User talk:Daniel M Jordan: Difference between revisions
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==[[Harvard:Biophysics_101/2009:Assignments|Biophysics 101 Assignments]]== | ==[[Harvard:Biophysics_101/2009:Assignments|Biophysics 101 Assignments]]== | ||
===Exponentials (Due Tue 15-Sep-2009)=== | ===My Observations on Exponentials (Due Tue 15-Sep-2009)=== | ||
* The exponentials for values near k = 1 look like what I expect exponentials to look like, including the fact that they are inverted when k < 1 and get steeper as k gets higher above 1. The logistic functions at the same values look like s-shaped curves, also inverted for k < 1 (though there may not actually be an s there, it's difficult to tell) and getting steeper as k gets higher. | * The exponentials for values near k = 1 look like what I expect exponentials to look like, including the fact that they are inverted when k < 1 and get steeper as k gets higher above 1. The logistic functions at the same values look like s-shaped curves, also inverted for k < 1 (though there may not actually be an s there, it's difficult to tell) and getting steeper as k gets higher. | ||
* For k = 3 and above, they start to look less like smooth curves and more like sharp spikes, but still with the same general shape. It's also very difficult to tell the three curves with high k apart by eye. By contrast, the logistic curves at high k have totally different behavior from the ones at low k — they rise up to what would be the top of the s-curve and then start oscillating, with wider oscillations for higher k. | * For k = 3 and above, they start to look less like smooth curves and more like sharp spikes, but still with the same general shape. It's also very difficult to tell the three curves with high k apart by eye. By contrast, the logistic curves at high k have totally different behavior from the ones at low k — they rise up to what would be the top of the s-curve and then start oscillating, with wider oscillations for higher k. | ||
* When the oscillations are large enough to take the population down to 0, the oscillation stops dead, even though it was at 1 a moment before. This models the fact that a population that's gone extinct will never come back | * When the oscillations are large enough to take the population down to 0, the oscillation stops dead, even though it was at 1 a moment before. This models the fact that a population that's gone extinct will never come back — an obvious conclusion if we're thinking in terms of populations. | ||
* The exponential function we're modeling seems to be k<sup>t</sup>, where t is time. The logistic function, according to google, is (1 + e<sup>-t</sup>)<sup>-1</sup>, but I can't figure out where the k comes in (which would have made writing it in python easier. | * The exponential function we're modeling seems to be k<sup>t</sup>, where t is time. The logistic function, according to google, is (1 + e<sup>-t</sup>)<sup>-1</sup>, but I can't figure out where the k comes in (which would have made writing it in python easier). | ||
Revision as of 19:46, 14 September 2009
Biophysics 101 Assignments
My Observations on Exponentials (Due Tue 15-Sep-2009)
- The exponentials for values near k = 1 look like what I expect exponentials to look like, including the fact that they are inverted when k < 1 and get steeper as k gets higher above 1. The logistic functions at the same values look like s-shaped curves, also inverted for k < 1 (though there may not actually be an s there, it's difficult to tell) and getting steeper as k gets higher.
- For k = 3 and above, they start to look less like smooth curves and more like sharp spikes, but still with the same general shape. It's also very difficult to tell the three curves with high k apart by eye. By contrast, the logistic curves at high k have totally different behavior from the ones at low k — they rise up to what would be the top of the s-curve and then start oscillating, with wider oscillations for higher k.
- When the oscillations are large enough to take the population down to 0, the oscillation stops dead, even though it was at 1 a moment before. This models the fact that a population that's gone extinct will never come back — an obvious conclusion if we're thinking in terms of populations.
- The exponential function we're modeling seems to be kt, where t is time. The logistic function, according to google, is (1 + e-t)-1, but I can't figure out where the k comes in (which would have made writing it in python easier).
