THE LESLIE MODEL
Age-Specific Population Model
The imaginary brown rat, Rattus Norvegicus1
The life-span of rats is up to 15-18 months, and they have their first litter at
approximately 3 months and continue to reproduce every 3 months until they reach senior citizen
status at 15 months. Assume the proportion of males is approximately equal to the proportion of
females in the population and the birth rates and survival rates of the female population remain
constant over time. The birth rate is the average # of daughters born per hundred females in each
age group. The survival rat is the probability that a rat will survive and move into the next age
group.
The birth and survival rate for each age-specific category is:
Age (months) Birth rate Survival Rate
0–3
0
0.6
3–6
0.3
0.9
6–9
0.8
0.9
9 – 12
0.7
0.9
12 – 15
0.4
0.6
15 – 18
0
0
And the initial population, P0, for each age category is:
0 – 3 3 – 6 6 – 9 9 – 12 12 – 15 15 – 18
15
9
13
5
0
0
1. How many newborn rats will there be after the first three months? Explain how you
found this number. We will call the first 3-month period P1 to represent the first iteration
of the population.
2. In order to get into the 3 – 6 month category, a rat would have to survive from the 0 – 3
moth bracket. Since the survival rate is 0.6, or 60% or the population, how many rats
will be in the 3 – 6 month category after the first iteration?
3. Similarly, determine the number of rats in each of the 3-month age-specific categories
after the first iteration.
0 – 3 3 – 6 6 – 9 9 – 12 12 – 15 15 – 18
a. What is the total number of rats in P1? ______________
Armontrout:
Exeter ’95
Recursion of Matrices: Leslie Model 1
Define the following matrices:
Initial Population Matrix, P0
BR
The Leslie Matrix, [L]
SR1 SR 2 SR3 SR 4 SR5
0  3  0 0.6
3  6  0.3 0
15 9 13 5 0 0
6  9  0.8 0

9  12  0.7 0
12  15  0.4 0

15  18  0
0
4. Why are there only 5 survival rates for the Leslie Matrix?
0
0
0
0.9
0
0
0
0.9
0
0
0
0.8
0
0
0
0
0
0
0 
0 
0 

0 
0.6 

0 
Put the initial population matrix P0 in matrix [A] in your calculator and the Leslie Matrix
[L] in matrix [B]. Of course you need to know the dimensions of the matrices. P0 is a 1×6
and [L] is a 66. Explain why you can multiply [A][B]. Can you multiply [B][A]? When
you multiply [A][B], what is your result in terms of this population problem?
On the TI-83, 83-Plus, or 84-Plus: {Import screen shots here}
5. What does P2 above represent in terms of this population?
a. What is the total # of rats in P2? ___________
6. What will be the distribution of the rat population at the end of 1 year (12 months)? In
other words, how many female rats can we expect in each age category? At the end of 18
months? after 2 years? What is the total number of female rats in P4, P6, and P8?
a. What seems to be happening to the population in each category as the number of
iterations increases? To the total population?
Armontrout:
Exeter ’95
Recursion of Matrices: Leslie Model 2
To calculate the population in each category at iteration Pn, you must know the population at
time n – 1. In other words,
P1 = P0 * [L]
P2 = P1 * [L], and
Pn = Pn-1* [L].
But P1 = P0 *[L], so P2 = (P0 * [L]) * [L] = P0 * ([L] * [L]) = P0 * [L]2
So, the closed form for the Leslie Population Model must be:
Pn = P0 * [L]n. Try it for the P4, P6, and P8 you got on the previous page.
To find the total number of rats in each cycle, multiply by a column matrix [J] of six 1’s.
RECALL: Pn is going to be a one-row matrix. Multiplying Pn by a column matrix of ones
will multiply each element in Pn by 1 and adding them up.
Professor Leslie made some important claims about the long-term growth of age-specific
populations. We have determined that, if left alone with no interference, each category will
continue to grow larger and the total population will also grow larger. But, what about the
rate of growth? Calculate P19, P20, and P21 and determine the growth rates of the individual
categories as well as for the total populations.
P19
P20
P21
0 – 3 months
0 – 3 months
0 – 3 months
3 – 6 months
3 – 6 months
3 – 6 months
6 – 9 months
6 – 9 months
6 – 9 months
9 – 12 months
9 – 12 months
9 – 12 months
12 – 15 months
12 – 15 months
12 – 15 months
15 – 18 months
15 – 18 months
15 – 18 months
Total
Total
Total
Growth Rates:
P20  P19
P21  P19
0 – 3 months
0 – 3 months
3 – 6 months
3 – 6 months
6 – 9 months
6 – 9 months
9 – 12 months
9 – 12 months
12 – 15 months
12 – 15 months
15 – 18 months
15 – 18 months
Total
Total
Armontrout:
Exeter ’95
Recall that the growth rate is
new pop.  old pop.
r=
 100
old pop.
Recursion of Matrices: Leslie Model 3
7. What do you observe? Look at P22 – P21 and P31 – P30 to test your conjectures.
8. Is the long-term growth rate for this rat population predetermined by the initial
population or will all populations stabilize to this same rate? Experiment and write up
your conjectures.
9. Suppose the carrying capacity of the environment is 200 rats. How many months will it
take before the population starts dying off due to overcrowding?
Harvesting the Population2:
To avoid overpopulation, a harvesting strategy is designed to control the rat population.
Since the total population stabilizes at a fixed growth rate r, the Leslie model Pn = Pn-1 * [L]
can be thought of the linear iteration model used in the study of financial mathematics for
CD’s: i.e. Pn = Pn-1 * (1 + r) for large values of n.
In order to maintain a population level at P0, a proportion of the population must be
harvested (transplanted, etc.) to match the growth rate, in other words, remove a percentage
of the population to return the population to its previous level. Let h be this percentage to be
harvested, so (1 – h) would be the percentage of the population remaining after the harvest.
We would like to see Pn (1 + r) (1 – h) [the population at the end of the nth iteration] to be
returned to the Pn level, or Pn (1 + r) (1 – h) = Pn. Solving for h, we get h  1  11r . Since
the long-term population growth rate for the rat population is 3.04%, h  1  1.1034  0.0295 ,
and 1 – h = 0.9705. Therefore, the harvesting strategy is to allow the population to reach the
desired limit, then at regular intervals harvest 2.95% of the population.
By the 20th iteration (3-month period), the growth rate has stabilized to 3.04%. if we now
decide to harvest the population using the 2.95% model, we have:
P20 = , and under the original model p+21would be = . However, using our harvesting
strategy, P21 =
1
Froelich, Crisler, and Fisher, Discrete Mathematics for High School Students, Freeman
Publishing Co., 1994
2
Lowther, Martha, The Tatnall School, paper presented at the University of Delaware.
Armontrout:
Exeter ’95
Recursion of Matrices: Leslie Model 4