A herd of American buffalo (bison) can be modeled by a stage matrix similar to that for the spotted owls. The females can be divided into calves (up to 1 year old), yearlings (1 to 2 years), and adults. Suppose an average of 42 female calves are born each year per 100 adult females. (Only adults produce offspring.) Each year, about 60% of the calves survive, 75% of the yearlings survive, and 95% of the adults survive. For k > 0, let xk = (ck, yk, ak), where the entries in xk are the numbers of females in each life stage at year k.
a. Construct the stage-matrix A for the buffalo herd, such that xk+1 = Axk for k > 0.
b. Show that the buffalo herd is growing, determine the expected growth rate after many years, and give the expected numbers of calves and yearlings present per 100 adults.