Leslie Growth
Models
Part 1: Age-distributed
growth
In this module we study
the Leslie growth model, developed in the 1940's, a model for growth
of a population that is naturally segmented into age classes. The relevant
data for each age class are the reproduction rate and the rate of survival
into the next age class. This model is widely used for management of forests,
fisheries, and animal herds.
Here is a specific example,
which we will use throughout the module. The following table lists reproduction
and survivor rates for the female population of a certain species of domestic
sheep in New Zealand -- where sheep farming is a major segment of the economy.
(For animal populations, it is conventional to consider only the females,
since only they reproduce, and they are usually a fixed percentage of the
total population.) Sheep give birth only once a year, which dictates a
natural time step of one year. In the species under consideration, sheep
seldom if ever live longer than 12 years, which gives a natural stopping
point for the age classes.
Birth and Survival Rates for Female New Zealand Sheep
[from G. Caughley, "Parameters for Seasonally Breeding
Populations," Ecology 48(1967)834-839]
Age (years) |
Birth Rate |
Survival Rate |
0-1 |
0.000 |
0.845 |
1-2 |
0.045 |
0.975 |
2-3 |
0.391 |
0.965 |
3-4 |
0.472 |
0.950 |
4-5 |
0.484 |
0.926 |
5-6 |
0.546 |
0.895 |
6-7 |
0.543 |
0.850 |
7-8 |
0.502 |
0.786 |
8-9 |
0.468 |
0.691 |
9-10 |
0.459 |
0.561 |
10-11 |
0.433 |
0.370 |
11-12 |
0.421 |
0.000 |
In any given year, a particular
population (e.g., a single herd) can be represented by a state vector
x = (x_{1}, x_{2}, ..., x_{11}, x_{12})^{T},
where x_{i} represents the number of female animals in the i-th age class.
If absolute numbers are not known, a state may be represented equally well
by a vector of fractions of the population in each age class, i.e.,
by a vector whose entries sum to 1. Another possibility is to set an arbitrary
number for some particular age class and represent the other classes relative
to the selected one. Thus, a state vector is determined only up to a scalar
multiple. In particular, any vector with nonnegative entries can be a state
vector.
The Leslie growth matrix
for the population is the transition matrix L from the state in one
year to the state in the next year. Thus, if x is the state vector
in a given year, the state vector after one year's growth is Lx
and the growth in that year (distributed in age classes) is Lx - x.
In one year's time, only
two types of transitions are possible:
- A sheep gives birth, adding
one to the youngest age class. Thus, the rates in the birth column are
entries of the first row of L.
- A sheep survives the year
to enter the next age class. Thus, the survival rates are entries of the
form L_{j+1,j}, that is, entries just below the main diagonal of
L.
All the other entries of
L are 0.
- Enter a symbolic Leslie
matrix L. The birth rates are entered in this matrix as a_{1} through
a_{12} and the survival rates as b_{1} through b_{11}.
- Now enter the command to
construct the characteristic polynomial. The highest order coefficient
is 1. Describe the coefficient of each of the other powers of lambda.
- The next set of commands
constructs an auxiliary function q. Explain why p(lambda) = 0 if and only
if q(lambda) = 1.
- Observe the structure of
q: in each term, the numerator is a nonnegative constant, and the denominator
is a positive power of lambda. Explain why, for positive values of lambda,
q must be a strictly decreasing function. Then explain why L must have
exactly one positive eigenvalue.
- Next, we assign the birth
and survival rates from the table above as the values of a_{1}
through a_{12} and b_{1} through b_{11}, and we
plot both q and p for this specific case. Explain how each of these plots
confirms that the specific Leslie matrix L has exactly one positive eigenvalue.
Estimate this eigenvalue. What is its multiplicity?
- Now use a command to compute
eigenvalues of L directly. Explain what you see as output. In particular,
find the unique positive eigenvalue.
- If x is a state
vector and also an eigenvector for the unique positive eigenvalue, what
is the percentage growth in the total population over the next year? How
does the distribution into age classes change over that year?
- If your computer algebra
system is capable of finding an eigenvector for the unique positive eigenvalue,
find one. Modify your eigenvector (if necessary) to make it a state vector,
and describe what it tells you about a distribution into age classes. (Hint: Any multiple of an eigenvector is also an eigenvector. Your state vector should have entries which sum to 1.) [If you cannot easily find an eigenvector,
go on to the next section, where we will find one another way.]
- If
the state vector you found in Step 8 is the age distribution in a given year, what will the distribution
be in one year? in ten years?
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