Age Structured Population
The population size at time t + 1 depends on the population structure at time t. For
example, you’d expect different results if 90% of the individuals at time t were of
breeding age versus if 90% of the individuals were infants.
In our discussion below we develop a discrete-time model of an age-structured
population which describes development, mortality, and reproduction of organisms.
We use the following variables and parameters in the model:

N n, t = number of organisms of age n at time t (age is measured in the same

units as time t). Usually, only females are considered and males are ignored
because, as a rule, the number of males does not affect population growth.
s n = survival-rate of organisms in age interval from n to n + 1.
Fn = average number of female offspring produced by 1 female in age interval
from n to n + 1 (mortality of parent and/or offspring organisms is included)

We now have the following two equations:
N
 N n, t  s n
n1, t 1
[1]
T
  N
F
0, t 1
n, t n
n0
[2]
N
Equation [1] represents development and mortality where the number of individuals in
age n + 1 in time t + 1 equals to the number of individuals in the previous age and
previous time multiplied by age-specific survival rate s n .
Equation [2] represents reproduction and specifies the number of individuals in the first
age class whereas equation [1] specifies the number of individuals in all other age
classes.
In the equation [2] the number of new-born organisms equals to the number of mothers,
N n, t , multiplied by the numbers of offspring produced, Fn
(called the
fecundity of class N n, t . The number of offsprings is summed over all ages of mothers.
For T = 3, the number of individuals in each class at time t + 1 are:
 F N
F N F N
F N
0 0, t
1 1, t
2 2, t
3 3, t
N
s N
1, t 1 0 0, t
N
s N
2, t 1 1 1, t
N
s N
3, t 1 2 2, t
N
0, t 1
We rewrite these equations as the system of linear equations:
 F N
F N F N
F N
0 0, t
1 1, t
2 2, t
3 3, t
N
s N
 0 N  0 N
 0 N
1, t 1 0 0, t
1, t
2, t
3, t
N
 0 N
 s  N  0 N
 0 N
2, t 1
0, t 1 1, t
2, t
3, t
N
 0 N
 0 N  s  N
 0 N
3, t 1
0, t
1, t 2 2, t
3, t
N
0, t 1
We may write the population sizes at time t and time t + 1 as the column matrices
N 0 
N 
 1  (t )
N 2 
 
N3 
and
N0 
N 
 1  (t  1) , respectively.
N 2 
 
N3 
We denote the coefficient matrix (also called the growth matrix) of the system of
equations as
F F F F 
2 3
 0 1
s
0 0 0 
0


L
0 s 0
0 
1


0 0 s
0 
2


The system can now be written as one matrix equation
F F F F 
2 3 N 0 
N0 
 0 1
N 
s
0 0 0  N 
 1  (t  1) =  0
  1  (t )
N 2 
0 s 0
0  N 2 
1
 

  
N3 
0 0 s
0  N3 
2


If we denote the population size matrices as N, we have
Nt +1 = LNt
As before
NT = LTN0
How to read matrix models?
Each column specifies the fate of organisms in specific state. The number in the
intersection of column i and row j indicates how many organisms in state j are produced
by one organism in state i. In the Leslie model, organisms' state is defined by age only.
For example, the third column corresponds to age a=2. An organism in age 2 produces m
offspring of age 0 (first cell in the column), and goes to age class 3 with probability s (the
cell under main diagonal).
Example:
Use the following values: F1=0, F2=0, F3=2, F4=4, S1=0.5, s2=1.0, s3=0.5
 F1
s
L=  1
0

0
F2
F3
0
0
s2
0
0
s3
F4   0
0  0.5
=
0  0
 
0  0
4
0 
0

0 0.5 0 
0
0
1
2
0
0
Start with a population of only 100 1 year olds
N 0 
100 
N 
 
 1  (t ) =  0 
N 2 
 0 
 
 
 0 
N3 
What would the population be after 1 year?
N0 
0
N 
 
 1  (t  1) = 50 
N 2 
0
 
 
0
N3 
One year olds do not produce any offspring, and half survive to become 2 year olds.
What about the next year?
N0 
0
N 
 
 1  (t  2) =  0 
N 2 
50 
 
 
0
N3 
Two year olds do not produce any offspring, but all survive to become 3 year olds. One
more year…
N0 
100 
N 
 
 1  (t  3) =  0 
N 2 
 0 
 
 
 25 
N3 
And finally…
N0 
100 
N 
 
 1  (t  4) =  50 
N 2 
 0 
 
 
 0 
N3 
Look at the total population: It went from 100 to 50 to 50 to 125 to 150. It’s fluctuating
up and down; it will eventually smooth out to a more even increase.
1000000
100000
10000
1000
100
10
1
0
10
20
30
40
50
The individual age groups also fluctuate before evening out
100000
10000
Series1
1000
Series2
Series3
100
Series4
10
1
0
10
20
30
40
50
Remember, earlier when we plotted these graphs the slope of the line was equal to the
rate of increase λ. These plots also have a straight line; although our basic values are a
matrix, there is a single value λ which describes the overall rate of increase (although not
the age structures). This can be found as the dominant eigenvalue of L; we may not teach
you how to calculate it here, but it is something you would learn in a full matrix algebra
course.
Let’s take another case: both simpler and more complicated. It’s simpler because we’ll
just have two age groups: young (N1) and old (N2)
 N1 
 N  (t)
 2
Now, let’s say that young survive to become old with rate s1. The rate of old survival
will be s2. Unlike before, not all of the oldest group have to die every year. The
fecundity of the old group will be F2. The fecundity of the young group will be F1; some
of the young may reproduce as well. Our L matrix becomes
F
L=  1
 s1
F2 
s2 
When we use age groups rather than actual age, we call it a Lefkovitch matrix (1965)
rather than a Leslie matrix, but it works the same way.
NT = LTN0
The complication is the equations are not as simple as they were before, because this
matrix is harder to multiply out. This is where we really need to learn a little matrix
algebra.