Modeling Population Growth?

📌 CONCEPT: To model population growth, we need to analyze the factors affecting the population size over time, including birth rates, death rates, and carrying capacities.

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

💡 WORKED EXAMPLE: Suppose we have a population of rabbits with a growth rate of 0.5 rabbits per rabbit per year and a carrying capacity of 1000 rabbits. The initial population is 100 rabbits. To analyze the effect of introducing an inhibitor that reduces the birth rate by 20% when the population exceeds 500 rabbits, we adjust the growth rate to r = 0.5 * (1 - 0.2) when P > 500. The logistic differential equation becomes dP/dt = 0.4P(1 - P/1000) for P > 500.

⚠️ COMMON MISTAKE: Students often forget to adjust the growth rate or carrying capacity when introducing an inhibitor or other external factors that affect population growth.