Question

Exercise 4: Assume that a population is governed by a logistic equation with carrying capacity K intrinsic growth rate r, and initial population size K is subjected to constant effort harvesting: (a) Determine the population size, N(t) (b) Verify that if E< r, the population size will approach the positive steady state, Ni, the carrying capacity K if Erand if E>r, the population will approach the zero steady state, No, astoo. (c) Find the maximum sustainable yield of the population.

Answer 1

dN/dt = r N (1 - N/k) - E N dN/ r N(1 - N/k) - E N = dt 1/(r.E) (r.E)/N (r - E - r/k N) dn = dt, 1/(r - E) [(r - E - rN/k) + rN/k /N (r.E - r N/k)] dN = dt 1/(r - E) [1/N + r/k (r - E - r N/k] dN = dt integrating both sides [ln N + ln (r - E - rN/k/(- rk)] + ln A = t (r - E) ln N - ln (r- E - rN/k) + ln A = t (r . E) N(0) = k ln AN/(r - E - rN/k) = t (r - E) = 0 since AK/(r - E - r K/K) = e degree = 1 AK = -E A = -E/K ln N - ln (r - E r N/k) + ln (- E/K) = t (r - E) ln (- NE/k/r - E - rN/k = t (r - E) \r\n - NE/k = [(r - E) - r N/k] e N/K (r -