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6.5. Logistic Growth Model. Bears. Years. Greg Kelly, Hanford High School, Richland, Washington. We have used the exponential growth equation to represent population growth. The exponential growth equation occurs when the rate of growth is proportional to the amount present.

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6.5

Logistic Growth Model

Bears

Years

Greg Kelly, Hanford High School, Richland, Washington


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We have used the exponential growth equation

to represent population growth.

The exponential growth equation occurs when the rate of growth is proportional to the amount present.

If we use P to represent the population, the differential equation becomes:

The constant k is called the relative growth rate.


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The population growth model becomes:

A more realistic model is the logistic growth model where growth rate is proportional to both the amount present (P) and the fraction of the carrying capacity that remains:

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.


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Logistics Differential Equation

The equation then becomes:

Our book writes it this way:

We can solve this differential equation to find the logistics growth model.


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Partial

Fractions

Logistics Differential Equation


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Logistics Differential Equation


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Logistics Differential Equation


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Logistics Growth Model


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Example:

Logistic Growth Model

Ten grizzly bears were introduced to a national park 10 years ago. There are 23 bears in the park at the present time. The park can support a maximum of 100 bears.

Assuming a logistic growth model, when will the bear population reach 50? 75? 100?


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Ten grizzly bears were introduced to a national park 10 years ago. There are 23 bears in the park at the present time. The park can support a maximum of 100 bears.

Assuming a logistic growth model, when will the bear population reach 50? 75? 100?


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At time zero, the population is 10.


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After 10 years, the population is 23.


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Bears

Years

We can graph this equation and use “trace” to find the solutions.

y=50 at 22 years

y=75 at 33 years

y=100 at 75 years

p


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