to each day of observation. This method is most accurate, however, when births are normally distributed and when there are no in- fant deaths during the sampling period or birth season. The latter constraint may be negligible when birth seasons are brief, but it is limiting when the birth season exends over several months during which infant mortality may be high. See Caughley (1977) or Caughley and Caughley (1974) for an outline of the method.

142 TECHNIQUES IN PRIMATE POPULATION ECOLOGY TESTING SEX RATIO AT BIRTH WITH THE BINOMIAL TEST Owing to differential mortality by sex among young infants, a census taken several months following the birth season may pro- duce an estimate different from the true sex ratio at birth. It is desirable, therefore, to census as close to the birthdate as possi- ble. Most primate and other vertebrate populations have an equal sex ratio at birth. The significance of a digression from a 1:1 ratio can be tested by using a binomial test. The toque monkeys at Polonnaruwa were sampled, and 208 females and 211 males were counted among 419 animals. The equation for the binomial test is where x = the smaller number of observed births (208) N = the total number of observed births (419) P = the proportion of births expected in one category or sex Q = 1 — P = the proportion of births expected in the other sex. Since, according to our null-hypothesis the expected proportion of births of one sex is one-half P = Q = 0.5 and (208 + 0.5) - (419X0.5) 208.5 - 209.5 -1 z — - V(419)(0.5)(0.5) V104.75 10.23 z = -0.1. The probability associated with the value of z is obtained from a table of probabilities for z as is found in most statistics texts, for example Sokal and Rohlf (1969) or Siegel (1956). In the above ex- ample, p = 0.4602 for a one-tailed test and 0.9204 for a two-tailed test. That is, the probability that a digression from a 1:1 sex ratio as large as the one observed in this example could have arisen by chance alone is 0.9204. For samples over 25, a significant (p g

Primate Population Analysis 143 0.05) deviation from a 1:1 sex ratio is obtained whenever z ^ 1.96. For sample sizes less than 25, a different binomial formula and table of probabilities are applied as outlined in the texts op. cit. BIRTHRATES AND FECUNDITY Natality, or birthrate, is the number of infants produced in 1 yr by a population. It can be expressed on the basis of the popula- tion as a whole (the crude birthrate), or on a subset of the popula- tion—for example, birthrate per adult female. The term "fecun- dity" is generally used to convey differences in natality between females of different ages. In analysis it is often convenient to con- sider the female segment of the population only, or at least sepa- rately from the male segment. Hence the statistic symbolized by "m" is defined as age-specific fecundity and refers to the average number of female infants (m) born to females of the age class "x." The raw data required for calculating birthrates are: (1) num- bers of infants born per year and (2) the total number of animals in that part of the population (total, segment, or female age class) for which the birthrate is expressed. Since the birthrate in pri- mates is conveniently measured per annum, the observation pe- riod should span at least 1 yr. Adult females are generally easy to recognize and count in a census, and birthrate is most reliably ex- pressed per adult female. As in collecting other birth data, fre- quent censusing reduces the likelihood of missing infants that died shortly after birth. The choice of the three methods for estimating birthrate and fecundity rate described below depends on the duration of the ob- servation period and on the degree of refinement in the raw data. Method 1 is an estimate based on known births in 1 yr. If the observation period spans only 1 yr and females are individually identified, the birthrate is estimated as follows: Birthrate, b = I,/Ft, where 7, = total number of infants born in 1 yr and F, = total number of females observed throughout 1 yr. If females are clas- sified according to different age classes (x), then the birthrate can be expressed specific to an age class (bx):

144 TECHNIQUES IN PRIMATE POPULATION ECOLOGY Age-specific birthrate, bx = IX/FX, where Ix = number of infants born to females of age class x and Fx = total number of females of age class x. Given a sex ratio at birth of 1:1, the age-specific fecundity is one-half of the birthrate, or: Age-specific fecundity, mx — IX/2FX. Similar data from several years of observation can be treated in the same way for each year, and the results can be averaged over all years. Method 2 is an estimate based on interbirth intervals. The in- terbirth interval in most of the larger primates exceeds 1 yr. Three conditions must be met to use the interbirth interval for calculating the birthrate. First, the observation period must be more than the interbirth interval, probably at least 2 yr for most primates. Second, the females must be individually identified so that a given female's second birth is correctly linked to her first for calculating the interbirth interval. Third, the dates of suc- cessive births must be known fairly accurately because the size of the error in estimating the interval will be the sum of the errors in estimates of both birth dates. Under these conditions, the birthrate, b = l/t, where t = in- terbirth interval. Assuming that the interval (tx) for females of age class x is known, then mx = \/2tx. Method 3 is an estimate based on female-years of observation. In most field studies the size of the census sample increases with observation time, and the quality of the data generally improves with experience. Therefore data may vary in information content between years. Assume, for example, that the first two conditions outlined under Method 2 are fulfilled and that several identified females are known to have had infants but the birthdates are too vaguely known to accurately estimate interbirth intervals. The es- timation of mx from such data is given in Table 7-2 for middle- aged adult female toque monkeys. Within each age class, x, the number of years of observation differed between females. The data can be expressed, however, in

Primate Population Analysis 145 TABLE 7-2 Estimating the Mean Fecundity for Middle-Aged (15-19 yr) Adult Female Toque Monkeys at Polonnaruwa on the Basis of Female-Years of Observation Name of No. Infants No. Years Mean Adult Born to Female was Fecundity Female Female Observed m15-19 Cres 4 4 Fut 1 3 Su 3 3 Nar 2 2 Zip 1 2 Halo 2 2 Total 13 16 13 Source: W. P. J. Dittus, unpublished data. female-years. For example, 3 female-years represent 3 years of observation of one female. As illustrated in Table 7-2, the age-specific fecundity is calcu- lated from mx = 1/2 the total number of infants that born to all females in age class x, divided by the sum of the female-years for all females in age class x. The event of first birth may be used as a criterion for defining the age of primiparous females. To avoid circularity in analysis, one should base estimates of mx for young adult females on at least 2 yr of observation after the first birth, or on the interbirth interval (Method 2). The fecundity pattern of a population is generally displayed in Table 7-3, which is a fecundity table that lists the mean fecundity of females of each age class. ESTIMATING RATES OF POPULATION GROWTH PER-CAPITA RATE OF INCREASE A simple measure of population growth is given by the per-capita rate or finite rate of increase, X.

