Example

Find GRR, R0, and G. Estimate r and then find its true value with the Euler equation. Verify the predicted stable age distribution (SAD). Project this population as described above. After reading the next section, use the Leslie matrix to project the population.

Age

lx

mx

Px

q*

l* x m *

x x lx x mx

Euler based on r = 0.152

Euler based on r = 0.154

lx x e-rx

Cx of SAD

0

1.00

0

0.250

0.750

0.00

0.00

0

0

1.000

0.733

1

0.25

0

0.400

0.600

0.00

0.00

0

0

0.214

0.157

2

0.10

7.0

0.800

0.200

0.70

1.40

0.517

0.514

0.073

0.054

3

0.08

7.5

0.500

0.500

0.60

1.80

0.380

0.378

0.050

0.037

4

0.04

5.0

0.250

0.750

0.20

0.80

0.109

0.108

0.022

0.016

5

0.01

0

0.000

1.000

0.00

0

0

0

0.005

0.003

6

0

0

-

-

0.00

0

0

0

0.000

0.000

E

GRR =

R0 = 1.50

4.00

1.006

1.000

1.364

1.000

19.5

G =

Estimated

Predicted

4.00/1.50

value of

value of

= 2.67

r = 0.152

l = er

= 1.17

Age

nx at

Cx at

nx at

Cx at

nx at

Cx at

nx at

Cx at

nx at

Cx at

t = 0

t = 0

t = 1

t = 1

t = 2

t = 2

t = 3

t = 3

t = 4

t = 4

0

250

0.714

318

0.743

359

0.729

420.6

0.733

492.10

0.733

1

60

0.171

62.5

0.146

79.5

0.162

89.75

0.156

105.15

0.157

2

20

0.057

24

0.056

25

0.051

31.8

0.055

35.90

0.054

3

12

0.034

16

0.037

19.2

0.039

20

0.035

25.44

0.038

4

6

0.017

6

0.014

8

0.016

9.6

0.017

10.00

0.015

5

2

0.006

1.5

0.004

1.5

0.003

2

0.003

2.40

0.003

6

0

0.000

0

0.000

0

0.000

0

0.000

0

0

N = 350

1.000

N = 428

1.000

N = 492.2

1.000

N = 573.75

1.000

N = 670.99

1.000

l =

l =

l =

l =

428/350

492/428

574/492

671/574

= 1.22

= 1.15

= 1.17

= 1.17

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