Keyfitz N., Caswell H. Applied Mathematical Demography

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3.3. The Leslie Matrix and the Life Table 55

historians, have a strong belief that processes are important but ini-

tial conditions are historical accidents. On the other hand, if a model

can be shown not to be ergodic, then it may be used to explain dif-

ferences in dynamics in situations where the underlying processes are

apparently the same.

Transient analysis. Short-term dynamics can be very diﬀerent from

asymptotic dynamics. Transient analysis, focusing on short-term

behavior, may be more relevant than asymptotic analysis in char-

acterizing the response of a population to perturbations.

Perturbation analysis. No matter how carefully it is constructed, a

model always leaves things out, and the data from which parame-

ter values are estimated are always imprecise. Any conclusions that

depend on those exact values are immediately suspect. Furthermore,

we usually want to extrapolate the results to other populations, other

species, or other environments. Thus it is important to know how

sensitive conclusions are to changes in the model. This kind of inves-

tigation is called perturbation analysis or sensitivity analysis.Inthe

previous examples we applied it to the linear model to see how the

growth rate changed when we modiﬁed survival and fertility.

Any study that does not address asymptotic analysis, transient analysis,

ergodicity, and the results of perturbations has not completely explored its

model.

3.3 The Leslie Matrix and the Life Table

The parameters in an age-classiﬁed matrix model are derived from the life

table. We begin by distinguishing birth-ﬂow populations, in which births

occur continuously over the projection interval, and birth-pulse popula-

tions, in which reproduction is concentrated in a short breeding season

within the interval (Caughley 1977). Humans are an example of a birth-

ﬂow population, whereas mammals, birds, and many other organisms in

seasonal environments are more accurately described as birth-pulse popu-

lations. These patterns of reproduction produce diﬀerent distributions of

individuals within age classes, and lead to diﬀerent approximations for the

survival probabilities and fertilities.

3.3.1 Birth-Flow Populations

3.3.1.1 Birth-Flow Survival Probabilities

is the probability that an individual in age class i will survive from t to

t + 1. This depends on the age of the individual within the age-class; the

56 3. The Matrix Model Framework

probability of survival from precise age x to x+1 is l(x +1)/l(x). However,

in forming age classes, we have given up all knowledge of age within the age

class. Therefore, we approximate l(x) within each age class by its average

over the interval i − 1 ≤ x ≤ i,sothat



i+1

l(x)dx



i−1

l(x)dx

. (3.3.1)

In the notation of (2.1.1), this is

i−1

, (3.3.2)

which might be crudely approximated as

≈

l(i)+l(i +1)

l(i − 1) + l(i)

. (3.3.3)

Here, as is customary in matrix population models, time has been scaled

in terms of the projection interval, so that the 5-year intervals of age and

time in Chapter 2 appear here as 1 projection interval.

Other alternatives are possible. Assuming a constant force of mortality

within the age interval suggests using the geometric rather than arithmetic

mean as the approximation within the interval, in which case



l(i)l(i +1)

l(i − 1)l(i)



1/2

(3.3.4)



l(i +1)

l(i − 1)



1/2

. (3.3.5)

A second alternative would be to calculate the probability of survival for

each age and then average over the age interval:



i−1

l(x +1)

l(x)

dx (3.3.6)

≈



l(i)

l(i − 1)

l(i +1)

l(i)



. (3.3.7)

Comparison of (3.3.3), (3.3.5), and (3.3.7) for several life tables suggests

that the diﬀerences in the P

are small (e.g., less than 2 percent in a life

table for United States females using 5-year age classes), and probably

irrelevant for most applications.

3.3.1.2 Birth-Flow Fertilities

The formulae for fertilities depend on the distribution of births and deaths

within an age class (Leslie 1945, Keyﬁtz 1968). The F

are deﬁned by the

3.3. The Leslie Matrix and the Life Table 57

ﬁrst row of the projection matrix:

(t +1)=



(t). (3.3.8)

Let B

(t,t+1)

denote the total number of births in the interval (t, t + 1),

and let n(x, t) be the number of individuals aged (x, x + dx)attimet

(remember that x is a continuous variable). At time t, individuals of age x

reproduce at the rate m(x)n(x, t), where m(x)dx is the expected number

of female oﬀspring produced by a female of age x in the interval (x, x +dx).