146 TECHNIQUES IN PRIMATE POPULATION ECOLOGY x - tf. + l \ — « where Nt = numbers in a population at time, t. For example, if population size at time zero were 450, N0 = 450, and 1 yr later there are 622 individuals, TV, = 622, then X = 622/450 = 1.38. This value of lambda indicates that the population has grown at the rate of 1.38 per individual per year. When X is greater than 1, the population has increased in the period t to t + 1; when X is less than 1, the population size has decreased; and when X = 1, the population size has remained constant. OBSERVED EXPONENTIAL RATE OF INCREASE Another measure of population growth is the observed exponen- tial rate of increase, r. It is related to X by X = er, where e = base of natural (Naperian) logs = 2.7183. Thus, loge X = loge er = r. From the example above, er = X = 1.38; therefore, r = loge 1.38 = 0.32. The expression of r in a population is N, — N0e". From this we see that if r = 0, then A', = N0, or the population is stable in size. Increasing populations have a positive r ( + r), whereas decreas- ing ones have a negative r ( — r). TABLE 7-3 Estimating Mean Age-specific Birthrates (bx) and Fecundity Rates (mx) of Females in the Population of Toque Monkeys at Polonnaruwa Between 1968 and 1972" Mean Mean Estimated Age Birthrate Fecundity Age Class (years) bx mx Infant and juvenile 0-4 0.000 0.000 Young adult 5-9 0.704 0.352 Young to middle-aged adult 10-14 0.724 0.362 Middle-aged adult 15-19 0.813 0.406 Old adult 20-24 0.885 0.442 Senile adult 25-29 0.286 0.143 Source: Dittus, 1975.

Primate Population Analysis 147 The observed exponential rate of increase is sometimes useful in calculations involving vital statistics and is often referred to in abbreviated form as the rate of population increase or the growth rate. It is generally used in place of X. It is not the same as the in- nate or intrinsic rate of population increases, rm, which is com- monly quoted in ecology textbooks. The latter is primarily a theo- retical construct that is measurable only, if at all, under specified or experimentally controlled conditions. The observed r calcu- lated from X cannot be used as an estimate of rm. ESTIMATING VITAL STATISTICS FROM A STABLE STANDING AGE DISTRIBUTION CONSTRUCTION OF AGE-STRUCTURE TABLES A standing age distribution is the number of animals of different age and sex observed at any one point in time. It is stable when the proportions of individuals at different ages remain constant through time. Age distributions are usually tabulated separately for each sex in age-structure tables, as illustrated in Table 7-4. In this table, most ages are grouped into classes and are tabulated according to the birth pulse model discussed under "Birth Seasons." The importance of the censusing methods described earlier can now be understood in terms of the assumptions underlying the construction of an age-structure table, the accuracy of which de- pends on the adequacy of the sample of the population, the com- pleteness and accuracy of the age and sex class identifications, and the accuracy of the estimate of the birth season. Sampling primate populations is relatively easy (compared with sampling herds of ungulates, for example) because most primates live in closed social groups of manageable sizes. A population sample is a set of these mutually exclusive troops. For estimating vital sta- tistics, data on the age and sex structure of all troops must be combined in order to obtain a sample size sufficiently large for valid analysis. Since the aim is to count all members of a population, care should be taken to identify all animals in each troop and to in- clude any all-male groups and peripheral males in this tabula-

148 TECHNIQUES IN PRIMATE POPULATION ECOLOGY TABLE 7-4 Age Structure for the Population of Toque Monkeys at Polonnaruwa in 1971 Age Class Estimated Age (years) No. Females No. Males Infant '/2-<l 21 30 Young juvenile 1 18 23 Old juvenile 2-4 36 60 Subadult 5-6 0 23 Young adult male 7-9 0 12 Young adult female 5-9 29 0 Young to middle-aged 10-14 26 16 Middle-aged 15-19 24 13 Old 20-24 20 6 Senile 25-29 11 1 Total 185 184 Source: After Dittus, 1975. tion. The chances of noting peripheral males that may be tem- porarily absent from a troop are increased by visiting the same troop on different days during the count period. The com- pleteness of counts within any troop can be enhanced by concen- trating on one age and sex class at a time and by distinguishing between individuals even if this can only be done temporarily dur- ing a visit to the troop. The timing of a count in relation to the birth season determines whether the ages of most individuals will be close to multiples of whole years or will have an added large fraction of a year. Since analysis by the birth pulse model has fewer constraints (because of the greater accuracy in assigning ages), it is advantageous to census close to the median or mean birth day of the population. These considerations assume, of course, that the average birth day of the population is the same every year and that the standard deviation from the average birth day is low. Individual troops may differ consistently in their average birth day from that of the population as a whole. The variance in birth days usually is less within a troop than for the entire population (W. P. J. Dittus, un- published). If such information is available, the time of censusing a troop might be adjusted to coincide with its average birth day.

