In the logistic model, the population grows fastest when P equals which value?

In the logistic model, the population grows fastest when P equals which value?

• A. L/2
• B. L
• C. 0
• D. L/4

Answer: (A)

In the logistic model the growth rate is dP/dt = rP(1 - P/L), which is a downward-opening quadratic in P. The maximum occurs at the vertex, found by setting the derivative with respect to P to zero: 1 - 2P/L = 0, so P = L/2. This is the point where resources are still available but crowding starts to limit growth, giving the fastest increase. For example, at P = L/2 you get dP/dt = r(L/2)(1/2) = rL/4, which is larger than the rate at P = L/4 or P = L (the latter being zero).

In the logistic model the growth rate is dP/dt = rP(1 - P/L), which is a downward-opening quadratic in P. The maximum occurs at the vertex, found by setting the derivative with respect to P to zero: 1 - 2P/L = 0, so P = L/2. This is the point where resources are still available but crowding starts to limit growth, giving the fastest increase. For example, at P = L/2 you get dP/dt = r(L/2)(1/2) = rL/4, which is larger than the rate at P = L/4 or P = L (the latter being zero).