Discussion
The population (N) of an ecosystem obeys the growth equation \(\frac{dN}{dt} = \alpha N - \beta N^{2}\) where \(\alpha\) and \(\beta\) are constants. The maximum sustainable yield of this ecosystem will be
Answer: C
The keyed answer is alpha divided by 2 beta. The growth equation dN/dt = alpha N - beta N^2 is the logistic model with carrying capacity K = alpha/beta. The growth rate dN/dt is greatest when N = K/2, which is alpha/(2 beta), and harvesting at that population gives the maximum sustainable yield. So the keyed option is the population size at which the maximum yield is obtained. The yield itself would be alpha^2/(4 beta), which is not listed, so the question is loosely worded. The expression alpha - beta N is the per capita growth rate, alpha/beta is the carrying capacity, and alpha^2/beta is not a meaningful quantity here.
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