Primate Population Analysis 149 CONSTRUCTION OF A LIFE TABLE BASED ON THE BIRTH PULSE MODEL A life table delineates the survivorship schedule of a population. The population is treated in an abstract fashion as though all ani- mals in it were born simultaneously and their numbers were pro- gressively depleted by mortality as they grow older. Such a group is called a cohort. Usually males and females are considered sepa- rate cohorts. The size of the cohort at its inception is 100% and is equal to the total number of births in the sample. One visualizes following the history of this cohort throughout life and recording the numbers still living at each birth day. These records would be equivalent to an age-structure table (but see limitations on p. 158), and the numbers alive at each age are the basis for calculat- ing other statistics. Vital statistics are most conveniently expressed as probabili- ties. The usual statistics given in a life table are as follows: lx = survivorship (or survival), or the probability at birth of surviving to the exact age x. dx = mortality, or the probability of dying during the age in- terval x to x + 1 . qx = rate of mortality, or the proportion of animals alive at age x that will die before age x + 1 . px = rate of survival, or the proportion of animals alive at age x that will survive to age x + 1 . It is important to note that given any one statistic the others can be calculated; each statistic merely displays the same data in a different format. The statistics are related to one another as follows: = dx/lx Occasionally, life-table data are expressed on a per- 1,000 or per-100 basis, the assumption being that the cohort started to-

150 «J O " u ~— Q£ oo SS>-5 5 H 2 2 | -o u R .2 c > « OJ w < S CO v § « JJ I 1 < S m u ii .5 oo "Q « r < 3 £l 1C „ e o op's g u S 3 -2 00 = = e < ? o c < z < o S -2 Z 0.0 £8 <u a oo JS (NfOf^-fN^^roCrO ooddooddo r*) fs (N r^ ^T ^o rNi/^i/)OooO-^ irt^-CSOOOOO ddddddcsdo (Ni/^^1—' — rs^i/i i/5^-^HOOOOO ddddddddo -n'oddddddo oq IN oo o r^j OOOOfNt/^uS'^-'^'fNO O ~H (N — < 0«(NiOOiAojq7J

Primate Population Analysis 151 gether as 1,000 (or 100) animals and their numbers decreased through time. Instead of recording probabilities of survivorship, the life table would portray numbers still alive, or dead with in- creasing age. Converting the probabilities to such whole numbers involves simple multiplication—for example, 1,000 lx, or 100 lx, and so on. In this chapter all life-table statistics are expressed as probabilities because these are easier to work with in analysis. The raw data for construction of a life table can be either the number dead per age class (death-based) or the number alive per age class (life-based). Most primate census data lend themselves best to life-based analysis, when the assumptions described under "Estimating Vital Statistics from Individual Life-History Rec- ords" are met (p. 163). Transformation of raw data from an age-structure table to its hypothetical life-table equivalent requires two manipulations. First, unless the population's birth season conforms exactly to the theoretical birth pulse model and the census was taken on the day of all births for the population for that year, the number of in- fants born in that year or during the year preceding the census will be underestimated. This can be corrected for, however, by eliminating from the raw data the number of infants less than 1 yr old and substituting an estimate of the number that were born in the season preceding the census. This is calculated as the product of the number of adult females in the census and their average birthrate. In the example from the population of toque monkeys (Table 7-4), the number of adult females in the population was 110 and their average birthrate over 4 yr was 0.688 infants/adult female/year. The estimated number of infants born at the theo- retical pulse prior to the census is 110 X 0.688 = 76 infants. The sex ratio at birth is 1:1; therefore, there are 38 infants of each sex (Table 7-5). Life-table data are given in Table 7-5 for female toque mon- keys. The number of females observed for each age class (fx) is taken directly from the age-structure table (Table 7-5), except that the estimated number of female births (/o) is substituted for the number of female infants recorded at the time of the census. The second manipulation requires the transformation of the observed frequencies of females, fx, to survivorship, lx, on a per- total-number-born basis: lx = fx/fo-

152 TECHNIQUES IN PRIMATE POPULATION ECOLOGY For example, for age class x = 1, in Table 7-5: /j = 18/38 = 0.474. If lx values have been computed for each year of age, x = 1, x = 2, x = 3, and so on, the statistics dx, qx, andpx can be de- rived directly for each year by using the formulas indicated above. The statistics are then tabulated for each year of life in a format similar to that in Table 7-5 to give a life table. ESTIMATING VITAL STATISTICS FROM AVERAGED AGE CLASS FREQUENCIES Theoretically, one would wish to have the observed frequencies of females,/,., for each year of life. This requires that the ages of all animals in the census be estimated to the nearest year. Unfor- tunately, in most primate censuses such accuracy is elusive. Data are generally presented according to age classes whose limits are estimated with confidence. In Table 7-5, fx is tabulated per age class, x, and an average/,, is computed to express/^ on a per- annum basis within each age class. These averages represent fx at an age midway between the birth pulse limits or age limits of an age class. The procedure is equivalent to smoothing fx between years. Thus, where ax = number of birth pulses of ages (in years) encom- passed by age class jr. For example, from Table 7-5, «2-4 3 The values of fx are then transformed to the average number of survivors in each age class in the usual manner:

Primate Population Analysis 153 For example, for the age class encompassing x = 2, 3, 4, T2_4 = -^ = 0.316. To find dx one first calculates d per age class or dx where y is the age class next older to age class x; then, the annual average number of animals dying per age class jc is For the macaque data, for example, t/2-4 = *2-4 ~ /s-9 = 0.316 - 0.153 = 0.163 and «2-4 Note that the sum of all deaths E"=0 dx or Z(dx)(ax) is 1.000, the size of the cohort at birth. The standard notation E"=0 represents the limits of the summation—in this case, the sum of values for jc from 0 to infinity or the sum of all age classes. Following conven- tion, the limits of the summation will not be shown when all age classes are summed. Age-specific mortality, qx, also is expressed as an average per annum for each age class: = ^=L and px = 1.000 - qx. 'x

