User talk:Benjamin Leibowicz

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Biophysics 101: Assignment 3

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Biophysics 101: Assignment 1

Our first assignment was to practice using spreadsheet software (Microsoft Excel) and the programming language Python by examining the exponential and logistic functions. First I used Excel to plot both functions using recursive relations where each value of the function is calculated from the previous one. The exponential function is given by the recursive relation [math]\displaystyle{ x_n=k*x_{n-1} }[/math] and the logistic function is given by the recursive relation [math]\displaystyle{ x_n=k*x_{n-1}*(1-x_{n-1}) }[/math]. Another function examined was a modified logistic expression [math]\displaystyle{ x_n=MAX(k*x_{n-1}*(1-x_{n-1}),0) }[/math]. All of these functions were plotted recursively for a list of k values including 0.9, 1.01, 1.1, 1.5, 3, 3.67859, 4, and 4.03. The results are shown below:

The exponential functions look as expected, with the higher k values resulting in growth curves that grow faster than for lower k values. The recursive representation of the logistic function does well in approximating the continuous form of the function for low k values. Once k reaches 3 the periodic behavior of this recursive relation becomes very noticeable, and for still higher k values the function takes off in the negative direction and must be restrained by the MAX condition.

Once I successfully installed Python and got it working, plotting the continuous forms of the exponential and logistic functions for different k values was fairly straightforward. The continuous form of the exponential function is [math]\displaystyle{ y=e^{kx} }[/math] and the continuous form of the logistic function for a starting value of [math]\displaystyle{ x_0=0.5 }[/math] when [math]\displaystyle{ t=0 }[/math] is [math]\displaystyle{ y=1/(1+e^{-kt}) }[/math]. The plots for a k value of 1.5 are shown below:

When experimenting with different k values, the exponential function again grows faster as k increases. As k increases the logistic function settles to its final value of 1 in less and less time.

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