untitled

and so forth for the other age groups 2 and 3. This would obviously be rather cum-

bersome to calculate by hand for very long. Fortunately, there is a simple way to

automate the procedure, using matrix operations:

n<t+^1 >=A·n<t>

The “<>” notation used here refers to the age-specific abundance in year t. Hence,

n<^2 >stands for the abundances n0,2, n1,2, n2,2, and n3,2 present in year 2.

To add some flesh to these theoretical bones, we can supply some arbitrary values

for the age-specific rates of survival (p) and fecundity (m):

This leads to the age-specific totals shown in the matrix called n. The first column

of the matrix is the initial density of individuals in each age group, with newborns

on the top row and old individuals on the bottom row. Each column thereafter

represents densities in successively later years:

Model predictions of total population density, obtained by summing the densities of

all age groups present at any point in time (Nt=∑nx,t), are shown in Fig. 14.1.

After an initial period of adjustment, the population settles into a pattern of geo-

metric (i.e. exponential) population growth, whose finite rate of increase depends on

the integrated combination of age-specific parameters. As the population settles into

a geometrical growth pattern, the proportions of each age class in the population (Wx,t)

n

.........

.........

.........

.......

=

9 3 19 51 22 36 32 01 43 75 58 94 80 97 110 01 150 05

5 8 4 65 9 76 11 18 16 01 21 88 29 47 40 49 55 01

35 522 419 878 1006 1441 1969 2652 3644

17 192 28 237 463 549 77575 10 62..14 32

⎛

⎝

⎜

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

⎟

pm

.

.

.

.

.

.

.

.

=

⎛

⎝

⎜

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

⎟

=

⎛

⎝

⎜

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

⎟

05

09

05

01

05

22

29

37

AGE AND STAGE STRUCTURE 245

200

150

100

50

0

02468

t

N

t

Fig. 14.1Population
growth over time
predicted by the Leslie
matrix model with
constant survival and
reproduction discussed
in the text.