Consider a lake fish population that follows the logistic growth model with constant harvest:
$$\frac{dN}{dt}= r N\left(1-\frac{N}{K}\right)-H,$$
where $r = 0.4\ \text{yr}^{-1}$, $K = 20\,000$ fish and the initial stock is $N_0 = 8\,000$ fish.

(a) Determine the maximum constant harvest $H_{\max}$ (in fish per year) that allows the population to persist.

(b) Find the constant harvest level $H$ (as a percentage of $H_{\max}$) that gives two positive equilibrium stock sizes, one at 8 000 fish and the other at 12 000 fish.

(c) State which of these equilibria is stable and which is unstable.

Question

Consider a lake fish population that follows the logistic growth model with constant harvest:
$$\frac{dN}{dt}= r N\left(1-\frac{N}{K}\right)-H,$$
where $r = 0.4\ \text{yr}^{-1}$, $K = 20\,000$ fish and the initial stock is $N_0 = 8\,000$ fish.

(a) Determine the maximum constant harvest $H_{\max}$ (in fish per year) that allows the population to persist.

(b) Find the constant harvest level $H$ (as a percentage of $H_{\max}$) that gives two positive equilibrium stock sizes, one at 8 000 fish and the other at 12 000 fish.

(c) State which of these equilibria is stable and which is unstable.

Accepted Answer

H_max = 2,000 fish/yr; H = 1,920 fish/yr (96 % of H_max); 8,000‑fish equilibrium unstable, 12,000‑fish equilibrium stable

Neco Model Questions Vol1 2024 Agricultural Science Question 21

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