Using the life table to construct a population growth model with age structure:

The Leslie matrix

1) Need an N for every age class: N0, N1, N2, N3

2) Next rates that govern the addition and subtraction of individuals by age class:

• Individuals are added by birth or by aging. • Individuals are subtracted by death or by aging.

N4(t+1) = 0

x l(x) g(x) b(x)

0 1 0.8 0

1 0.8 0.5 2

2 0.4 0.25 3

3 0.1 0 1

4 0 0 0

pi : probability of surviving from age i to age i+1: i

ii llp 1

N0(t+1) = b1N1(t)+b2N2(t)+b3N3(t)

N3(t+1) = p2*N2(t) p2=0.25

N2(t+1) = p1*N1(t) p1=0.50

N1(t+1) = p0*N0(t) p0=0.80

N0(t+1) = b1N1(t)+b2N2(t)+b3N3(t)

N3(t+1) = p2*N2(t)

N2(t+1) = p1*N1(t)

N1(t+1) = p0*N0(t)

One model, four equations:

Another way to write these equations is in “matrix form”:

N0(t+1)N1(t+1)N2(t+1)N3(t+1)

b0 b1 b2 b3 p0 0 0 00 p1 0 00 0 p2 0

N0(t)N1(t)N2(t)N3(t)

=

Leslie Matrix

Age

cla

ss

3210

No of individuals

10050

1520

Time 0: Time 1:

N0(t+1) = 2*N1(t)+3*N2(t)+N3(t)

N3(t+1) = 0.25*N2(t)N2(t+1) = 0.5*N1(t)N1(t+1) = 0.8*N0(t)

Age

cla

ss

3210

No of individuals

100+60+1580

525

For:b1 =2, b2 = 3, b3 =1andp0=0.8, p1=0.5, p2=0.25

Plug & Play

Excel Worksheets:

• Leslie Matrix

0

2E+10

4E+10

6E+10

8E+10

1E+11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

age 1

age 2

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age 4

all

No of individuals

time

1

100

10000

1000000

1E+08

1E+10

1E+12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

age 1

age 2

age 3

age 4

all

No of individuals (log scale)

time

0.000.100.200.300.400.500.600.700.800.90

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

age 1

age 2

age 3

age 4

Frequency of individuals

time

1

100

10000

1000000

1E+08

1E+10

1E+12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

age 1

age 2

age 3

age 4

all

No of individuals (log scale)

time

STABLE AGE STRUCTURE:

A condition wherein the proportions of individuals of certain ages do not change as the population as whole increases or declines.

In age-structured population models (Leslie), the stable age structure is dictated by the model

parameters (age-specific birth rates and survivorships), and it often (but not always)

develops spontaneously.

N1(t+1) = N1(t)

N4(t+1) = N4(t)

N3(t+1) = N3(t)

N2(t+1) = N2(t)

At stable age structure, the Leslie Matrix model can be simplified:

equivalent to:

N1(t+1)N2(t+1)N3(t+1)N4(t+1)

N1(t)N2(t)N3(t)N4(t)

=

r = ln(R0)/G

r = ln()/

Summary:

1. Life table information can be used to write a population growth model for populations with overlapping generations.

2. Leslie Matrix models are density-independent, giving rise only to either exponential growth, zero growth, or exponential decay.

3. These models predict that often, but not always, populations approach a stable age distribution.

4. At that stable age distribution, all age classes grow or decline with equal rate.

Even though the Malthusian model can approach the output of age-structure models, responses to

perturbation are very different:

Define: Demographic Inertia

= the resistance of a population to change growth rate, due to the population’s age structure.

Plug & Play

Excel Worksheets:

• Demographic Inertia

A population in decline: JapanTransitioning from positive to zero or negative growth has

implications for the social organization of society.

Even though the Malthusian model can approach the output of age-structure models, responses to

perturbation are very different:

Structured populations require more time to adjust to new conditions. The results are time-lags between regime change

and the attainment of the new steady-state condition.