154 TECHNIQUES IN PRIMATE POPULATION ECOLOGY From Table 7-5, for example, d2 4 0.0543 and = 1.000 - q2_4 = 1.000 - 0.172 = 0.828. To express lx, dx, "qx, or px on the basis of the age class as a whole, one multiplies each statistic by the appropriate ax. Using very old females (25-29 yr old) from Table 7-5 as an example, we find: 925-29 = (925-29)(a25-29) = 0.200 X 5 = 1.000. That is, all females of the age class x = 25-29 will, on the average, die before their 30th birthdays. When ^ = 1, as for x = 0 and x = 1 in Table 7-5, then Tx = lx, dx = dx, and so on. Another statistic that is often displayed in life tables is ex = mean expectation of life remaining for individuals at the start of an age x. The statistic ex = (Lx+n + . . . Lx+2 + Lx+l + Lx)/lx, where Lx — average survivorship during the age interval jc tox + 1: - X lx From Table 7-5, for example, L0 = /„ + /,/2 = 1.000 + 0.553/2 = 0.777. The sum of L is obtained by starting with the oldest age class and adding from the bottom of the life table until the speci- fied age class is reached, in this example—L^. Because x varies between the limits from the specified age class to infinity rather than from 0 to infinity, the limits are shown as E£°=^ Ly. The data shown in Table 7-6 are given to exemplify estimation of ex for a

Primate Population Analysis 155 portion of the life table only. Thus, for 25-yr-old female toque monkeys - (!„ + £28 + L21 + Z26 + £25) - 0.029 + 0.058 + 0.058 + 0.058 + 0.058 0.058 and Zv 0.261 625 - The average life expectancy for females in the senile age class is e25-29 — a25-29 4.5 + 3.5 + 2.5 + 1.5 + 0.5 5 12.5 Or, a female of the age class x = 25-29 will, on the average, live another 2.5 yr. The average life expectancies of individuals of different age classes are given in Table 7-7 for female toque monkeys. The life expectancy at birth, e0, is the average life expectancy of the entire cohort; for the female toque monkey, e0 = 4.8 yr. It is so low because of high mortality among infants and juveniles. GRAPHIC REPRESENTATION OF LIFE TABLES Vital statistics can be displayed graphically in a plot of age on the abscissa and the selected statistic on the ordinate. The most use-

156 TECHNIQUES IN PRIMATE POPULATION ECOLOGY TABLE 7-6 Estimating the Life Expectancy, ex, for a Portion of a Life Table for Female Toque Monkeys Age* lx Lx El, Jx 0-22 not tabulated 23 0.105 0.105 0.448 4.3 24 0.105 0.0815 0.343 3.3 25 0.058 0.058 0.261 4.5 26 0.058 0.058 0.203 3.5 27 0.058 0.058 0.145 2.5 28 0.058 0.058 0.087 1.5 29 0.058 0.029 0.029 0.5 30 0.000 Source: Adapted from Dittus, 1975. ful schedules are survivorship (/.,.) and age-specific mortality rate (qx). The advantage of such graphs is that mortality rate and sur- vivorship schedules can be easily assessed at a glance. Figures 7-2 and 7-3 show survivorship and mortality rate curves for male and female toque monkeys. The survivorship curves appear in a stepwise fashion because statistics were computed as averages that apply to all ages within TABLE 7-7 Average Life Expectancies for Female Toque Monkeys of Different Age Classes Average Life Expectancy Age Class (years) ex 0 4.8 1 8.6 2-4 10.7 5-9 16.6 10-14 13.0 15-19 9.3 20-24 5.2 25-29 2.5 Source: Adapted from Dittus, 1975.

Primate Population Analysis 157 1.000 .800 .600 —i .400 - .200 - Female ' ' l ' ' ' ' l ' ' l ' ' ' i ' ' ! Age (years) FIGURE 7-2 Survival curves for male and female toque monkeys at Polon- naruwa from 1968 to 1972. After Dirtus, 1975. any one age class. Theoretically, one would expect the transitions from one age to the next to be smoother than they appear here. Other graphic representations include age pyramids for each sex, or histograms made by plotting lx and </r against age (see Eberhardt, 1968, for examples). LIFE TABLES BASED ON THE BIRTH FLOW MODEL The life-table calculations above assumed a birth pulse model. In a birth flow situation, the census data are divided into intervals of age x to x + 1 starting with age class 0.5-1.5. The cohort is assumed to begin at age 0.5 yr. To estimate the size of the cohort at age 0.5, one must know the mortality in the interval 0-0.5 yr. Given the cohort strength at birth, 10 (as calculated from fecun- dity), mortality in the interval 0-0.5 can be estimated if the num- ber of infants 6 mo old is known.

158 TECHNIQUES IN PRIMATE POPULATION ECOLOGY 1.000 -i .800 - X cr oT .600 - o o .400 - 4- L. O Z .200 - Female Male i 1 l l | l l 1 i | 1 l l l | I l l 1 | l l l l | l l l l | 0 10 20 30 Age (years) FIGURE 7-3 Age-specific mortality rate curves for male and female toque monkeys. After Dittus, 1975. LIMITATIONS OF AGE DISTRIBUTION DATA FOR CALCULATING VITAL STATISTICS The Effect of Age Distribution on Life-Table Estimates The frequency distribution of ages in a population sample at the time of the census is known as a "standing age distribution" and is usually shown in an age-structure table. In estimating life-table statistics from such data, one assumes that the standing age dis- tribution is equivalent to a "temporal age distribution" of a cohort. The latter is an abstract distribution of the number of survivors remaining each year from a group of animals that started life together and thereafter faced the same risks of death at each age, although these risks changed with successive years or ages. In a real population, however, natality differences between years by themselves may influence the numbers recruited from 1

