A discrete dynamical system model for the population of cheetahs and gazelles in Namibia is given by the following pair of equations:
.3Ck + .4 Gk = Ck+1
-pCk + 1.3 Gk = Gk+1
where Ck measures the number of cheetahs present in a certain Namibian game reserve at time K, Gk gives the number of gazelles (measured in tens), and k is measured in months.
a) Find a value for p that guarantees a steady- state outcome for this model, then determine the number of cheetahs present for every 1000 gazelles (in the long run).
b) find a value for p that will guarantee 2% growth in both populations. In the long-run, what is the ratio of gazelles to cheetahs?
c) What must be true about the eigenvalues of the transition matrix if both populations die out in the long- run?