### 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.

 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 = (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. 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 Lj+1,j, that is, entries just below the main diagonal of L.

All the other entries of L are 0.

1. Enter a symbolic Leslie matrix L. The birth rates are entered in this matrix as a1 through a12 and the survival rates as b1 through b11.
2. 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.
3. The next set of commands constructs an auxiliary function q. Explain why p(lambda) = 0 if and only if q(lambda) = 1.
4. 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.
5. Next, we assign the birth and survival rates from the table above as the values of a1 through a12 and b1 through b11, 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?
6. Now use a command to compute eigenvalues of L directly. Explain what you see as output. In particular, find the unique positive eigenvalue.
7. 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?
8. 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.]
9. 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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