Primate Population Analysis 159 year (or age) to another. Also, at any point in time the ages of all animals in the population are not equal as is assumed in the imagined cohort model. Thus if the environment changes to the extent of altering the risks of death from one calendar year to another, not all animals in a real population would face the same risks of death in passing through a particular age. The assumptions of a cohort model hold true only if the popu- lation sampled is a "stationary" one so that over many years the schedules of fecundity and survivorship have been constant and the exponential rate of population increase (r) has been zero. This can be shown as follows. The standing age distribution, Sx, can be expressed as the number of animals in an age class relative to the number of new- born (Caughley, 1977). It is estimated from a census sample as Sx = fx/fo. It is related to survivorship lx by Sx — lxe~rx, where r = rate of population increase. It is clear from this relationship that unless r = 0, the standing age distribution will differ from lx. Therefore, Sx can be used to estimate lx only in populations where r is and has been zero for some time. Or, intuitively, if r is con- stantly changing, the numbers recruited or dying at different ages are changing and are thereby altering the age distribution from year to year. In a "stable age distribution" the survivorship and fecundity schedules have also been constant for some time so that the pro- portions of one age or age class to another are constant. Under these conditions r will also converge toward a constant value, but not necessarily r = 0. However, if the value of r is known and can be shown to have been constant for a long time then, from the re- lationship of r to Sx and lx, lx may be computed given Sx. Estimation of life-table statistics from a standing age distribu- tion is therefore restricted to only those populations whose mor- tality and fecundity schedules have been constant for two or three generations and whose rate of population increase (r) is known and has been constant for a long time. Assessing Population Stability Stable populations have stable age distributions. They are "sta- tionary" when r = 0. But stable populations need not be sta-

160 TECHNIQUES IN PRIMATE POPULATION ECOLOGY tionary; they can have any r. Given any r, however, a population requires two or three generations to converge toward stability. Hence a good deal of information is needed about the history of a population before Sx can be used to estimate lx. Any information indicative of the state of population growth (r) might be useful for rejecting or accepting Sx as a basis for estimating vital statistics. Population growth rate (r) may be assessed directly or indi- rectly. A direct assessment requires that r be estimated at intervals over a period of two or three generations. Short-term estimates, as over a few years of a field study, are by themselves inadequate to predict whether long-term stability will occur. Indirect assessments of population growth rate (r) deal with gauging the stability of the environment, which influences popu- lation growth. In a stable environment the seasonal changes are of the same kind and magnitude every year. Most primates are /("-selected, and their numbers are attuned to the carrying capac- ity of their environments (see Chapter 8). A population inhabiting an environment that has been stable over a long period probably has an average annual r = 0 and a fairly stable and stationary age distribution. The following conditions are indicative of long-term en- vironmental stability: • Forest and other vegetation types that are climax and undis- turbed. • Constant climatic regime. A climatic regime in which tem- perature and amount and seasonal distribution of rainfall follow the same pattern over many years would tend to induce predict- able phenological and productivity patterns in the forest. Most primate populations are attuned to the predictable seasonal flux in resources. • A balanced flora and fauna relative to environment type. Unstable environments would tend to alter r so that popula- tions inhabiting such areas probably do not have a stable age dis- tribution. Unless an environmental change is of the same kind and magnitude over a long period of time, it is unlikely that any value of r ^ 0 would be constant for a long period. Conditions

Primate Population Analysis 161 that are indicative of unstable environments or populations are as follows: • Forests that have been disturbed recently, either through na- tural events, such as fire, cyclones, or changes in soil salinity, or through felling by people. • Change in cultivation practices. The balance of populations in an area where they are partly or wholly dependent on agricul- ture would be changed by a change in long-established cultivation practices in the area, such as the introduction of a new crop or of the use of pesticides. • Unnatural disease. For example, the introduction of yellow fever to South America resulted in a marked decline of howler monkey populations (Collias and Southwick, 1952). • Hunting or trapping by man. • Highly irregular climatic regimes. • The introduction of human settlements. • An impoverished flora and fauna relative to environmental conditions. Data from several sources may assist in a final judgment about stability. For example, the population of toque monkeys at Polonnaruwa was judged to have had a fairly stable history based on forest ecology, a very constant long-term climatic regime, and short-term population stability. Errors Due to Inaccurate Age Estimates Probably the greatest source of error in estimating vital statistics is faulty age estimation. Since estimates of mortality are a func- tion of (lx — lx+l), the ability to distinguish individuals aged x from those aged jc + 1 is obviously crucial in determining the numbers in each age class. Indirect age estimates are often based on body-size differences (see Chapter 5). The relationship between body size, age, and numbers of ani- mals is shown in Figure 7-4. In this figure, A and B are physical growth curves for two cohorts having equal birthrates, sex ratios at birth, and survivorship schedules. The cumulative number of living individuals by age groups is the same for both cohorts. The

162 TECHNIQUES IN PRIMATE POPULATION ECOLOGY E 3 O "- 0 "-0 Age - x FIGURE 7-4 The relationship between growth in body size, age, and cumulative survivorship in two imaginary cohorts. The imaginary cohorts, A and B, have equal survivorship and sex ratios at birth, but they have different growth rates, with B growing faster than A (see text). curves serve to illustrate the type of error that can arise in popula- tion analyses from faulty age estimation. An error that might occur is estimating the ages of slow- growing animals (A) on the basis of the size of fast-growing ones (B). Assigning an age xh to an A animal of size b would underestimate the true age, xa, of A, and would overestimate the number of A animals, /„, relative to their assigned age, x/,, and relative to the number of B animals, fb, of the same age, xb. In analysis, such an error would underestimate the mortality of A animals relative to their true ages and relative to B animals of the same age. In primate studies this error may arise if the size-to-age rela- tionships of fast-growing captive animals fed on nutritious diets are used to estimate the ages of equally sized but slower-growing wild animals. Similarly, male infants and juveniles often grow faster than their female age mates. Using the size-to-age relation- ship of one sex to estimate the ages of the other, or an average growth curve to estimate the ages of either, would tend to overes- timate the mortality of the faster-growing males relative to their slower-growing female age mates.

