4.6.5 Comments and exercises

Exercises

1. Prove that for a Leslie matrix, there is a unique positive dominant eigenvalue and that the corresponding eigenvector has components that can all be taken to be non-negative.
2. Show that the natural rate of increase r of the population of Blue Whales is given by where is the eigenvalue of greatest absolute size. State the value which must exceed if the population is to increase. State the value of which implies a stable population.
3. Show that the population of Blue Whales is, in general, increasing from N to lN over one perioo of time and that as a consequence the harvest H that can be taken from the population is given by



where H is a percentage of the total population.

Note that the above results will apply to any Leslie matrix model of a population.

Problem 1

Assume that the initial population is p₀ and that the population after n time periods is p_n. We know that the population after one year is Mp₀ where M is our Leslie matrix. In general we have:

p₁ = Mp₀

p₂ = Mp₁ = M²p₀

.

.

.

p_n = Mp_n-1 = Mⁿp₀

But

where x_i are the eigenvectors of M.

HHnce

where are the eigenvalues of M.

But for every corresponding eigenvalue and eigenvector x₀ we have

so that

and in general and so p_n may be written as

One eigenvalue, say is dominant so that we may factor from each term and write

Hence, as

The limiting population is given by taking the limit as so that all the terms except the first approach zero.

We can now write

so that we have

This shows that we are multiplying our initial population by a constant factor ( ) and so the population increase is exponential with a natural rate of increase r given by .

Hence

The rate of increase of the population will be positive if .

The population will remain static if .

Problem 2

Assuming an appropriate value of , the population will increase from N to , this is clear from Problem 1.

Hence the harvesting is given by the expressions

Problem 3

The exact results depend on your harvesting strategies - there is no simple correct answer to modelling problems which require you to decide on a strategy and implement it!