Logistic growth The population of a rabbit community is governed by the initial value problem
P′(t) = 0.2 P (1 − P/1200), P(0) = 50
d. What is the population when the growth rate is a maximum?

Verified step by step guidance

1. Recognize that the given differential equation models logistic growth: \(P'(t) = 0.2 P \left(1 - \frac{P}{1200}\right)\) with initial condition \(P(0) = 50\).
2. Recall that the growth rate \(P'(t)\) reaches its maximum when the derivative of \(P'(t)\) with respect to \(P\) is zero, because \(P'(t)\) depends on \(P\) directly.
3. Express the growth rate function as \(f(P) = 0.2 P \left(1 - \frac{P}{1200}\right)\) and find its critical points by differentiating with respect to \(P\): compute \(f'(P)\).
4. Set \(f'(P) = 0\) and solve for \(P\) to find the population values where the growth rate could be maximum or minimum.
5. Determine which critical point corresponds to the maximum growth rate by analyzing the sign of \(f'(P)\) around the critical points or using the second derivative test.

Key Concepts

Logistic Growth Model

The logistic growth model describes population growth that starts exponentially but slows as the population approaches a carrying capacity. It is represented by the differential equation P'(t) = rP(1 - P/K), where r is the growth rate and K is the carrying capacity. This model captures limited resources affecting growth.

Growth Rate and Its Maximum

The growth rate P'(t) represents how fast the population changes over time. To find when this rate is maximum, we analyze P'(t) as a function of P and determine the population size that maximizes it, often by setting the derivative of P'(t) with respect to P to zero.

Initial Value Problem and Solution Behavior

An initial value problem specifies the starting population P(0) and governs the population's evolution over time. Understanding the initial condition helps predict the population trajectory and ensures the solution to the differential equation is unique and applicable to the given scenario.