Primate Population Analysis 163 When one does not have empirically derived growth curves for both sexes in a census population, it is advisable to use the same growth-age relationship to estimate the ages of both sexes, keep- ing in mind that if growth differences by sex occurred, the mor- tality of the faster-growing sex would tend to be underestimated among infants and juveniles. In conclusion, the use of standing age distributions to estimate life-table functions should be approached with caution and should be used only when it is possible to approximate the condi- tions of stable age distribution, long-term zero population growth rate or known r, and accurate aging in the census. Two alternative methods can be used to estimate vital statistics when no assumptions are required concerning the stability of the age distributions or the rate of population increase at any time. (The methods are given in the next two sections.) The first method involves identifying a large number of individuals and tracing their fates from birth. The second involves identifying only a set of troops and comparing their standing age distribu- tions at yearly intervals. ESTIMATING VITAL STATISTICS FROM INDIVIDUAL LIFE-HISTORY RECORDS To estimate vital statistics from individual life-history records one must have, as raw data, (1) numbers of individuals that died in the age interval jc to x + 1 or (2) numbers of individuals alive at age x. The method used in obtaining both kinds of data is similar and requires that all individuals in the population be identified so that their fates can be traced and their birth days determined. From records of the age at death of individuals the number of deaths, f'x, in each age interval x to x + 1 is obtained. From records of numbers living at census time, t, the number surviving, ff, at age x is obtained. The accuracy of both age estimates will depend on the census interval. If an estimate to the nearest interval of 1 yr (x to jc + 1) is sought, then each individual should be censused at least on every birth day. In practice, of course, any one census will sample many individuals whose birth days vary. Therefore

164 TECHNIQUES IN PRIMATE POPULATION ECOLOGY several censuses are needed, especially during the birth seasons. In populations whose average birth days have a large variance, or whose birth schedule approximates the birth flow model, a regular census schedule at intervals of 1-3 mo is desirable. When the original data are/^ = number of deaths in each age interval jc to x + 1 , the following calculations are used to derive vital statistics (after Caughley, 1977, p. 91): x-\ lx = 1.000 - £ dx o <7, = —T- and Px = 1.000 - qx. When the original data are in the form offx = number of sur- vivors at age x, the following calculations apply (same source): h and qx andpx are estimated in the same fashion as above. The use of individual life-history records provides the most ac- curate estimates of life-table statistics; their main disadvantage is the large investment of time and effort needed for their applica- tion. ESTIMATING VITAL STATISTICS FROM STANDING AGE DISTRIBUTIONS SAMPLED AT YEARLY INTERVALS When a standing age distribution is obtained from the same pop- ulation two or more times at intervals of 1 yr, any two successive samples may be used to calculate life-table statistics. The number of survivors, fx, at the time of the first sample is

Primate Population Analysis 165 compared with the number still living, fx+i, 1 yr later in age class x + 1. The differencefx ~ fx+l is the number that died in the in- terval x tojc + 1, or in the year between census dates. The follow- ing calculations are used to determine qx for each age interval (from Caughley, 1977): /t(at time 0) -/x+i(at time 1) q* ~ fx(at time 0) This method has the following constraints: • The same proportion of the population must be taken at any two samplings. In primate populations this condition is readily fulfilled by sampling the same set of troops each time. Hence, in- dividual troops making up the sample must be identified. Sam- pling the same number of troops making up the sample does not meet the stated condition because troops vary in size and compo- sition. • Any two successive samples should be taken at the same time of year, preferably during the birth season, so that in a birth pulse population the ages of animals will be distributed near their birth days. • Since most births are unlikely to occur on the day of sam- pling (t = 0), the estimate for the number of births,/0, might be based on a series of birth censuses. • The accuracy of estimates of fx will reflect the accuracy of ag- ing individuals. DERIVING VITAL STATISTICS FROM FECUNDITY AND SURVIVORSHIP SCHEDULES The rate at which individuals replace themselves depends on how well they survive and reproduce. Several important measures based on the relationship between fecundity and survivorship are considered below. ESTIMATING NET REPRODUCTION RATE The net reproductive rate, RQ, is the rate of population increase per generation. It is a special case of \, where the time interval t

166 TECHNIQUES IN PRIMATE POPULATION ECOLOGY to t + 1 over which the population increase is measured is equal to one generation. R0 can also be defined as the average number of female off- spring produced by a female during her entire lifetime (Wilson and Bossert, 1971). Thus defined, it is easy to visualize how R0 would depend on the survivorship and fecundity of females. R0 is calculated from fecundity and survivorship schedules as follows: D — V7 *».» ^0 — Llxmx- In Table 7-8 the toque monkey data are used to illustrate the estimation of R0. Since lx and mx are expressed as averages (lx and inx) per annum within each age class, and we are interested in the sum of lxmx for each year of life, the lx and mx must be multiplied by the number of ages, ax, in each age class. For grouped data, RQ = Elxmx = Elxmxax. When R0 = 1, the population is stationary per generation; when RQ is less than 1, it is decreasing; and when R0 is greater than 1, the population is increasing. If lx is calculated with the assumption that lx = Sx and r = 0 (as in Table 7-9), then R0 must = 1. Therefore, RQ can be estimated from lxmx only if the lx schedule was calculated by methods not assuming r = 0. TABLE 7-8 Calculation of Net Reproductive Rate, R0, from Fecundity and Survivorship Schedules of Female Toque Monkeys Age Class (years) Age or Years per Age Class ax I m* lxmx = ajxmx 0 1 1.000 0.000 0.000 1 1 0.474 0.000 0.000 2-4 3 0.316 0.000 0.000 5-9 5 0.153 0.352 0.269 10-14 5 0.137 0.362 0.248 15-19 5 0.126 0.406 0.256 20-24 5 0.105 0.442 0.232 25-29 5 0.058 0.143 0.041 KO = =/X» ix= 1.046 *The values of mx (Table 7-3) are based on small sample sizes. If the average fecundity (inx = 0.344) for all females is based on a large sample size and is used to estimate R0, then R0 = 0.9959. Source: W. P. J. Dittus, unpublished data.

