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 k > 0, let xk = (jk, ak), 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 xk+1 = Axk for k > 0.
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. 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).