Modeling Population Growth

📌 CONCEPT: The differential equation for population growth is given by dP/dt = kP, where P is the population at time t and k is the growth rate constant.

📐 RULE / FORMULA: To solve the differential equation dP/dt = kP, we use the formula P(t) = P0 * e^(kt), where P0 is the initial population and e is the base of the natural logarithm.

💡 WORKED EXAMPLE: Given an initial population of 100 and a growth rate of 25% per year, we can find the population after 5 years. First, we convert the growth rate to a decimal (k = 0.25). Then, we use the formula P(5) = 100 * e^(0.25*5) = 100 * e^1.25 ≈ 100 * 3.491 ≈ 349.1. Therefore, the population after 5 years is approximately 349.1.

⚠️ COMMON MISTAKE: Students often forget to convert the growth rate from a percentage to a decimal or use the wrong base for the natural logarithm. Additionally, they may not consider the implications of exponential growth on the ecosystem, such as the potential depletion of resources and disruption of the food chain.