Primate Population Analysis 167 ESTIMATING MEAN GENERATION LENGTH The mean generation length, T, can be defined as the mean period elapsing between the birth of a mother and the birth of her offspring (Laughlin, 1965). This has also been called the mean cohort generation length. In mammals such as primates, where a female gives birth to many offspring over a long period, the mean generation length might be more easily understood as the mean interval between the birth of a mother and the birth of a daughter that survives to re- produce herself. Stated in this way, it expresses the average age that a female must attain in order to replace herself once in the population. Of course, if the population is increasing, the mother will replace herself by more than a single daughter. The formulas used for calculating mean generation length, T, for populations with overlapping generations (after Caughley, 1977) are as follows. For a birth pulse population, -= And for a birth flow population, where mxx+1 is the fecundity between age x and age x + 1. In a birth pulse population where ages are expressed in classes, T — Y,Sxmxxax, where x = average age of age class x and ax = number of birth pulses of ages in years encompassed by age class x and Sx = fx/fo. ESTIMATING EXPONENTIAL RATE OF POPULATION INCREASE The exponential rate of increase, r, may be estimated from r = log,. RQ/T. This estimate of r may be used only if in the derivation of lx, which is used to compute R0 and T, it was not assumed that r = 0.

168 TECHNIQUES IN PRIMATE POPULATION ECOLOGY TABLE 7-9 Estimation of Reproductive Values, vx, for Female Toque Monkeys at Polonnaruwa (1968-1972) when r = 0 Age 0 1.000 0.000 0.000 1.046 1.000 1.0 1 0.474 0.000 0.000 1.046 2.110 2.2 2-4 0.316 0.000 0.000 1.046 3.165 3.3 5-9 0.153 0.352 0.269 1.046 6.546 6.8 10-14 0.137 0.362 0.248 0.777 7.299 5.7 15-19 0.126 0.406 0.256 0.529 7.937 4.2 20-24 0.105 0.442 0.232 0.273 9.524 2.6 25-29 0.058 0.143 0.041 0.041 17.241 0.7 »,«, = 1.046 ESTIMATING REPRODUCTIVE VALUE The reproductive value, v^., is an estimate of the number of female offspring that remain to be born to a female of age x. It is a unitary measure expressing how survivorship and fecundity in- teract to determine how many offspring a female is likely to pro- duce before dying. Young adult females that are just beginning to reproduce have a long life expectancy and will produce more off- spring before dying than will older females. Similarly, juvenile females and very old females with zero or low fecundity have low reproductive values. The "exact" vx will of course be determined by how survivorship and fecundity interact. Following Wilson and Bossert (1971), vx is calculated for each age x as v, = - E e' y=x where y — all the ages that a female has yet to pass through from age x to infinity (death). Assume, as in the example from the toque monkey in Table 7-9, that r = 0. Then, the formula for the calculation of vx simpli- fies to

Primate Population Analysis 169 1 v / (x y=* y y To find EJIL, /^.w^, for any age x, we first assign the sum of the lxmx schedule to age 0. Thereafter, for each age (or age class) E"=x lymy is computed by subtracting lxmx of the preceding age from the E"=A. lymy of the preceding age. For example, from Table 7-9 £y"=jt lymy of age class x = 10-14: E lymy = 1.046 - 0.269 = 0.777. The average v^ per age class is the product of the reciprocal of lx and Ey*=x lymy appropriate to each age class. For female toque monkeys of age class x = 10-14, for example, = —^— X 0.777 = 5.7. The values of vx are plotted against age in Figure 7-5. At birth, v0 = 1; the average female replaces herself once per generation in this population for which we have assumed r = 0. For a more detailed discussion of the estimation of reproduc- tive values, see Wilson and Bossert (1971). Reproductive value is preeminently of genetic and evolutionary interest because it indexes the force of selection acting on behav- ioral and other traits that differ by age. DISPERSAL The process of dispersal whereby animals leave their natal group to live and reproduce elsewhere is of interest for two reasons. First, the disappearance of an individual through dispersal must be distinguished from its death if estimates of mortality are to be correct. Second, the rates at which individuals of different age and sex disperse are of interest in themselves from the point of view of genetic interchange between populations. From the perspective of the biologist studying a population, movements of individuals can be considered at two levels: intra-

170 TECHNIQUES IN PRIMATE POPULATION ECOLOGY 10 o > £ 5 u 3 l l 1 1 | l 1 1 1 | l l 1 1 [ 1 l l l 1 l l 1 1T l1" 1 10 20 30 Age (years) FIGURE 7-5 The reproductive values, vx, of female toque monkeys at different ages. After Dittus, 1979. population and interpopulation movements. Primatologists have referred to the former as intergroup transfer of individuals. The distinction between intra- and interpopulation movements is merely one of sampling convenience; biologically they are similar, except perhaps for the difference in the distances moved and the genetic implications this may have. For interpreting demographic events within the sample population, it is important to know rates of movement into and out of the population. For example, if a high proportion of males leaves the study population every year, and they are not replaced by an influx of an equal number of males, the mortality of males in the population might be overesti- mated. Intrapopulation movements have little influence on the demo- graphic sample and therefore have little bearing on estimates of life-table statistics. However, knowledge of such movements is important for correctly assessing interpopulation movements.

