Harvesting an Age-Distributed Population
Part 1: Sustainable harvesting
Recall that in the Leslie
Growth Models module, we saw that in any particular year, a
population (e.g., a single herd of New Zealand sheep) can be
represented by a state vector
x = (x1, x2, ..., x11, x12)T,
where xi 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. 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. The entries
a1,..., a12 of the first row
of L represent the rates of birth for each age class, while the
subdiagonal entries b1,...,b11 represent the survival
rates for each class. 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 the Leslie Growth Models
module, we saw that
a New Zealand sheep population will increase by about 17.6% per year --
and approach a stable age distribution -- if left alone to do nothing but
reproduce (and perhaps get sheared once in a while). However, New Zealand's
sheep farmers cannot live entirely on their income from wool, especially
if they have to keep feeding ever more sheep. A desirable goal for management
of a sheep herd (or any renewable resource) is to find a stable configuration
from which one can harvest the growth at regular intervals -- thereby
producing income and returning the population to its previous configuration.
A sustainable harvesting
policy is a plan for harvesting on a regular schedule in such a way
that the harvest is always the same and the state of the population after
harvesting is always the same.
Suppose we let hi
be the fraction of the i-th age group that will be harvested at the end
of each growth period, and we let H be the diagonal matrix whose entries
are the hi's. If we start a growth period with age-distribution
state x, then the state after growth will be Lx. The harvest after growth
will be HLx, and that will reduce the population to Lx - HLx, or (I - H)Lx.
To be sustainable, the population state after harvest must match the starting
state, i.e., (I - H)Lx = x. That is, x must be an eigenvector for eigenvalue
1 for the matrix (I - H)L. In this part of the module, we explore some
of the implications of this observation.
- Enter the Leslie matrix L in
symbolic form (i.e., without specific numbers assigned to the a's and b's),
and compute (I - H)L. You should find that (I - H)L is another Leslie matrix.
It differs from L in that the i-th row of L has been mulitplied by 1 - hi.
In the previous module, we defined the characteristic polynomial p of the Leslie matrix L, and an auxillary function q (which depended on p). Refresh your memory of these equations by entering in your worksheet the commands which define p and q.
Recall that lambda is an eigenvalue for L if and only if p(lambda) = 0, and p(lambda) = 0 if and only if q(lambda) = 1. Recall that the dominant eigenvalue lambda1
of a Leslie matrix is the unique positive solution of q(lambda) = 1.
To get zero population
growth, we must have q(1)=1, so that the largest eigenvalue (and only positive
one) turns out to be 1. Substitute lambda
= 1 in the definition for q to get an explicit condition on birth and survival
rates for having zero growth. You don't have to enter this in your worksheet
-- just write enough of the condition on paper to see how it goes. Now
think about what that condition would look like for the Leslie matrix (I
- H)L, in which the i-th row of L has been multiplied by 1 - hi.
The result is a very complicated condition -- but a single equation in
12 unknowns (the h's) that can be satisfied in many different ways. Thus,
there are infinitely many ways to construct a sustainable harvesting policy.
In the rest of the module, we consider three of those ways, one in this
part and the others in subsequent parts.
Before we continue, we need to recompute the dominant eigenvalue lambda1 for the Leslie matrix associated to our New Zealand sheep population. The commands which assign the birth and survival rates to the entries in our symbolic matrix, as well as the command for finding the eigenvalues of the matrix, are in your worksheet. Evaluate them and specify lambda1.
A uniform harvesting
policy is one in which the same fraction h is harvested from each age group.
In this case, we must have (1 - h)Lx = x. Explain why this
means that h must satisfy lambda1
= 1/(1-h), so h = 1 - 1/lambda1.
Use this observation to find a uniform fraction of the New Zealand sheep
population that can be harvested every year and leave the population distribution
the same at the start of each year.
Explain why the harvest rate is not the same as the growth rate of approximately 17.6%.
modules at math.duke.edu
