Keyfitz N., Caswell H. Applied Mathematical Demography

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7.3. Transient Dynamics and Convergence 165

In practice, when confronted by a reducible matrix, examine the life cycle

structure to see which subspaces are of most biological interest. In a popu-

lation with postreproductive age classes, it is obvious that the reproductive

part of the population dominates the dynamics, and that submatrix is of

main interest. In a metapopulation model the entire matrix is of interest,

and you must take care in interpreting the eigenvalues and eigenvectors.

7.3 Transient Dynamics and Convergence

The dominant eigenvalue of A gives the asymptotic population growth rate.

The population would grow at this rate if present environmental conditions

were maintained indeﬁnitely. That is unlikely to happen, so it is often useful

to calculate measures of the short-term or transient dynamics. The simplest

approach is numerical projection, which shows exactly what happens to the

population from any speciﬁc initial condition.

7.3.1 The Damping Ratio and Convergence

The solution (7.1.18) contains information on the rate of convergence to the

stable population structure and the oscillations produced by the subdom-

inant eigenvalues during convergence (e.g., Lefkovitch 1971, Usher 1976,

Horst 1977, Longstaﬀ 1984, Rago and Goodyear 1987). The asymptotic

rate of convergence is governed by the eigenvalue(s) with the second largest

magnitude. From (7.2.3) it is clear that, all else being equal, convergence

will be more rapid the larger λ

is relative to the other eigenvalues. This

leads to the deﬁnition of the damping ratio

ρ = λ

/|λ

|. (7.3.1)

From (7.2.3) it follows that

lim

t→∞



n(t)

− c

−t



=0.

Thus, for large t

n(t)

− c

≤ kρ

−t

= ke

−t log ρ

(7.3.2)

for some constant k. That is, convergence to the stable structure is asymp-

totically exponential, at a rate at least as fast as log ρ. Convergence could

be faster; for example, an initial population for which c

= 0 would con-

verge at a rate at least as fast as log(λ

/|λ

|). And, to take the extreme

case, an initial condition that is proportional to w

has already converged.

166 7. Birth and Population Increase from Matrix Population Models

The time t

required for the contribution of λ

to become x times as

greatasthatofλ

can be calculated as



|λ



= x (7.3.3)

from which

= log(x)/ log(ρ). (7.3.4)

For a method that considers all the subdominant eigenvalues, see Horst

(1977) and Rago and Goodyear (1987).

7.3.1.1 Entropy and Convergence

Demetrius (e.g., 1974, 1983) introduced a quantity he calls population en-

tropy, H, which is related to the rate of convergence. The derivation of H is

based on subtle connections between the dynamics of the age structure in a

matrix population model and the dynamics of the probability distribution

of “genealogies,” that is, of pathways that an individual, its ancestors, and

its descendants may take through the life cycle graph, and is claimed to

have important evolutionary implications.

The result, for an age-classiﬁed model, is deﬁned in terms of the discrete

net fertility function

= P

···P

i−1

(7.3.5)

Since the characteristic equation (7.1.11) is



−i

=1, (7.3.6)

the quantity φ

−i

can be treated as a probability distribution, and one

can calculate the entropy

H = −



−i

log(φ

−i

). (7.3.7)

Demetrius actually uses

H = −



−i

log(φ

−i

), (7.3.8)

where T =



iφ

−i

is a measure of mean generation length.

H measures the extent to which reproduction is spread through the life

cycle. For a semelparous life cycle, H = 0, while H is maximized by spread-

ing expected reproduction out evenly over the life cycle. For a related index

measuring the curvature of the survivorship curve, see Section 4.3.

Tuljapurkar (1982, 1993) has studied the relations between population

entropy and measures of convergence based on the eigenvalues. He shows

that H gives a lower bound on the rate of convergence to the stable age

7.3. Transient Dynamics and Convergence 167

distribution. In the second paper, he discusses the case of an iteroparous

but imprimitive matrix, in which the entropy is nonzero, but the population

does not converge to the stable structure. In these cases, H measures the

rate at which the age distribution converges to the d-dimensional subspace

of the state space within which the oscillating population structure must

lie (Tuljapurkar 1993).