Integrating over time and age gives the total oﬀspring production:

(t,t+1)



∞

m(x)



t+1

n(x, z) dz dx. (3.3.9)

We, however, are ignorant of the detailed dynamics of n(x, t) within the

time interval, so we approximate



t+1

n(x, z)dz by the arithmetic mean of

n(x, t)andn(x, t + 1). With this approximation,

(t,t+1)

≈



∞

m(x)



n(x, t)+n(x, t +1)



dx. (3.3.10)

Next, we approximate the continuous variables m(x)andn(x, t) by con-

stant values (e.g., their means) m

and n

(t) over the age interval i − 1 ≤

x ≤ i. Given these approximations,

(t,t+1)

≈

∞



i=1



(t)+n

(t +1)



. (3.3.11)

However, n

(t +1) =P

i−1

(t)fori ≥ 2. Substituting this expression

into (3.3.11) and rearranging terms gives the number of births

(t,t+1)

≈

∞



i=1

+ P

i+1

) n

(t). (3.3.12)

Thenumberofbirthsisnotquiteequalton

(t+1); some of those oﬀspring

will not survive to time t+1. Those born just after t must survive almost an

entire projection interval to be included in n

(t+1). Those born just before

t + 1 are at risk of mortality for only a brief time. An average individual

must survive for one-half of the projection interval, the probability of which

is l(0.5). Thus

= l(0.5)



+ P

i+1



. (3.3.13)

If l(0.5) is not known directly, it can be estimated using linear interpolation

(Keyﬁtz 1968):

l(0.5) ≈

l(0) + l(1)

. (3.3.14)

58 3. The Matrix Model Framework

Most organisms, however, have relatively high neonatal mortality; in such

cases logarithmic interpolation is more accurate. Suppose that µ(x)=µ

for 0 ≤ x ≤ 1. Then l(1) = l(0) exp(−µ)and

l(0.5) = l(0)e

−µ/2

(3.3.15)

= l(0)



l(1). (3.3.16)

If detailed survival information within the ﬁrst age class is available, we

can use l(0.5) =



l(x)dx (Keyﬁtz 1968).

According to (3.3.13), the typical individual in age class i produces oﬀ-

spring at a rate that is the average of the maternity function for that age

class and the subsequent age class, the latter weighted by the probability of

survival to the subsequent age class. The oﬀspring produced must survive

for one-half time unit to be counted in the population at time t +1. An

alternative interpretation of (3.3.13) is in terms of a typical individual in

the center of age class i. It spends half of the projection interval producing

oﬀspring at the rate m

, and, if it survives, passes into the next age class

and produces oﬀspring at the rate m

i+1

for the rest of the interval.

3.3.2 Birth-Pulse Populations

In an ideal birth-pulse population, individuals reproduce on their birthday;

thus the maternity function is a discontinuous series of delta functions (Fig-

ure 3.6). The age distribution at any time is also a discontinuous series of

pulses (Figure 3.7). The age of these pulses depends on when the population

is counted, relative to the time of the pulse of breeding. Let p (0 <p<1)

denote the fraction of the time interval that elapses between the pulse of

reproduction and the census. At census time, the age distribution consists

of a series of pulses of individuals aged p,1+p,2+p, ....

In many studies (e.g., birds, large mammals; see Caughley 1977) cen-

suses are carried out either just before or just after breeding. These cases

correspond to the limits as p → 1andp → 0, and are called prebreeding and

postbreeding censuses, respectively. The formulae for survival and fertility

depend on which kind of census is assumed.

3.3.2.1 Birth-Pulse Survival Probabilities

In calculating birth-pulse survival probabilities, we no longer have to use ap-

proximate survival probabilities for “typical” individuals; every individual

in age class i is identical, aged i − 1+p.Thus

= P [survival from age i − 1+p to i + p] (3.3.17)

l(i + p)

l(i − 1+p)

. (3.3.18)

3.3. The Leslie Matrix and the Life Table 59

01 2 3

Age (

)

(

)

Figure 3.6. The maternity function for a birth-pulse population. Individuals

reproduce only on their birthday.

Abundance

Age (

)

1 –

1023

Figure 3.7. The age distribution at the time of census for a birth-pulse population.

The time of the census is deﬁned by p, the proportion of the time interval elapsing

between the pulse of reproduction and the census.

Thus, as shown in Figure 3.8, the survival probabilities are calculated as

⎧

⎪

⎨

⎪

⎩

l(i)

l(i − 1)

postbreeding (p → 0)

l(i +1)

l(i)

prebreeding (p → 1)

. (3.3.19)

Mortality during the ﬁrst age interval (say, the ﬁrst year) appears in two dif-

ferent forms in prebreeding and postbreeding censuses. For a postbreeding

census, P

= l(1)/l(0), and includes ﬁrst-year mortality. For a prebreeding

census, P

= l(2)/l(1), and excludes ﬁrst-year mortality; in this case the

missing mortality is incorporated into the fertility coeﬃcients.

