# 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

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