7.3.1.2 Factors Inﬂuencing the Damping Ratio

The factors determining ρ in age-classiﬁed populations have been studied

by Coale (1972) for humans and Taylor (1979) for insects. Both studies

used Lotka’s continuous time model (7.5.1) for the birth series B(t). They

measured the damping ratio by r

− u

where u

is the real part of the

second root of Lotka’s equation; this measure is equivalent to log ρ.

Human Populations

Coale’s (1972) calculations are based on the net fertility function φ(x)=

l(x)m(x), which gives the age-speciﬁc expected reproductive output of a

newborn individual. He assumed that φ(x) is approximately symmetrical,

with mean µ

(µ

is a measure of mean generation time; see Section 11.3.5).

He measured the concentration of reproduction near µ

by the proportion

of φ(x) in the interval 3µ

/4 ≤ x ≤ 5µ

/4, and measured the asymme-

try of φ(x) by the ratio of the median to the mean. Based on 47 human

life tables, he concluded that log ρ was negatively correlated with both

the concentration and the asymmetry of fertility. Thus, populations in

which reproduction is spread out symmetrically over a wide range of ages

should converge to the stable age distribution more rapidly than those

with tightly restricted ages at reproduction, or skewed distributions of age

at reproduction. See Section 14.6 for a more analysis.

Insect Populations

Taylor (1979) carried out an extensive investigation of r

− u

in insect

populations. He used an age-classiﬁed model after converting calendar time

to units of “degree-days” to compensate for the temperature dependence of

development rate in insects. He used t

, the time required for contribution

of the second root to decline to 5 percent of that of the dominant root [see

(7.3.4)], to measure of the time required for convergence to the stable age

distribution.

Based on statistical models of the l(x)andm(x) functions, Taylor con-

cluded that the time to convergence was most strongly aﬀected by the age at

ﬁrst reproduction (earlier reproduction leads to more rapid convergence)

and the variance in m(x) (greater variance leads to more rapid conver-

gence). The time to convergence was nearly independent of survivorship

and the amount of reproduction.

168 7. Birth and Population Increase from Matrix Population Models

Taylor also calculated t

from data for 36 populations of 30 species of

insects and mites. He found values of t

ranging from 280 degree-days

(for the aphid Myzus persicae) to 115,120 degree-days (for each of two

species of moths). Typical ﬁgures for the duration of a growing season are

on the order of 1000–3000 degree-days, which would allow 40–70 percent of

the populations Taylor examined to converge to within 5 percent of their

stable age distributions.

Of course, this conclusion assumes constant vital rates during the course

of the growing season. As Taylor notes, variation in the vital rates would

slow down the process of convergence.

Taylor concludes that “the greater part of insect species existing in sea-

sonal environments never experience, or spend a small proportion of their

time in, a stable age distribution” (Taylor 1979, p. 527). This conclusion

may be too strong. There is nothing magical about a 5 percent contribution

of the second root as representing convergence. If a 10 percent threshold

is used instead, 55–75 percent of the populations would have time to con-

verge in a typical growing season. A 20 percent threshold would allow 65–80

percent of the populations to converge.

Carey (1983) provides some additional insight into the process of conver-

gence. He collected information on the stage distribution (egg, immature,

and mature) of a tetranychid mite on cotton plants throughout a growing

season [mites were among the most rapidly converging species in Taylor’s

(1979) tabulation]. He compared these distributions with the stable age dis-

tribution calculated on the basis of laboratory life table experiments. The

laboratory data predicted stable age distributions for increasing, station-

ary, and declining populations diﬀering only in their fertilities. He found

that age distributions of the increasing, stationary, and declining phases of

the ﬁeld population tended to converge to the predicted stable values. This

indicates that even if populations do not reach a constant stable age distri-

bution, the patterns of convergence to and deviation from that distribution

may provide useful biological information.

Multi-regional and Age-Size Models.

In age-size or multi-regional models individuals are characterized by mul-

tiple criteria. In this context, one can ask whether the age distribution

converges more or less rapidly than the size or region distribution (Liaw

1980, Law and Edley 1990). The answer seems to depend on the details of

the model.