60 3. The Matrix Model Framework

3.3.2.2 Birth-Pulse Fertilities

We begin by calculating the number of births in the interval (t, t+1). These

births occur when the individuals celebrate their next birthday, so

(t,t+1)

∞



i=1

(t) m

, (3.3.20)

where m

is the reproductive output of an individual upon reaching its ith

birthday and φ

is the probability that an individual in age class i survives

from the census to its next birthday. This probability can be approximated

by assuming a constant force of mortality over the interval (t, t + 1). Since

the probability of survival from t to t +1 is P

, the probability of surviving

for a fraction 1 − p of a time unit is φ

= P

1−p

. (Detailed information on

seasonal mortality rates within the year could, of course, be used if it were

available.)

Once reproduction has taken place, the oﬀspring must survive a fraction

p of a time unit in order to be counted in n

(t+1). That probability is given

by l(p); it may be estimated by interpolation, as l(0.5) was for birth-ﬂow

populations [Equations (3.3.14) and (3.3.16)]. Thus the fertility coeﬃcient

for a birth-pulse population is given by

= l(p)P

1−p

(3.3.21)



postbreeding (p → 0)

l(1)m

prebreeding (p → 1)

. (3.3.22)

Example 3.4 Parameterization of an age-classiﬁed matrix

Consider the following hypothetical life table for a population in

which all individuals die by their fourth birthday.

xl(x)

01.0

10.8

20.5

30.1

40.0

Application of (3.3.3) and (3.3.19) yields the following estimates for

the P

3.3. The Leslie Matrix and the Life Table 61

Birth-pulse

i Birth-ﬂow p → 0 p → 1

0.8+0.5

1.0+0.8

=0.722

0.8

1.0

=0.800

0.5

0.8

=0.625

0.5+0.1

0.8+0.5

=0.462

0.5

0.8

=0.625

0.1

0.5

=0.200

0.1+0.0

0.5+0.1

=0.167

0.1

0.5

=0.200

0.0

0.1

00—

Notice that in the prebreeding census (p → 1) there is no survival

from the third to the fourth age class, because individuals in the

fourth age class at the time of the census would be beginning to

celebrate their fourth birthday, but the life table implies that none of

them survives to do so. In the postbreeding (p → 0) case, however,

individuals in the fourth age class have just celebrated their third

birthday; hence P

> 0.

Given the reproductive outputs for each age class, we can use (3.3.13)

and (3.3.22) to calculate the fertility coeﬃcients F

Birth-pulse

i m

Birth-ﬂow p → 0 p → 1

1 00.9



0+2(0.722)



=0.650 (0.8)(0) = 0 (0.8)(0) = 0

20.9



2+6(0.462)



=2.052 (0.625)(2) = 1.250 (0.8)(2) = 1.6

60.9



6+3(0.167)



=2.926 (0.200)(6) = 1.200 (0.8)(6) = 4.8

30.9



3+0(0.000)



=1.350 (0)(3) = 0 (0.8)(3) = 2.4

The parameterization of the matrix aﬀects the results of the analysis.

For example, the eventual rates of exponential growth (calculated

using the methods of Section 7.1) implied by the three matrices in

this example are

Birth-ﬂow 1.793

Birth-pulse, p → 0 1.221

Birth-pulse, p → 1 1.221

The birth-pulse model yields the same growth rate regardless of the

time of census, but the birth-ﬂow and birth-pulse models predict quite

diﬀerent dynamics.

62 3. The Matrix Model Framework

Abundance

Age (

)

1 –

1023

= 2

= 3

Postbreeding census

(2)

1023

(1)

= 2

= 3

Prebreeding census

(3)

1023

(2)

Figure 3.8. The calculation of survival probabilities for a birth-pulse population.

The calculation of P

is shown; that of survival at other ages is similar. In a

postbreeding census, individuals in age class 2 are just barely 1 year old; to

survive for a year they must live to 2 years of age. In a prebreeding census,

individuals in age class 2 are almost 2 years of age, and to survive a year they

must live to reach 3 years of age.

3.4 Assumptions: Projection Versus Forecasting

Ecologists are, if anything, even more concerned than human demographers

about the logic and assumptions underlying these analyses. So far, we have

made three kinds of assumptions.

• We have assumed that it is appropriate to classify individuals by

age. This is not an innocuous assumption; the demography of many

organisms depends more on size or developmental stage than on age

(for statistical analyses of the relative importance of age and other

state variables see MPM Chapter 3).

• Because the model is discrete, it discards all information on the ages

of individuals within age classes. When we choose age classes (or

any other categories), we implicitly assume that this within-class

information is irrelevant.

