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
a. Find the equilibrium solutions.

Verified step by step guidance

1. Identify the differential equation given: \(P\'(t) = 0.2 P \left(1 - \frac{P}{1200}\right)\).
2. Recall that equilibrium solutions occur when the population does not change over time, meaning \(P\'(t) = 0\).
3. Set the right-hand side of the differential equation equal to zero: \$0.2 P \left(1 - \frac{P}{1200}\right) = 0$.
4. Solve the equation \$0.2 P \left(1 - \frac{P}{1200}\right) = 0\( by setting each factor equal to zero separately: \)P = 0\( or \)1 - \frac{P}{1200} = 0$.
5. From \$1 - \frac{P}{1200} = 0\(, solve for \)P\( to find the second equilibrium solution: \)P = 1200$.

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 reflects limited resources affecting growth.

Equilibrium Solutions of Differential Equations

Equilibrium solutions occur when the rate of change is zero, meaning P'(t) = 0. For population models, these solutions represent steady states where the population remains constant over time. Finding equilibria involves setting the differential equation's right side to zero and solving for P.

Initial Value Problem (IVP)

An initial value problem specifies a differential equation along with an initial condition, such as P(0) = 50. This condition helps determine a unique solution curve from the family of possible solutions. Understanding IVPs is essential for applying and interpreting models in real-world contexts.