Liaw (1980) found that the age distribution in a multi-regional model for

the human population of Canada converged more rapidly than the region

distribution. He interpreted this in terms of the magnitudes of the subdom-

inant eigenvalues of the matrix. Keyﬁtz (1980) likened the phenomenon to

the convergence of temperature in a set of interconnected rooms; the higher

rate of mixing homogenizes temperature within each room before the tem-

7.3. Transient Dynamics and Convergence 169

perature diﬀerences between rooms can decay. This interpretation depends,

of course, on the fact that migration between regions is generally small.

Law and Edley (1990) draw the opposite conclusion for age-size matrices.

They conclude that the size distribution should converge more rapidly than

the age distribution. This is partly a function of their assumption that all

births take place in the smallest size class. Since all individuals are born at

the same size, each cohort proceeds through the life cycle graph with the

same distribution of size-at-age. Thus, once the individuals in the initial

population have died out, the size distribution no longer changes. This will

not be true in a multi-state model in which individuals may be born into

more than one size class, region, or other state.

7.3.2 The Period of Oscillation

When complex eigenvalues are raised to powers, they produce oscillations

in the stage distribution, the period of which is given by

2π

(7.3.9)

2π

tan

−1



(λ

)

(λ

)



, (7.3.10)

where θ

is the angle formed by λ

in the complex plane and (λ

)and

(λ

) are the real and imaginary parts of λ

, respectively.

The longest-lasting of the oscillatory components is that associated with

. In age-classiﬁed models, P

is approximately equal to the mean age

of childbearing in the stable population (Lotka 1945, Coale 1972). Thus

we would expect that perturbations to the stable age distribution would be

followed by damped oscillations with a period about equal to the generation

time.

In complex life cycles, P

cannot be identiﬁed with the mean age of re-

production, but it still measures the period of the oscillations contributed

by the most important subdominant eigenvalue. Resonance between these

oscillations and environmental ﬂuctuations is an important factor in the dy-

namics of populations in periodic environments (Nisbet and Gurney 1982,

Tuljapurkar 1985).

7.3.3 Measuring the Distance to the Stable Stage Distribution

It is useful to be able to measure the distance between two stage distri-

butions, or between an observed stage distribution and the stable stage

distribution. We will discuss two such measures here; a third (the Hilbert

projective pseudometric) is useful in the study of convergence in variable

environments (Golubitsky et al. 1975; MPM, Chapter 13), but has received

little actual use in demography.

170 7. Birth and Population Increase from Matrix Population Models

We wish to measure the distance between n(t) and the stable population

w. Without loss of generality, w can be scaled so that



=1,andwe

can transform n(t)intox(t)=n(t)/



(t), so that both vectors describe

the proportions of the population in the diﬀerent stages.

7.3.3.1 Keyﬁtz’s ∆

Keyﬁtz (1968, p. 47) proposed a measure that is equivalent to

∆(x, w)=



− w

|, (7.3.11)

which is a standard measure of the distance between probability vectors. Its

maximum value is 1 and its minimum is 0 when the vectors are identical.

7.3.3.2 Cohen’s Cumulative Distance

Keyﬁtz’s ∆ measures the distance between n and w independent of the

path by which n would actually converge to w. Cohen (1979b) proposed

two indices that measure the distance between the two vectors along the

pathway by which convergence takes place. We know from (7.2.4) that

n(t)/λ

will converge to c

. Suppose that n(0) = n

. Cohen’s indices are

obtained by accumulating the diﬀerences between n(t)/λ

and c

s (A, n

,t)=



i=0



n(i)

− c



(7.3.12)

r (A, n

,t)=



i=0



n(i)

− c



. (7.3.13)

The vector s(t) accumulates the diﬀerence between n(t)/λ

and c

whereas r(t) accumulates the absolute value of those diﬀerences.

As a measure of the cumulative distance between an initial population

and its eventual limiting distribution, Cohen proposes calculating the

limit as t →∞of s(t)andr(t), and then adding the absolute values of the

elements of the vectors.



lim

t→∞

(A, n

,t)| (7.3.14)



lim

t→∞

(A, n

,t)|. (7.3.15)

Cohen actually proposes scaling these indices by multiplying by a constant

(50 in his example) to make their magnitude comparable to that of ∆, but

this is not necessary.

