Population Model with Differential Growth?

📌 CONCEPT: A differential equation is used to model the population growth of the rabbit species on the isolated island, where the growth rate is proportional to the product of the present population and the difference between the maximum capacity of the island and the current population.

📐 RULE / FORMULA: The differential equation for this scenario is dP/dt = kP(M - P), where P is the current population, M is the maximum capacity of the island, and k is a constant representing the growth rate.

💡 WORKED EXAMPLE: Let's assume the initial population of rabbits is 100, the maximum capacity of the island is 1000, and the growth rate k is 0.1. Using the differential equation, we can calculate the population after 1 year as dP/dt = 0.1P(1000 - P), and solving this equation, we get P = 1000 / (1 + 0.1*1000) ≈ 909.09. This means the population will increase to approximately 909 rabbits after 1 year.

⚠️ COMMON MISTAKE: Students may incorrectly assume that the population will grow exponentially without considering the carrying capacity of the island, leading to an unrealistic model.