Question
Let y 0 be the initial size of an insect population, and let y n be its size after n years. The growth rate R
Let y0 be the initial size of an insect population, and let yn be its size after n years. The growth rate R is the relative population increase per year given by R = (yn+1 yn )/yn. 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 yn, i.e., there exists a constant K > 0 such that R = K(N yn). Set r = 1 + KN and xn = (K/r)yn .
* What is the lower bound for r, and what are the upper bound and lower bounds for xn? Again, Let fr (x) = rx(1 x).
xn+1 = fr (xn) = rxn(1 xn) ---------- (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)
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