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Team:Evry/LogisticFunctions
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<p>When we started to model biological behaviors, we realised very soon that we were going to need a function that simulates a non-exponential evolution, that would include a simple speed control and a maximum value. A smooth step function.</p> | <p>When we started to model biological behaviors, we realised very soon that we were going to need a function that simulates a non-exponential evolution, that would include a simple speed control and a maximum value. A smooth step function.</p> | ||
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<h2>Differential form:</h2> | <h2>Differential form:</h2> | ||
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Let x be a temporal function, and y be a x-related logistic function. In order to integrate y into a temporal ODE, we need to write it differently:<br/> | Let x be a temporal function, and y be a x-related logistic function. In order to integrate y into a temporal ODE, we need to write it differently:<br/> | ||
<img src="https://static.igem.org/mediawiki/2013/4/40/Logistic_calcul1.jpg"/><br/> | <img src="https://static.igem.org/mediawiki/2013/4/40/Logistic_calcul1.jpg"/><br/> | ||
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| + | But this equation can't be integrated in a temporal system like other equations. Because y depend on x. In our model, x is a state variable of the system. To implement this equation, we solve it before the entire system. | ||
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If <img src="https://static.igem.org/mediawiki/2013/a/a0/Txdet.jpg"/> is a continuous real function, then:<br/> | If <img src="https://static.igem.org/mediawiki/2013/a/a0/Txdet.jpg"/> is a continuous real function, then:<br/> | ||
<img src="https://static.igem.org/mediawiki/2013/5/5e/Yxtegalyt.jpg"/><br/> | <img src="https://static.igem.org/mediawiki/2013/5/5e/Yxtegalyt.jpg"/><br/> | ||
Finally,<br/> | Finally,<br/> | ||
<img src="https://static.igem.org/mediawiki/2013/b/bd/Logistic_calcul2.jpg"/> | <img src="https://static.igem.org/mediawiki/2013/b/bd/Logistic_calcul2.jpg"/> | ||
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Latest revision as of 02:42, 29 October 2013
Logistic functions
When we started to model biological behaviors, we realised very soon that we were going to need a function that simulates a non-exponential evolution, that would include a simple speed control and a maximum value. A smooth step function.
Such functions, named logistic functions were introduced around 1840 by M. Verhulst.
These functions looked perfect, but we needed more control : we needed to set a starting value and a precision.
Parameters:
- Q : Magnitude.
The limit of g as x approaches infinity is Q. - d : Threshold.
The value of x from which we consider the start of the phenomenon. - p : Precision.
Since the function never reaches 0 nor Q, we have to set an approximation for 0 or Q. - k : Efficiency.
This parameter influences the length of the phenomenon.
Differential form:
Let the following be a Cauchy problem:
The solution of this Cauchy problem is as below:
Here is our logistic function. Yet, differential equations are not always time-related.
Let x be a temporal function, and y be a x-related logistic function. In order to integrate y into a temporal ODE, we need to write it differently:
But this equation can't be integrated in a temporal system like other equations. Because y depend on x. In our model, x is a state variable of the system. To implement this equation, we solve it before the entire system.