Primate Population Analysis 171 The best estimates of intra- and interpopulation movements are based on individual life-history records. With respect to any one social group or population, there are two kinds of move- ments: emigration and immigration. For analysis, a distinction between the following types is useful: Type 1: In intrapopulation emigrations, an individual is known to have left one social group and has joined another within the sample population. Type 2: Intrapopulation immigrations are the same as intra- population emigrations. Type 3: In interpopulation emigrations, an individual is known to have left a group within the sample population and has joined a group outside the sample population. Type 4: In interpopulation immigrations, a new individual not previously seen in the population appears in a census. Type 5: Disappearances occur when a previously censused individual is no longer sighted and its fate is unknown. Note that in these considerations the membership of the sam- ple population is defined by the sum of the memberships of all the social groups in the population. With these distinctions in mind, estimates might be made of rates of transfer between social groups within a population and of those between populations. ESTIMATING RATES OF TRANSFER BETWEEN SOCIAL GROUPS The raw data for estimating rates of intergroup transfer are (1) number of individuals whose life histories are being traced, (2) duration for which their histories are being traced, and (3) the number of transfers each individual has made during the period of observation. Since there is a category of "unknown fates" (Type 5), the exact rates of movements cannot be estimated, but a minimum and maximum can be estimated. To find rates, the minimum and maximum numbers are found first: Minimum number of emigrations = total of Type 1 plus Type 3. Maximum number of emigrations = total of Types 1, 3, and 5.

172 M 0) c 13 g fN O OO o u S S s 0 M g — u — .« c -2 3 Si B* u 2 ^ 1 2 u 8 M i2 H i s i e I n C; in C < o, ca w ^ i/l ro O O O 1 .§> .1 1 1 p o 2 « -SP O O O O CQ C c R S 05 -0 CtS UJ s o o o o rt cfl OS oo 3 *H IA M I g | 1 2 C! •" ° ,2 & n & il d •o m i o r- o o ^H •3 c .S § ^5 "o >, c A PL, S .2 'S -g I"8! If s s i O tU *° LD r^ n y^ 1 ! z >• o • 03 uf 5r 3 o ™ u u *p g TABLE 7-1 of Different « 1 1 % * Illll Q 0 I M 1 "3 "o "3 « •*; •o -o -o -a ° < .< .< «S 4

Primate Population Analysis 173 Minimum number of immigrations = total of Type 2 plus Type 4. Maximum number of immigrations = same as the minimum. To calculate rates, one finds the total number of monkey-years of observations per individual, or for a class of individuals. Esti- mates of emigration are given in Table 7-10 for different age and sex classes of toque monkeys at Polonnaruwa. Rates are found by dividing the number of immigrations or emigrations by the num- ber of monkey-years observed for a class or individual. An exam- ple taken from Table 7-10 shows that the minimum rate of emi- gration for adult male toque monkeys is 10 emigrations per 58.73 adult male monkey-years, which is 0.170 emigrations per monkey per year. The reciprocal of such rates gives the average interval of time between successive emigrations. Adult male toque monkeys, for example, emigrate maximally at an average of once every 1/0.170 = 5.9 yr. ESTIMATING RATES OF MOVEMENT BETWEEN POPULATIONS The dynamics of dispersal between populations is likely to be the same in kind as between social groups. Knowledge of the latter therefore is useful for predicting movements between popula- tions. For example, from Table 7-10 it is highly unlikely that adult females or infants and young juveniles shift between popu- lations because they appear not to leave their social groups. If we consider a set of troops to constitute the sample popula- tion, the number of individuals moving into and out of the popu- lation is estimated as follows: Minimum number of emigrations = total of Type 3. Maximum number of emigrations = total of Type 3 plus Type 5. Minimum number of immigrations = total of Type 4. Maximum number of immigrations = same as the minimum. These numbers are expressed as rates by dividing them by the total number of monkey-years of observation appropriate to the class in question.

174 TECHNIQUES IN PRIMATE POPULATION ECOLOGY Finding individuals that have left the population is probably the most difficult task in population analysis. Methods for assess- ing such movements accurately are beyond the scope of this chap- ter; see Caughley (1977) for general approaches to this problem. An indirect assessment of the fate of animals that frequently disappear might be obtained by censusing populations adjacent to the study population. If an unusually high proportion of the age-sex class whose members frequently disappear shows up in an adjacent population, emigration from the sample population might be seriously suspected. MORTALITY VERSUS EMIGRATION Estimates of intrapopulation movement indicate which age and sex classes are subject to emigration. The disappearance of indi- viduals of a class that does not emigrate can be safely assumed to have died. With some notable exceptions, female and young juve- nile primates do not emigrate from their natal groups.* The males of many species, however, do emigrate from their natal groups as adolescents, and as adults they continue to shift between groups (Dittus, 1980). If estimates of mortality are based on individual life-history records, the disappearance of a male may be inter- preted as either death or emigration. In populations where it can be assumed that immigration and emigration are equal, mortality can be estimated independently of dispersal by applying methods based on standing age distribu- tions. (See "Estimating Vital Statistics from a Stable Standing Age Distribution" and "Estimating Vital Statistics from Stand- ing Age Distributions Sampled at Yearly Intervals," above.) The survivorship and mortality schedules of male toque monkeys (Figures 7-2 and 7-3) are based on such independent derivations. In speculations concerning the fate of individuals it is generally considered "conservative" to attribute such losses to emigration. Whether this is conservative in the sense of being the "most likely" biological outcome or of projecting human values to *For the exceptions, see Olivers (1974)—siamang gibbons; Harcourt et al. (1976)—gorilla; Kawanaka and Nishida (1975) and van Lawick-Goodall (1973)— chimpanzee; and Marsh (1979) and Struhsaker and Leland (1979)—red colobus.

Primate Population Analysis 175 demographic events is open to question. If all males that disap- peared from a population had merely emigrated and such emi- gration was common in other populations as well, then some- v.nere there must be a large reservoir of emigrated males. Such a population has yet to be found in primate demographic studies Lone males and all-male groups must, of course, be incorporated into any valid demographic sample. SUGGESTED READINGS The following are excellent introductions to the study of animal populations: Caughley (1977), Eberhardt (1968), and Wilson and Bossert (1971). The book by Caughley is especially recommended as a clear and practical manual. Some other important publications are: Andrewartha and Caughley (1966, 1967), Deevey (1947), and Pielou

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