Cohen gives an analytical expression for the limit in (7.3.14). Let B =



,andletZ =(I + B − A/λ)

−1

.Then

lim

t→∞

s (A, n

,t)=(Z − B)n

. (7.3.16)

7.3. Transient Dynamics and Convergence 171

The limit in D

, however, must be calculated numerically. When n

= w,

= 1 and D

= D

= 0. There is no well-deﬁned upper bound for either

distance.

Example 7.4 Calculation of D

and D

Consider the matrix

A =

⎛

⎝

0.4271 0.8498 0.1273

0.9924 0 0

00.9826 0

⎞

⎠

(7.3.17)

(for the 1965 U.S. population with 15-year age intervals, not that it

matters here). The stable age distribution and reproductive value,

scaled so that bov

∗

=1,are

⎛

⎝

0.4020

0.3299

0.2681

⎞

⎠

⎛

⎝

1.4487

1.1419

0.1525

⎞

⎠

(7.3.18)

and the observed age distribution, which we take as the initial

population, is

⎛

⎝

0.4304

0.3056

0.2640

⎞

⎠

. (7.3.19)

Thus c

= v



=1.0127.

The cumulative deviation vectors s (A, n

,t)andr (A, n

,t)are

r (A, n

,t)

s (A, n

,t)

0 0.0232 0.0285 0.0075 0.0232 −0.0285 −0.0075

0.0358 0.0476 0.0306 0.0106 −0.0094 −0.0306

0.0424 0.0579 0.0461

0.0172 −0.0198 −0.0151

0.0457 0.0633 0.0545 0.0138 −0.0144 −0.0235

0.0474 0.0660 0.0589 0.0155 −0.0171 −0.0192

0.0483 0.0674 0.0611

0.0147 −0.0157 −0.0214

0.0487 0.0681 0.0622 0.0151 −0.0165 −0.0203

0.0489 0.0685 0.0628 0.0149 −0.0161 −0.0209

0.0490 0.0686 0.0631 0.0150 −0.0163 −0.0206

0.0491 0.0687 0.0633 0.0149 −0.0162 −0.0207

0.0491 0.0688 0.0633 0.0150 −0.0162 −0.0206

By t = 10, both r(·)ands(·) have converged to four decimal places.

The distance indices are D

=0.0518 and D

=0.1814.

Unlike ∆, both D

and D

are functions not only of n

, but also of the

projection matrix A, which determines the pathway by which n(t)

will converge. Consider the following three age-classiﬁed projection

172 7. Birth and Population Increase from Matrix Population Models

matrices:

⎛

⎝

0.3063 0.6094 0.0913

0.9924 0 0

00.9826 0

⎞

⎠

⎛

⎝

00.8784 0.1316

0.9924 0 0

00.9826 0

⎞

⎠

⎛

⎝

00.0641 0.9603

0.9924 0 0

00.9826 0

⎞

⎠

These matrices all have the same dominant eigenvalue (λ

=1)and

the same stable age distribution,

w =

⎛

⎝

0.3370

0.3344

0.3286

⎞

⎠

, (7.3.20)

but A

and A

have reproduction concentrated in the last two and

the last age class, respectively.

Suppose that all three populations begin with the common initial

vector (7.3.19). The distance between n

and w as measured by ∆ is

∆=0.093, regardless of which projection matrix is used. However,

the values of D

and D

vary dramatically depending on the pattern

of reproduction:

Matrix D

0.1995 0.6058

0.1652 1.5646

0.1066 5.6866

When measured by D

, the cumulative distance from n

to w

de-

creases as reproduction is concentrated in a single age class. When

measured by D

, the cumulative distance increases as reproduction is

concentrated. This reﬂects the increasing oscillations in the age distri-

bution as reproduction is concentrated. Since s (A, n

,t) accumulates

positive and negative deviations from the stable distribution, oscilla-

tions tend to cancel each other out. On the other hand, r (A, n

,t)

accumulates the absolute value of the deviations, and the oscillations

are reﬂected in larger values of D

7.3.4 Population Momentum

Consider a population growing subject to a set of vital rates, and imagine

that you instantly and permanently change those rates to a set of stationary

rates; i.e., rates that yield λ

= 1. The population will eventually stop

7.3. Transient Dynamics and Convergence 173

growing, but unless its structure at the moment of the change happens to

match the stable stage distribution of the new vital rates, it will not stop

immediately. Hence, its ﬁnal size will diﬀer from its size when the vital

rates were changed. Keyﬁtz (1971b; see Sections 8.6 and 12.3) introduced

the term population momentum to describe this diﬀerence.

