A fish population in a lake follows logistic growth described by $\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)$. The intrinsic growth rate is $r = 0.4\ \text{month}^{-1}$ and the carrying capacity is $K = 10\,000$ fish.

(a) Determine the maximum sustainable harvest rate $H_{\max}$ (in fish per month) that can be removed constantly while keeping the population at equilibrium, and state the corresponding equilibrium population size.

(b) If a drought reduces the carrying capacity to $K' = 8\,000$ fish while $r$ remains unchanged, calculate the new maximum sustainable harvest $H'_{\max}$ and the percentage change in the sustainable harvest compared with the original value.

Question

A fish population in a lake follows logistic growth described by $\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)$. The intrinsic growth rate is $r = 0.4\ \text{month}^{-1}$ and the carrying capacity is $K = 10\,000$ fish.

(a) Determine the maximum sustainable harvest rate $H_{\max}$ (in fish per month) that can be removed constantly while keeping the population at equilibrium, and state the corresponding equilibrium population size.

(b) If a drought reduces the carrying capacity to $K' = 8\,000$ fish while $r$ remains unchanged, calculate the new maximum sustainable harvest $H'_{\max}$ and the percentage change in the sustainable harvest compared with the original value.

Accepted Answer

(a) $H_{\max}=1000$ fish/month, equilibrium $N=5000$ fish; (b) $H'_{\max}=800$ fish/month, a 20 % decrease

Igcse Model Questions Vol1 2024 Biology Question 23

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