• The linear time-invariant model in Example 3.1 projects the popula-

tion without changing the vital rates. This seems to assume that the

fertilities and survival probabilities remain constant over time.

3.4. Assumptions: Projection Versus Forecasting 63

The third assumption seems absurd. The vital rates of most organisms

vary conspicuously in time and space, and eﬀects of density on population

dynamics are well documented. How, then, can we justify analyses such as

those we have just discussed? Worse yet, how can we justify the even more

complex analyses of equally simple models that will follow in subsequent

chapters?

The answer to this question is fundamental to the interpretation of demo-

graphic analyses. It turns on an important distinction between projection

and forecasting or prediction (Keyﬁtz 1972a; see Chapter 12). A forecast

predicts what will happen. A projection describes what would happen, given

certain hypotheses. Grammatically speaking, forecasting uses the indicative

mood and projection the subjunctive. For example, a projection matrix A

implies an eventual population growth rate and structure. A population

described by A would eventually grow at that rate with that structure, if

nothing happened to change the vital rates. Ecological use of this purely

analytical result is sometimes criticized as if it asserted that the environ-

ment is constant. However, one must assume a constant environment as a

fact only if the model is being used as a forecast. No such assumption is re-

quired to interpret the growth rate and population structure as answers to

the hypothetical question: how would the population behave if the present

conditions were to be maintained indeﬁnitely?

Population projections reveal something about present conditions (more

precisely, about the relation between present conditions and the population

experiencing them), not about the future behavior of the population. As

Keyﬁtz (1972a) pointed out, one of the most powerful ways to study present

conditions is to examine their consequences were they to remain as they

are. A speedometer works the same way. A reading of 60 miles per hour

predicts that, in one hour, the car will be found 60 miles in a straight line

from its present location. As a forecast, this is almost always false. But as

a projection, it provides valuable information about the present situation

of the automobile.

So it is with demographic projections. They are particularly revealing be-

cause they integrate the impact of environmental conditions on vital rates

throughout the life cycle. To know the survival probabilities and fertilities

of every age class under a particular set of circumstances is to possess a

great deal of biological information about those circumstances. This in-

formation is most valuable when coupled with a comparative approach,

in which the vital rates are measured under two or more diﬀerent condi-

tions (e.g., Chapter 13). The use of demographic analysis in these studies

does not depend on assumptions of density independence or environmental

constancy.

64 3. The Matrix Model Framework

3.5 State Variables and Alternatives to

Age-Classiﬁcation

Age-speciﬁc survival and fertility rates are not always suﬃcient to de-

termine population dynamics. Even in human populations, for which

age-classiﬁed demography was originally developed, factors other than age

(sex, marital status, location) are known to aﬀect the vital rates. In or-

ganisms with more complex life cycles, age is even less adequate, and

demographic models should classify individuals by a more appropriate set

of life cycle stages.

One of the ﬁrst steps in an analysis is thus to choose a variable in terms

of which to describe population structure. This choice can be understood

in terms of the formal notion of “state” in dynamical system theory. The

“state” of a system provides the information necessary to predict the re-

sponse of the system. In Newtonian mechanics the state of a system is

given by the positions and momenta of its component particles, because

that information is suﬃcient to determine the response to any force. In

ethology, the ideas of “motivation” or “drive” are used to describe the

state of individual organisms, because they determine the response to a

stimulus. Physiologists characterize individuals by their levels of energy

reserves, lipid storage, hormones, metabolites, and so on. In demographic

models the state of the population is usually given by the distribution of

individuals among a set of categories (e.g., age classes). We will begin by

examining the formal basis for these usually intuitive ideas.

3.5.1 State Variables in Population Models

Formal state theory was introduced into population ecology by Caswell

et al. (1972), Boling (1973), and Metz (1977; see Metz and Diekmann

1986). Because demographic models connect individuals and populations,

Metz and Diekmann (1986) recognized the need to begin with the state

of the individual, which they called an i-state. Examples of i-state vari-

ables include age, size, maturity, developmental stage, and physiological

condition. Papers by Hallam et al. (1990) and Gurney et al. (1990) and the

book by Kooijman (1993) exemplify how much detail can be included in

physiological i-states.

The i-state variable provides the information necessary to predict the

response of an individual to its environment. However, we are interested

in modelling the population and hence need a population state variable,

or p-state variable. Metz and Diekmann (1986, Metz and de Roos 1992)

gave two conditions suﬃcient to guarantee that the p-state variable can

be written as a distribution of individuals among i-states (e.g., by an age

distribution if individuals are characterized by age).

1. All individuals experience the same environment.