Solving the Logistic Growth Equation?

📌 CONCEPT: The logistic growth equation dP/dt = kP(1 - P/M) models the growth of a population with a carrying capacity M and growth rate k, where P is the population at time t.

📐 RULE / FORMULA: To solve the logistic growth equation, we first separate variables and then integrate both sides to find the population P as a function of time t.

💡 WORKED EXAMPLE: Given the initial population P(0) = 10,000, carrying capacity M = 10,000,000, and growth rate k, we separate variables to get dP/P(1 - P/M) = kdt. Integrating both sides, we get ∫dP/P(1 - P/M) = ∫kdt. Solving this integral, we get P(t) = M/(1 + Ae^(-kt)), where A is a constant determined by the initial condition. Substituting the initial condition, we get A = (M - P(0))/P(0).

⚠️ COMMON MISTAKE: Students often forget to separate variables correctly or overlook the initial condition when solving the logistic growth equation.