Population Growth Modelling?

📌 CONCEPT: The population growth model represents the rate of change of the population over time, assuming it grows at a rate proportional to the current population.

📐 RULE / FORMULA: The differential equation representing this growth model is dP/dt = kP, where P is the population at time t and k is the growth rate constant.

💡 WORKED EXAMPLE: Given an initial growth rate of 0.5% per annum, we can calculate k as 0.005. Substituting this into the differential equation gives dP/dt = 0.005P. To solve this equation, we can use separation of variables: P^-1 dP = 0.005 dt. Integrating both sides gives ln|P| = 0.005t + C. This implies that P = Ae^(0.005t), where A is a constant representing the initial population of 50,000.

⚠️ COMMON MISTAKE: Students often confuse the growth rate constant k with the initial population A, leading to incorrect solutions.