Suppose that we have a population that, in year n, has xn population of juveniles and yn population of adult females.

The units of xn and yn could be, for example, in millions of individuals (which would be typical for insect populations).

Also suppose that in year n + 1, the number of juveniles is αyn Exercise 11.15 is a simple example of how a Leslie matrix can be used to model a population. Leslie matrices are named after Patrick Leslie, an ecological statistician who published a famous paper in 1945 on the use of matrices in population models. This approach has since been widely used in the study of population dynamics. Typically, because males cannot give birth, the models only consider females. In the next few questions we’ll see slightly more complex examples of the same approach.

(i.e., each adult female produces α juveniles in a year) while the number of adult females is βxn (i.e., a fraction of juveniles survive a year and mature to become females, while all the females from the previous year have died. This means that each insect can live only two years, which isn’t unreasonable).

This gives the system xn+1 = αyn, yn+1 = βxn.

Just for convenience we’ll write 

xn yn



= pn.

a. Write this system as a matrix equation pn+1 = Apn.

b. If the population starts at p0 =



1 0



calculate p1, p2 and p4. What is p2n?

c. Find a formula for pn in terms of A and p0.

d. If the population reaches a steady state when it doesn’t change from year to year, then we must have pn+1 = pn.

Call this steady state p∞. What is the equation you need to solve to find p∞?

e. What conditions on α and β are required for a non-zero p∞ to exist?

f. Calculate p∞ as a function of α and β.