Problem 17E

Construct a stage-matrix model for an animal species that has two life stages: juvenile (up to 1 year old) and adult. Suppose the female adults give birth each year to an average of 1.6 female juveniles. Each year, 30% of the juveniles survive to become adults and 80% of the adults survive. For , where the entries in xk are the numbers of female juveniles and female adults in year k.

a. Construct the stage-matrix A such that for .

b. Show that the population is growing, compute the eventual growth rate of the population, and give the eventual ratio of juveniles to adults.

c. [M] Suppose that initially there are 15 juveniles and 10 adults in the population. Produce four graphs that show how the population changes over eight years: (a) the number of juveniles, (b) the number of adults, (c) the total population, and (d) the ratio of juveniles to adults (each year). When does the ratio in (d) seem to stabilize? Include a listing of the program or keystrokes used to produce the graphs for (c) and (d).

Solution 17E

Step 1 of 9

(a)

The animal species that has two life stages, named as Juvenile and adult.

Let be the number of juvenile and adult at the year .

Since each adults give birth each year to an average of 1.6 female juveniles, that is the number of juveniles count for the next year is equal to the 1.6th of adults count for the present year.

That is,

.

In the Juveniles, each year 30% will survive and become adults and only 80% of the adults will survive, that is the sum of 0.3rd of juvenile count, and 0.8th of adults count for the present year is equal to the adults count in the next year.

That is,