Calc06_5 - 6.5 Logistic Growth Model Bears 40 20 0 20 40...

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6.5 Logistic Growth Model Years Bears Greg Kelly, Hanford High School, Richland, Washington
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We have used the exponential growth equation to represent population growth. 0 kt y y e = 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: dP kP dt = The constant k is called the relative growth rate . / dP dt k P =
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The population growth model becomes: 0 kt P P e = 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 fraction of the carrying capacity that remains: M P M -
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The equation then becomes: dP M P kP dt M - = Our book writes it this way: Logistics Differential Equation ( 29 dP k P M P dt M = - We can solve this differential equation to find the logistics growth model.
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