Let t = 0 denote the instant when the vital rates are changed; the

population vector is n(0) = n

. The momentum M is

M = lim

t→∞

n(t)

n



, (7.3.21)

where n =



| is the total population size. When M>1 the popu-

lation stabilizes at a size larger than its size at t =0.WhenM<1, the

population shrinks before stabilizing.

Denote the old and new projection matrices by A

old

and A

new

,with

eigenvalues λ

(old)

and λ

(new)

. Assume that both matrices are primitive.

Since λ

(new)

= 1, we know from (7.1.30) that

lim

t→∞

n(t)

(new)

= lim

t→∞

n(t)



(new)∗



(new)

Thus

M =



(new)∗



(new)

, (7.3.22)

where e is a vector of 1s.

For age-classiﬁed models M can be calculated explicitly in terms of the

birth rate, life expectancy, mean age of childbearing, and net reproductive

rate (Section 8.6).

The momentum of declining populations may be of interest in conserva-

tion biology. Consider a declining endangered species (i.e., λ

(old)

< 1), and

suppose (optimistically) that a new management strategy instantaneously

brings the vital rates up to replacement level. If M<1, the population

will continue to decline before stabilizing, and the extent of this decline

might be an important management consideration. It turns out that stage-

classiﬁed models can yield results that are surprising from an age-classiﬁed

perspective.

Example 7.5 Population momentum in Calathea ovandensis

Calathea ovandensis is a perennial herb that grows in the under-

story of tropical forests. Horvitz and Schemske (1995) reported

size-classiﬁed matrices for four sites, over four years, in a Mexican rain

forest. Individuals were classiﬁed into eight stages: seeds, seedlings,

174 7. Birth and Population Increase from Matrix Population Models

juveniles, pre-adults, and small, medium, large, and extra-large

adults. Only the adult stages reproduce.

Here we focus on two of the 16 matrices reported by Horvitz and

Schemske (Plot 3, 1982–1983 and Plot 3, 1983–1984; we will call them

plots 3-82 and 3-83). The growth rates and net reproductive rates

†

are

Plot 3-82 1.1572 13.7360

Plot 3-83 0.8876 0.5771

The conditions in plot 3-82 support rapid population growth with a

high value of R

. Suppose that, concerned about a population explo-

sion of C. ovandensis, we implemented a pest control strategy that

instantly reduced fertility to replacement level by dividing each fer-

tility by R

. The momentum, calculated from (7.3.22), is M =0.107.

Rather than continuing to grow, the population would shrink to about

10 percent of its current size before stabilizing.

The conditions in plot 3-83 would, if maintained, lead to a population

decline of almost 12 percent per year and eventual extinction. Sup-

pose that, concerned about this endangered species, we introduced a

conservation strategy that instantly increased fertility to replacement

level. The resulting momentum is M =5.067. The population would

grow to ﬁve times its present size before stabilizing.

These results diﬀer from those familiar in age-classiﬁed models be-

cause of the stable stage distributions before and after the change in

the vital rates. These distributions are

Plot 3-82 Plot 3-83

(old)

(new)

old

new

(old)

(new)

old

new

0.875 0.737 1.186 0.942 0.971 0.970

0.104 0.102 1.025

0.031 0.028 1.093

0.009 0.010 0.885

0.004 0.001 5.482

0.002 0.003 0.693

0.007 0.000 36.820

0.002 0.008 0.288

0.015 0.000 54.864

0.000 0.002 0.241

0.002 0.000 62.810

0.002 0.039 0.055

00—

0.005 0.099 0.047

00—

In plot 3-82, the old stable stage distribution has too few individuals

in the adult stages (5–8). As a result, the population actually declines

as it converges to stationarity. In contrast, in plot 3-83, the old stable

stage distribution has too many individuals in the adult stages. The

†

See Section 11.3.4 for calculation of R

from stage-classiﬁed models.