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
age 3
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.
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