6. Section 9.4 Logistic Growth Model Bears Years

Essential Question: Why is logistic growth model more realistic than the exponential growth model we have learned?

http://www.otherwise.com/population/logistic.html What is the limiting factor? What is it called? Look at the graph of fish over time, describe some of its characteristics.

Mad Cow Disease

We have used the exponential growth equation to represent population growth. However, real-life populations do not increase forever. There is some limiting factor such as food, living space or waste disposal. There is a maximum population, or carrying capacity, M. A more realistic model is the logistic growth model where growth rate is proportional to both the amount present (P) and the amount of the carrying capacity that remains:

The growth differential equation then becomes: Logistics Differential Equation M is maximum population, or carrying capacity, k is constant of proportionality, P is current population We can solve this differential equation to find the logistics growth model.

Logistics Differential Equation Partial Fractions

Logistics Differential Equation

A is a constant determined by initial condition Logistics Growth Model A is a constant determined by initial condition Do you need to memorize this?

Important points about Logistics (the carrying capacity) This is true no matter what the initial population is The maximum rate of change will be at the point of inflection which will always be at ½ of M If initial population is less than ½ of M, the curve is increasing, there will be 1 point of inflection, curve will be concave up until that point and concave down after it

Important points about Logistics If initial population is more than ½ of M, the curve is always increasing, there is no point of inflection, and curve will always be concave down If initial population is more than M, the curve is decreasing and always concave up

Example Given the equation dP/dt = 0.008P(100-P), answer the following questions For what values of P will the growth rate of dP/dt be close to zero? What is carrying capacity? What is population when growth rate is maximized? If initial population is 30, what does curve look like? If initial population is 70, what does curve look like? If initial population is 120, what does curve look like?

Example Given the equation dP/dt = 0.0085P(1-P/10) and P(0)=3, answer the following questions Find the equation for P 2. What is carrying capacity? 3. Use the answer to 1 to find P when t = 3? 4. Use the answer to 1 to find t when P = 7.

Example – Regression in calculator The table shows the population of Aurora, CO for selected years between 1950 and 2003. Years after 1950 Population 11,421 20 74,974 30 158,588 40 222,103 50 275,923 53 290,418

Example – Regression in calculator Use logistic regression to find a logistic curve to model the data Based on the regression equation, what will the Aurora population approach in the long run? Based on the regression equation, when will the population of Aurora first exceed 300,000 people? Write a logistic differential equation in the form dP/dt = kP(M – P) that models the growth of the Aurora data in the table.

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