A fish population in a lake follows the logistic growth model $ \frac{dP}{dt}= rP\left(1-\frac{P}{K}\right) $ with intrinsic rate $ r = 0.5\ \text{yr}^{-1} $ and carrying capacity $ K = 10\,000 $ fish. Initially ($ t=0 $) the population size is $ P_0 = 2000 $ fish.

(a) Using the logistic solution, calculate the population size after 3 years (before any harvesting).

(b) After 3 years, a constant harvest of $ H $ fish per year is introduced, modifying the model to $ \frac{dP}{dt}= rP\left(1-\frac{P}{K}\right)-H $. Determine the maximum constant harvest $ H_{\max} $ (in fish per year) that can be sustained indefinitely without driving the population to extinction. Show all steps.

Question

A fish population in a lake follows the logistic growth model $ \frac{dP}{dt}= rP\left(1-\frac{P}{K}\right) $ with intrinsic rate $ r = 0.5\ \text{yr}^{-1} $ and carrying capacity $ K = 10\,000 $ fish. Initially ($ t=0 $) the population size is $ P_0 = 2000 $ fish.

(a) Using the logistic solution, calculate the population size after 3 years (before any harvesting).

(b) After 3 years, a constant harvest of $ H $ fish per year is introduced, modifying the model to $ \frac{dP}{dt}= rP\left(1-\frac{P}{K}\right)-H $. Determine the maximum constant harvest $ H_{\max} $ (in fish per year) that can be sustained indefinitely without driving the population to extinction. Show all steps.

Accepted Answer

P(3) ≈ 5 286 fish; H_max = 1 250 fish yr⁻¹.

Neco Model Questions Vol1 2017 Biology Question 48

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