### 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.

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 `

0.13

`
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-

0.45

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

- with(linalg):
- M:=array([[],[],....[]]);
- sys:=evalf(Eigenvals(M,vecs));
- x:=col(vecs,n);
- x:=evalm(x/x[n]*1000);

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}