Team:Evry/LogisticFunctions

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<h1>Logistic functions :</h1>
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<h1>Logistic functions</h1>
<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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<li><b>Q</b> : Magnitude.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<i>The limit of g as x approaches infinity is Q.</i></li>
<li><b>Q</b> : Magnitude.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<i>The limit of g as x approaches infinity is Q.</i></li>
<li><b>d</b> : Threshold.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<i>The value of x from which we consider the start of the phenomenon.</i></li>
<li><b>d</b> : Threshold.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<i>The value of x from which we consider the start of the phenomenon.</i></li>
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<li><b>p</b> : Precision.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<i>g(d)=Q*p Since the function never reaches 0 nor Q, we have to set an approximation for 0 or Q.</i></li>
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<li><b>p</b> : Precision.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<i><img src="https://static.igem.org/mediawiki/2013/3/3b/Gdp.jpg"/> Since the function never reaches 0 nor Q, we have to set an approximation for 0 or Q.</i></li>
<li><b>k</b> : Efficiency.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<i>This parameter influences the length of the phenomenon.</i></li>
<li><b>k</b> : Efficiency.<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<i>This parameter influences the length of the phenomenon.</i></li>
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<p><img width="80%" src="https://static.igem.org/mediawiki/2013/0/05/CourbeLogistique.png"/></p>
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<p><img width="70%" src="https://static.igem.org/mediawiki/2013/0/05/CourbeLogistique.png"/></p>
<h2>Differential form:</h2>
<h2>Differential form:</h2>
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<p>Here is our logistic function. Yet, differential equations are not always time-related.  
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<p>Here is our logistic function. Yet, differential equations are not always time-related.<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/>
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/>
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<img src="https://static.igem.org/mediawiki/2013/4/40/Logistic_calcul1.jpg"/><br/>
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dy/dx = by(x(t))(1 – y(x(t))/K)
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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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<=>( dy/dt)/(dx/dt) =  by(x(t))(1 – y(x(t))/K)
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<p>And so:<br/>
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dy/dt = dx/dt *  by(x(t))(1 – y(x(t))/K)
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If t → x(t) is a continuous real function, then:<br/>
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y(x(t)) = y(t)
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If <img src="https://static.igem.org/mediawiki/2013/a/a0/Txdet.jpg"/> is a continuous real function, then:<br/>
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<img src="https://static.igem.org/mediawiki/2013/5/5e/Yxtegalyt.jpg"/><br/>
Finally,<br/>
Finally,<br/>
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<img src="https://static.igem.org/mediawiki/2013/b/bd/Logistic_calcul2.jpg"/>
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dy/dt = dx/dt *  by(1 – y/K)
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<p id="references">References:</p>
 
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Latest revision as of 02:42, 29 October 2013

Iron coli project

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.

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