Population Growth Modelling

📌 CONCEPT: The logistic differential equation is used to model population growth, where the rate of change of the population is proportional to the product of the current population and its carrying capacity minus the current population.

📐 RULE / FORMULA: The logistic differential equation is given by dP/dt = rP(1 - P/K), where P is the population, r is the birth rate, K is the carrying capacity, and t is time.

💡 WORKED EXAMPLE: Let's consider a population of rabbits with an initial population of 100, a birth rate of 0.2, and a carrying capacity of 1000. Using the logistic differential equation, we can derive an expression for the population after 5 years: dP/dt = 0.2P(1 - P/1000). Assuming a constant birth rate and no deaths, we can integrate the equation to find the population after 5 years.

⚠️ COMMON MISTAKE: Students often forget to check the units of the variables and the birth rate, which can lead to incorrect solutions.