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If a fishery develops on a stock that is previously unfished, we may assume that the initial biomass of the stock is N(0) = K. A sustainable steady state harvest that maintains the population size at Ns will remove all of the biological production, so that if h is the harvest, we know h = rN^ 1 - Nf) (6-2°)

(a) Show that, in general, solving Eq. (6.20) for Ns leads to two steady states, one of which is dynamically unstable; to do this, it may be helpful to analyze the dynamical system N(t + 1) = N(t) + rN(t) [1 -(N(t)/K)] — h graphically.

(b) Now envision that the development of the fishery consists of two components. First, there is a ''bonus harvest'' in which the stock is harvested from K to Ns, which for simplicity we assume takes place in the first year. Second, there is the sustainable harvest in each subsequent year, given by Eq. (6.20). The harvest in year t after the bonus harvest is discounted by the factor 1/(1 + 8)'. (This is the common representation of discounting in discrete time models. To connect it with what we have done before, note that (1 + fi)-' = e-'lo8(1+fi) « e-fit when fi is small.) Combining these, the present value PV(Ns) of choosing the value Ns for the steady state population size is

Now we can factor h out of the summation and then you should verify that

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