Which of the following is true?

(You may assume we are talking about strictly "interior" steady states, where both X0 and Y0)

Question

Which of the following is true?

(You may assume we are talking about strictly "interior" steady states, where both X>0 and Y>0)

BlackTom AI · Accepted Answer

At an interior steady state the biological growth equals harvest, so f(X*) = q E X* (harvest = catchability q × effort E × stock X*), where f(X) is the resource growth function and Y* = q E X* is steady-state yield. 1) Relevant concepts - Steady state condition: f(X*) = q E X*. - Implicit differentiation to see how X* responds to E. - Stability/sign of the derivative: for an interior, stable steady state we have ∂(f(X) − q E X)/∂X = f'(X*) − q E < 0. 2) Step-by-step reasoning (implicit differentiation) Start from f(X*) − q E X* = 0. Differentiate both sides w.r.t. E: f'(X*) (dX*/dE) − q X* − q E (dX*/dE) = 0. Collect terms in dX*/dE: (f'(X*) − q E) dX*/dE = q X*. Thus dX*/dE = q X* / (f'(X*) − q E). 3) Intermediate steps and sign - Numerator q X* > 0 because q>0 and X*>0. - For a stable interior steady state, f'(X*) − q E < 0, so the denominator is negative. Therefore dX*/dE < 0: increasing effort necessarily reduces the steady-state stock X*. 4) Why the correct answer is correct The algebraic result shows a sign-definite negative derivative under the usual stability condition, so higher effort must decrease the steady-state resource stock. 5) Why the other option is incorrect Higher effort does not necessarily increase steady-state yield Y*. Differentiate Y* = q E X*: dY*/dE = q X* + q E dX*/dE = q X* [ f'(X*)/(f'(X*) − q E) ]. Because f'(X*) − qE < 0, the sign of dY*/dE depends on f'(X*) and can be positive or negative. Concrete example: logistic growth f(X)=rX(1−X/K) gives X* = K(1 − qE/r) (decreasing in E) and Y* = q E K(1 − qE/r), which increases with E up to E = r/(2q) and then falls. So yield can rise or fall with E. 6) Conclusion Therefore the correct statement is: Higher effort necessarily decreases the steady state level of the resource stock (matches the provided answer).

Which of the following is true? (You may assume we are talking about strictly "interior" steady states, where both X>0 and Y>0)Single choice

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