x

age

nx

Sx

dx lx sx

px

mx

qx

Nx Tx ex kx

0 200

1 100

2 50

3 20

4 0

x = age at beginning of the time interval

Sx = Number of survivors at start of interval

x

age

nx

Sx

dx lx sx

px

mx

qx

Nx Tx ex kx

0 200 100 1.00

1 100 50 0.50

2 50 30 .25

3 20 20 .10

4 0

dx = Number dying during interval

lx = Proportion surviving at start of age interval = nx/no

x

age

nx

Sx

dx lx sx

px

mx

qx

Nx Tx ex kx

0 200 100 1.00 0.50 0.50

1 100 50 0.50 0.50 0.50

2 50 30 .25 .40 .60

3 20 20 .10 0 1.00

4 0

px = sx = age specific survival rate (probability of surviving to the

end of the period) = (nx+1)/nx; = 1- qx

qx = mx = age specific mortality rate (probability of dying before

the end of the period) = (nx-(nx+1)) / nx = dx/nx = 1 - px

x

age

nx

Sx

dx lx sx

px

mx

qx

Nx Tx ex kx

0 200 100 1.00 0.50 0.50 150 270 1.35

1 100 50 0.50 0.50 0.50 75 120 1.20

2 50 30 .25 .40 .60 35 45 0.90

3 20 20 .10 0 1.00 10 10 0.50

4 0

Nx = Mean number of individuals alive during the period =

(nx+nx+1)/2

Tx = sum of the time to be lived by the nx currently alive

ex = age specific life expectancy

x

age

nx

Sx

dx lx sx

px

mx

qx

Nx Tx ex kx

0 200 100 1.00 0.50 0.50 150 270 1.35 .693

1 100 50 0.50 0.50 0.50 75 120 1.20 .693

2 50 30 .25 .40 .60 35 45 0.90 .916

3 20 20 .10 0 1.00 10 10 0.50 …

4 0

kx = Exponential mortality = ln sx

sx = e-kt

Age specific fecundity of human females

x nx lx bx Fx lxbx sx N0 N1 N2 N3 N4 N5

0 200 1.0 0

1 100 .5 1

2 50 .25 2

3 20 .1 2

4 0 0 0

GRR=5

x nx lx bx Fx lxbx sx N0 N1 N2 N3 N4 N5

0 200 1.0 0 0 0

1 100 .5 1 100 0.5

2 50 .25 2 100 0.5

3 20 .1 2 40 0.2

4 0 0 0 0

GRR=5 NRR=1.2

lxbx = Number of female individuals born during interval x.

Gross Reproductive Rate (potential number of offspring) = bx

Net Reproductive Rate (average number)= Ro = lxbx = ( Fx)/no

x nx lx bx Fx lxbx sx N0 N1 N2 N3 N4 N5

0 200 1.0 0 0 0 .5 10 50

1 100 .5 1 100 0.5 .5 10 5

2 50 .25 2 100 0.5 .4 10 5

3 20 .1 2 40 0.2 0 10 4

4 0 0 0 0 0

GRR=5 NRR=1.2

x nx lx bx Fx lxbx sx N0 N1 N2 N3 N4 N5

0 200 1.0 0 0 0 .5 10 50 23

1 100 .5 1 100 0.5 .5 10 5 25

2 50 .25 2 100 0.5 .4 10 5 2.5

3 20 .1 2 40 0.2 0 10 4 2

4 0 0 0 0 0

GRR=5 NRR=1.2

x nx lx bx Fx lxbx sx N0 N1 N2 N3 N4 N5

0 200 1.0 0 0 0 .5 10 50 23 34 38.5 38.5

1 100 .5 1 100 0.5 .5 10 5 25 11.5 17 19.2

2 50 .25 2 100 0.5 .4 10 5 2.5 12.5 5.75 8.5

3 20 .1 2 40 0.2 0 10 4 2 1 5 2.3

4 0 0 0 0 0

GRR=5 NRR=1.2

x nx lx bx Fx lxbx lxbxx0 200 1.0 0 0 0 0

1 100 .5 1 100 0.5 0.5

2 50 .25 2 100 0.5 1.0

3 20 .1 2 40 0.2 0.6

4 0 0 0 0

GRR=5 NRR=1.2

G = cohort generation length = mean period elapsing between birth of parent and birth of offspring.

 G = [(lxbxx)]/[(lxbx)] = 2.1/1.2 = 1.75 = (xFx)/Fx

 (The time it takes) / (number produced)

Calculation of r

 Recall Nt = Noe

rt;

set t=G NG/No = erG = Ro

 

r = ln(Ro)/G = ln(1.2)/1.75 = .104

  r =(RolnRo)/ (lxbxx)

 

Reproductive value

vx = [erx/lx][ (e-rxlyby)] ;

where summation over range y = x → ∞