Let y₀ be the initial size of an insect population, and let y_n be its size after n years. The growth rate R is the relative population increase per year given by R = (y^_n+1 y_n )/y_n. To limit the exponential growth in the model above, P.F. Verhulst stated in 1854 that the rate R must vary with the population size. Arguing that a given generation can only sustain a certain size N , he postulated that the growth rate R must be proportional to N y_n, i.e., there exists a constant K > 0 such that R = K(N y_n). Set r = 1 + KN and x_n = (K/r)y_{n .}

_* What is the lower bound for r, and what are the upper bound and lower bounds for x_n? Again, Let fr (x) = rx(1 x).

x_n+1 = fr (x_n) = rx_n(1 x_n) ---------- (5)

In order for the function fr to describe the population growth process (5), one has to restrict its domain and its range. What is a reasonable domain of fr in equation (5)? What is a reasonable range of fr in equation (5)? For what values of r does fr have this range? Graph the restriction of fr . (Please use Matlab)