Growth rate functions
a. Show that the logistic growth rate function f(P)=rP(1−P/K) has a maximum value of rK/4 at the point P=K/2.

Verified step by step guidance

1. Start with the logistic growth rate function given by \(f(P) = rP\left(1 - \frac{P}{K}\right)\), where \(r\) and \(K\) are constants, and \(P\) is the variable representing population.
2. Rewrite the function in a simplified form to make differentiation easier: \(f(P) = rP - \frac{rP^2}{K}\).
3. Find the critical points by taking the derivative of \(f(P)\) with respect to \(P\) and setting it equal to zero. Compute \(f'(P) = r - \frac{2rP}{K}\) and solve \(f'(P) = 0\) for \(P\).
4. Solve the equation \(r - \frac{2rP}{K} = 0\) to find the critical point \(P = \frac{K}{2}\).
5. Verify that this critical point corresponds to a maximum by checking the second derivative \(f''(P) = -\frac{2r}{K}\), which is negative, confirming a maximum at \(P = \frac{K}{2}\). Then, substitute \(P = \frac{K}{2}\) back into the original function to find the maximum value \(f\left(\frac{K}{2}\right) = \frac{rK}{4}\).

Key Concepts

Logistic Growth Function

The logistic growth function models population growth with a carrying capacity, expressed as f(P) = rP(1 - P/K), where r is the intrinsic growth rate, P is the population size, and K is the carrying capacity. It describes how growth slows as the population approaches K.

Finding Maximum Values Using Derivatives

To find the maximum value of a function, take its derivative, set it equal to zero, and solve for critical points. Then, use the second derivative test or analyze the function behavior to confirm if the critical point is a maximum.

Substitution and Evaluation of Functions

After finding the critical point, substitute it back into the original function to evaluate the maximum value. This step confirms the exact output of the function at the point where the growth rate is maximized.