In this problem we assume that fish are caught at a

Chapter 2, Problem 21

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In this problem we assume that fish are caught at a constant rate h independent of the sizeof the fish population. Then y satisfiesdy/dt = r(1 y/K)y h. (i)The assumption of a constant catch rate h may be reasonable when y is large but becomesless so when y is small.(a) If h < rK/4, show that Eq. (i) has two equilibrium points y1 and y2 with y1 < y2;determine these points.(b) Show that y1 is unstable and y2 is asymptotically stable.(c) From a plot of f(y) versus y, show that if the initial population y0 > y1, then y y2as t , but that if y0 < y1, then y decreases as t increases. Note that y = 0 is not anequilibrium point, so if y0 < y1, then extinction will be reached in a finite time.(d) If h > rK/4, show that y decreases to zero as t increases, regardless of the value of y0.(e) If h = rK/4, show that there is a single equilibrium point y = K/2 and that this pointis semistable (see 7). Thus the maximum sustainable yield is hm = rK/4, correspondingto the equilibrium value y = K/2. Observe that hm has the same value as Ymin 20(d). The fishery is considered to be overexploited if y is reduced to a levelbelow K/2.Epidemics. The use of mathematical methods to study the spread of contagious diseasesgoes back at least to some work by Daniel Bernoulli in 1760 on smallpox. In more recent yearsmany mathematical models have been proposed and studied for many different diseases.16 22 through 24 deal with a few of the simpler models and the conclusions that can bedrawn from them. Similar models have also been used to describe the spread of rumors andof consumer products.