untitled

converge on a stable age distribution, as illustrated in Fig. 14.2. Each age-specific

proportionWx,tis given by:

Wx,t=

So, over time, we can be sure of two things: (i) the population will eventually

grow geometrically; and (ii) once this happens, the proportions of individuals in

each age group will also become constant. The findings of geometric increase in N

and stable age distribution imply that the following mathematical statements are

equivalent: Nt+ 1 =λ*Ntand n<t+^1 >=A*n<t>, where Nand λare scalars(i.e. single,

countable numbers) and Aand nare matricesor vectors. In other words, a simple

model of geometric increase (Nt+ 1 =λNt) yields the same results as the Leslie matrix

model (n<t+^1 >=A·n<t>). This means that we can estimate λ(the finite annual rate

of increase) from the transition matrix A, by something called the dominant

(largest) eigenvalueof the transition matrix. The largest of the eigenvalues (1.36) is

the finite rate of population increase (λ) once the population has reached a stable

age distribution:

Hence, after the initial period of uncertainty, the total population would increase

by 36% per year (because λ=1.36). This period of uncertainty is generally two

or three generations, where generation is defined as the typical time that elapses

between a mother’s birth and that of her daughters (discussed in more detail in

Chapter 6).

Just as there is a simple means of estimating the eventual rate of population growth

(dominant eigenvalue), there is an equally simple way to predict the eventual pro-

portion of individuals in each age group. We calculate the so-called “right eigenvector”

corresponding to the “dominant eigenvalue” of the transition matrix A:

eigenvals( )A

.

..

..

.

=

−+

−−

−

⎛

⎝

⎜

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

⎟

136

049 038

049 038

003

i

i

nx,t

Nt

246 Chapter 14

0.8

0.6

0.4

0.2

0

024 6 8

t

S3, t

S1, t

S2, t

Fig. 14.2Changes over
time in the standing age
distribution predicted by
the Leslie matrix model
discussed in the text,
with the lines showing
different age groups.