Lecture 14: Life Table Analysis

Reading:  None.

Static Life Table (Vertical Table): Evaluation of all age classes at one point in time, this is a time specific analysis, also called an age structure table. Static life tables are imperfect (they may not correctly indicate the survival probabilities of a given cohort over time) but static life table data is often better than nothing.

Assumptions for a static life table to correctly indicate cohort survival probabilities:

1. Constant age distribution

2. No year to year variation in total births

3. No year to year variation in age-specific survival

Static life tables do show general patterns of survivorship. A static fecundity schedule shows the general pattern of age specific fecundity.

Cohort and static life tables are the same only when population size is constant.

If a population is increasing, then older age classes will be under-represented in a static life table.

If a population is declining, then older age classes will be over- represented in a static life table.

Survivorship Curves
        Semi-log plot of the number of survivorsas a function of age (x).

Idealized survivorship curves are shown in the figure below. Type I is typical of human populations, Type II is common in birds and some invertebrates, Type III is typical of many fishes and marine invertebrates.

Reproductive Rates, Generation Times, and Rates of Population Increase

Reproductive rate from a cohort life table (Basic or net reproductive rate):

Ro = S where and = mean # offspring/adult

This is an average rate per individual in a cohort life span, we need to derive a population growth rate.

Given a population which changes as follows:

10, 20, 40, 80, 160, 320 .........

then,

The term (lambda) combines births of new individuals and the survival of existing individuals. If >1 the population increases, if <1 the population decreases. We call the geometric growth rate.

Given the population growth sequence above:

N1 = 20 = No

N2 = 40 = N1

N3 = 80 = N2

 

so

and

Since Ro is the net reproductive rate during a given generation time (T):

ln(l) is termed the per capita growth rate, the change in population size per individual per unit time, also termed (r), so:

Lambda (l) was defined as: for T = 1 where r = ln l

          so the per capita rate of change (r) can be estimated from sequential population census data, and a life table is not necessary to make an estimate of population growth.

Exponential Growth Rate

The rate of geometric (exponential) population growth rate is:

exponential growth equation

The change in numbers of a population over time interval t is equal to the per capita rate of change (exponential growth rate) times the number of individuals present in the population at the beginning of the time interval.

Given any population with constant (between time intervals) age specific survival and constant age specific fecundity , a stable age distribution will be approached over time (see below Ricklefs, 1996, page 330 and 331, Tables 15.1 and 15.2), and the per capita rate of change (change in numbers over time) will stabilize to an ideal value termed the intrinsic rate of increase.

The intrinsic rate of increase for a population can be approximated from cohort life table data with the assumption of a stable age distribution (Ricklefs, 1996, page 337, Table 15.8). A cohort generation time can be used to estimate of true generation time (T). is the average length of time from the birth of an individual to the birth of its own offspring.

since

The exponential growth model is the simplest description of biological population change, but this kind of growth is observed in nature. Examples include species invading new habitats, bacteria, viruses, pest eruptions, algal blooms, and human populations.

Life table for a hypothetical population of 100 individuals.

Projection of population age classes and total size through time.

	0	1	2	3	4	5	6	7	8	Percent
	20	74	69	132	175	274	399	599	889	63.4
	10	10	37	34	61	87	137	199	299	21.3
	40	8	8	30	28	53	70	110	160	11.4
	30	20	4	4	15	14	26	35	55	3.9
N	100	112	118	200	279	428	632	943	1403	100
	1.12	1.05	1.69	1.40	1.53	1.48	1.49	1.49
*The population was projected by multiplying the number of individuals in an age class by the survival to obtain the number in the next older age class in the next time period. Thus, . Then the number of individuals in each age class was multiplied by its fecundity to obtain the number of newborns. Thus,

Estimation of exponential rate of increase from the life table above.

x
0	0.5	1.0	0	0.0	0.0
1	0.8	0.5	1	0.5	0.5
2	0.5	0.4	3	1.2	2.4
3	0.0	0.2	2	0.4	1.2
Net reproductive rate (Ro)				2.1
Expected number of births weighted by age					4.1
*The sums of the lxbx column (not reproductive rate) and the xlxbx column are used to estimate ra according to the equation given in the text. In this case, we caculate ra to be 0.38; this is equivalent to lambda = 1.46, close to the observed value of about 1.48 after the population achieved a stable age distribution.

World human population growth during the past 345 years. (data sources: Illustrated Atlas of the World. Rand McNally and Co., 1992, and Worldwatch Database. Worldwatch Institute, 1996)

Current world human population growth is at an annual rate of 1.56% (for 1995). This is the change in world population as a fraction of existing populations:

The values for this calculation in 1995 were:

1.56% annual growth rate = 0.0156 ˜ r

The annual growth rates or the per capita rates of change are deceptively small values.

The integrated form of the exponential equation permits calculation of population doubling times.

= population size at time t

= population size at an earlier time

e = natural base = 2.71878 (a constant)

r = per capita rate of population change

t = time in the same units as r

When any population doubles in size, the ratio of / would be 2.0, so:

2.0 = and ln(2.0) = ln() which is ln(2.0) = rt

ln(2.0) is a constant = 0.69315

so the time it takes for an exponentially growing population to exactly double in size, regardless of the absolute size of the population will be:

in the time units of r

Copyright 1999 Northern Arizona University
ALL RIGHTS RESERVED