Question

COMPUTING LESLIE MATRIX

Example After one year, we have only 250 fishes left. And then 125 have reached their reproduction rate. If we set f₃ = 8, then we are back to n = (1000, 0, 0): We see that n1 = (0, 250, 0), n2 = (0, 0, 125), n3 = (1000, 0, 0)

Exercise Write down the Leslie matrix for the previous example and calculate for various choices of n the population vectors ni. What do you observe?

Exercise Show that you can find some n such that n⁺ = Ln = n. If n = (a, b, c) then

n⁺ = (8c, 0. 25a, 0. 5b). Then (a, b, c) = (8c, 0. 25a, 0. 5b) determines a unique stable distribution n amongst the age groups. n itself is unique up to a factor.

Exercise Now change f₃ = 8 to numbers smaller as well as larger than 8, say 6 and 10. Then calculate again for various choices of n the population vectors ni. Can you still find some n such that n⁺ = n?

Answer 1

Consider the Leslie matrix is given by

In this example, . and

Now consider, the initial population is out of which 250 is left after one year and only 125 reproduce in the third year.

Comparing the populations in second year with the initial population of 1000 we have the population in the second year is of the population of the first year.

Thus, .

Similarly, for the third year,

The population in the third year is of the population of the second year.

Thus,

Substitute the calculated values in

for example,

First take

Thus, in the second year the population left is and in the third year the population is and again in the next year is .

So, , and  .

Secondly take

Thus, in the second year the population left is and in the third year the population is and again in the next year is .

So, , and .