Logistic-continuous

Logistic-continuous - -P c 2 e rt P = c 2 e rt-Pc 2 e rt P...

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The Logistic Flow (Continuous) Tom Carter http://astarte.csustan.edu/˜ tom/SFI-CSSS Complex Systems Summer School June, 2008 1
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Logistic flow . . . We all know that the discrete logistic map P n +1 = rP n (1 - P n ) exhibits interesting behavior of various sorts for various values of the parameter r , including chaos, etc. 2
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What kind of behavior can we expect for the continuous version of a logistic flow: dP dt = rP (1 - P ) ? Note that this is a non-linear ODE, but fortunately we can actually integrate . . . dP dt = rP (1 - P ) dP P (1 - P ) = rdt Thus: Z dP P (1 - P ) = Z rdt Z dP P (1 - P ) = rt + c 1 3
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By partial fractions, we have: Z dP P + Z dP (1 - P ) = rt + c 1 log( P ) - log(1 - P ) = rt + c 1 log( P 1 - P ) = rt + c 1 P 1 - P = e rt + c 1 P 1 - P = c 2 e rt P = (1 - P ) c 2 e rt 4
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And thus:
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Unformatted text preview: -P ) c 2 e rt P = c 2 e rt-Pc 2 e rt P + Pc 2 e rt = c 2 e rt P (1 + c 2 e rt ) = c 2 e rt giving us: P = c 2 e rt 1 + c 2 e rt and, dividing top and bottom by c 2 e rt , and simplifying, we have: P = 1 1 + ce-rt 5 This function just gives us the classic logistic/sigmoid curve: and changes in c and r make minor changes in the behavior near 0 . . . The difference between the behavior of the discrete and continuous logistic functions can give us some idea of the significance of working in the discrete regime . . . To top ← 6...
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