4.6.4 Application of the Leslie model in the case of blue whales

The following information has been obtained from studies of the blue whale and enables us to build a Leslie model of the population.

  1. Blue whale females reach maturity at between four and seven years of age;
  2. the gestation period for a pregnant female is about one year;
  3. a single calf is born and requires nursing by the mother for about seven months;
  4. nursing mothers do not become pregnant;
  5. observations show that in general not more than one calf is born to a female every two years;
  6. the observed male:female ratio in actual catches of blue whales is roughly 1:1;
  7. the rate of breeding for older whales is slightly less than that of whales in their prime;
  8. natural mortality appears to account for the deaths of about 13% of the whales in any given group except the final group where the mortality must be 100%;
  9. the maximum age of blue whales is about 40 years.

The following table gives the estimated fertility and mortality rates for the age distribution of the animal.

0.13

0.45

Age of Whale

Fertility Rate
calves/female born in
a two year period

Mortality Rate
Percentage of deaths
in each age group

0-

0.00

0.13

2-

0.00

0.13

4-

0.19

0.13

6-

0.44

0.13

8-

0.50

10-

0.50

0.13

12-

0.45

0.13

14-

0.45

0.13

16-

0.45

0.13

18-

0.45

0.13

20-

0.45

0.13

22-

0.45

0.13

24-

0.45

0.13

26-

0.45

0.13

28-

0.45

0.13

30-

0.45

0.13

32-

0.13

34-

0.45

0.13

36-

0.45

0.13

38-

0.00

1.00

The information in the table gives rise to the following Leslie matrix M of order 20.

Using the Maple commands

will give the Eigenvalues and Eigenvectors of the matrix M. The eigenvalues are shown and the command • x:= col(vecs,n); gives the eigenvector corresponding to the nth eigenvalue.

In our case M is the above Leslie matrix.

Running the command gives the results shown below:

Eigenvalue = 1.112535559

The corresponding eigenvector x ={-0.6233088368, . . . . . . . -0.007454178041, 0}

The corresponding population, normalised to give 1000 members in the first class is
{1000, 782, 612, 478, 374, 292, 229, 179, 140, 109, 86, 67 ,52, 41, 32, 25, 20, 15, 12, 0}