A fish population follows logistic growth $ \frac{dN}{dt}= r N\left(1-\frac{N}{K}\right) $ with intrinsic rate $r = 0.5\ \text{yr}^{-1}$ and carrying capacity $K = 10\,000$. A constant harvest $H$ (fish yr$^{-1}$) is applied.

(i) Determine the maximum constant harvest $H_{\max}$ that still permits a non‑zero equilibrium, and give the corresponding equilibrium population size.

(ii) If the harvest is set at 80 % of this maximum, calculate the stable equilibrium population size.

Question

A fish population follows logistic growth $ \frac{dN}{dt}= r N\left(1-\frac{N}{K}\right) $ with intrinsic rate $r = 0.5\ \text{yr}^{-1}$ and carrying capacity $K = 10\,000$. A constant harvest $H$ (fish yr$^{-1}$) is applied.

(i) Determine the maximum constant harvest $H_{\max}$ that still permits a non‑zero equilibrium, and give the corresponding equilibrium population size.

(ii) If the harvest is set at 80 % of this maximum, calculate the stable equilibrium population size.

Accepted Answer

H_max = 1250 fish/yr; equilibrium at H_max = 5000 fish; with 0.8 H_max, stable equilibrium ≈ 7236 fish

Gce Model Questions Vol1 2023 Biology Question 32

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