(Please make the correct selection to fill in the blank)

For steady states where the stable level of the resource stock is ____________ than X_MSY, the same steady-state yield can be achieved using a lower level of effort.

Question

(Please make the correct selection to fill in the blank)

For steady states where the stable level of the resource stock is ____________ than X_MSY, the same steady-state yield can be achieved using a lower level of effort.

BlackTom AI · Accepted Answer

Using the logistic growth model G(X) = r X (1 − X/K), the steady-state condition equates growth to harvest: q E X = G(X). Solve for effort E as a function of the steady-state stock X to see how effort changes with X. 1) Relevant concepts: - Sustainable (steady-state) harvest: harvest = growth. - Logistic growth: G(X) = rX(1 − X/K), with MSY at X_MSY = K/2. - Harvest function: H = q E X (catchability q, effort E, stock X). - For a given steady-state X, the required effort E satisfies q E X = G(X). 2) Algebraic steps (complete reasoning): q E X = r X (1 − X/K) Divide both sides by q X (X > 0): E = [r (1 − X/K)] / q 3) Intermediate/calculus step: Differentiate E with respect to X: dE/dX = [r (−1/K)] / q = − r /(q K) < 0 So E is a strictly decreasing linear function of X. 4) Why this implies the correct answer: Since E decreases as X increases, any steady-state with stock X greater than X_MSY (K/2) requires a smaller effort than the effort required at X_MSY to produce its corresponding steady-state yield. In particular, E(X_MSY) = r(1 − 1/2)/q = r/(2q); for X > X_MSY, E(X) < r/(2q). Therefore the same steady-state yield can be achieved with a lower effort when the equilibrium stock exceeds X_MSY. 5) Why the other option is incorrect: Choosing “less than” would imply that higher stock requires higher effort to achieve the same yield, contradicting the derived relation E = r(1 − X/K)/q and its negative slope; mathematically dE/dX < 0 rules that out. Conclusion: The correct completion is “greater than,” which matches the provided answer.

(Please make the correct selection to fill in the blank) For steady states where the stable level of the resource stock is ____________ than X_MSY, the same steady-state yield can be achieved using a lower level of effort.单项选择题

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