IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results

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('''Our Results''')
Current revision (09:40, 2 November 2006) (view source)
('''Our Results''')
 
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{{Template:IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analysis}}
{{Template:IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analysis}}
=='''Our Results'''==
=='''Our Results'''==
-
 
:During the run of the summer 2006, we had time to study six 2-dimensional Dynamical Systems. Unfortunately we lacked time to carry out a thorough analysis of the 3D model.In order of complexity, the 2D models are:
:During the run of the summer 2006, we had time to study six 2-dimensional Dynamical Systems. Unfortunately we lacked time to carry out a thorough analysis of the 3D model.In order of complexity, the 2D models are:
<br><br>
<br><br>
-
:<font size="4"> '''2D Model 1: Lotka – Volterra''' </font size="4">
+
:*<font size="4"> '''2D Model 1: Lotka – Volterra''' </font size="4">
:::[[Image:Model1.PNG]]
:::[[Image:Model1.PNG]]
::*Lotka-Volterra is the first (and most famous) model for prey-predator interactions and is notoriously endowed with some very appealing properties. Lotka-Volterra also was a major inspiration for the design of the molecular predation oscillator.
::*Lotka-Volterra is the first (and most famous) model for prey-predator interactions and is notoriously endowed with some very appealing properties. Lotka-Volterra also was a major inspiration for the design of the molecular predation oscillator.
-
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model1| Detailed Analysis for Lotka-volterra]]</b>
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model1| Detailed Analysis for Lotka-volterra]]</b>
<br><br>
<br><br>
-
:<font size="4"> '''2D Model 2: Bounded Prey Growth'''</font size="4">  
+
:*<font size="4"> '''2D Model 2: Bounded Prey Growth'''</font size="4">  
:::[[Image:Model2.PNG]]
:::[[Image:Model2.PNG]]
::*Lotka-Volterra is far too simple to yield essential results on the complex 2D model.  
::*Lotka-Volterra is far too simple to yield essential results on the complex 2D model.  
::*We start to investigate the influence of various components of the system by bounding the growth of the preys.
::*We start to investigate the influence of various components of the system by bounding the growth of the preys.
-
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model2| Detailed Analysis for Model with Bounded Prey Growth]]</b>
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model2| Detailed Analysis for Model with Bounded Prey Growth]]</b>
<br><br>
<br><br>
-
:<font size="4">  '''2D Model 3: Bounded Predator and Prey Growth'''</font size="4">  
+
:*<font size="4">  '''2D Model 3: Bounded Predator and Prey Growth'''</font size="4">  
:::[[Image:Model3.PNG]]
:::[[Image:Model3.PNG]]
::*Bounding the growth of the preys only stabilises the system to the extent we cannot make it oscillate anymore.  
::*Bounding the growth of the preys only stabilises the system to the extent we cannot make it oscillate anymore.  
::*We now seek ways to obtain oscillations by bounding the growth terms of both preys and predators.
::*We now seek ways to obtain oscillations by bounding the growth terms of both preys and predators.
-
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3| Detailed Analysis for Model with Bounded Growths]]</b>
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3| Detailed Analysis for Model with Bounded Growths]]</b>
<br><br>
<br><br>
-
:<font size="4">'''2D Model 3bis: Bounded  Prey Growth and Prey Killing '''</font size="4">   
+
[[Image:blockdiagram.jpg|thumb|600px|center|The path from Lotka-Volterra to the 2D model of the Predation Oscillator ]]
 +
<br>
 +
:*<font size="4">'''2D Model 3bis: Bounded  Prey Growth and Prey Killing '''</font size="4">   
:::[[Image:Model3a.PNG]]
:::[[Image:Model3a.PNG]]
::*We have studied this model in parallel with Model 3.
::*We have studied this model in parallel with Model 3.
::*Instead of bounding the production of the predator, we bound the degradation of preys
::*Instead of bounding the production of the predator, we bound the degradation of preys
::* In both cases the goal was to investigate whether the various terms of the model could balance each other and yield oscillations.   
::* In both cases the goal was to investigate whether the various terms of the model could balance each other and yield oscillations.   
-
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3a| Detailed Analysis for Model with bounded prey growth and degradation]]</b>
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3a| Detailed Analysis for Model with bounded prey growth and degradation]]</b>
<br><br>
<br><br>
-
:<font size="4"> '''2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys'''</font size="4">
+
:*<font size="4"> '''2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys'''</font size="4">
:::[[Image:Model4.PNG]]
:::[[Image:Model4.PNG]]
::* Bounding growth and killing yielded oscillations; bounding prey and predator growths did not.
::* Bounding growth and killing yielded oscillations; bounding prey and predator growths did not.
::* We now combine both previous models and get one step closer to the final system  
::* We now combine both previous models and get one step closer to the final system  
-
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model4| Detailed Analysis for Model 4]]</b>
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model4| Detailed Analysis for Model 4]]</b>
<br><br>
<br><br>
-
:* '''Final 2D Model : 2D Model 5'''
+
:* <font size="4">'''Final 2D Model : 2D Model 5'''</font size="4">
:::[[Image:Model5.PNG]]
:::[[Image:Model5.PNG]]
::*Model 4 can be made to oscillate but also exhibits some very unrealistic properties.
::*Model 4 can be made to oscillate but also exhibits some very unrealistic properties.
::* Fortunately experimental conditions lead us to introduce a final dissipative term –eU to the derivative of the prey population.
::* Fortunately experimental conditions lead us to introduce a final dissipative term –eU to the derivative of the prey population.
::*We investigate the properties of this final 2D model and prove that the new dissipative term confers it some very interesting characteristics.
::*We investigate the properties of this final 2D model and prove that the new dissipative term confers it some very interesting characteristics.
-
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model5| Detailed Analysis of the complete 2D Model]]</b>
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model5| Detailed Analysis of the complete 2D Model]]</b>
 +
 
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Current revision

Analysis of the Model of the Molecular Predation Oscillator


Our Results

During the run of the summer 2006, we had time to study six 2-dimensional Dynamical Systems. Unfortunately we lacked time to carry out a thorough analysis of the 3D model.In order of complexity, the 2D models are:



  • 2D Model 1: Lotka – Volterra
Image:Model1.PNG
  • Lotka-Volterra is the first (and most famous) model for prey-predator interactions and is notoriously endowed with some very appealing properties. Lotka-Volterra also was a major inspiration for the design of the molecular predation oscillator.



  • 2D Model 2: Bounded Prey Growth
Image:Model2.PNG



  • 2D Model 3: Bounded Predator and Prey Growth
Image:Model3.PNG
  • Bounding the growth of the preys only stabilises the system to the extent we cannot make it oscillate anymore.
  • We now seek ways to obtain oscillations by bounding the growth terms of both preys and predators.
  • Detailed Analysis for Model with Bounded Growths



The path from Lotka-Volterra to the 2D model of the Predation Oscillator
The path from Lotka-Volterra to the 2D model of the Predation Oscillator


  • 2D Model 3bis: Bounded Prey Growth and Prey Killing
Image:Model3a.PNG



  • 2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys
Image:Model4.PNG
  • Bounding growth and killing yielded oscillations; bounding prey and predator growths did not.
  • We now combine both previous models and get one step closer to the final system
  • Detailed Analysis for Model 4



  • Final 2D Model : 2D Model 5
Image:Model5.PNG
  • Model 4 can be made to oscillate but also exhibits some very unrealistic properties.
  • Fortunately experimental conditions lead us to introduce a final dissipative term –eU to the derivative of the prey population.
  • We investigate the properties of this final 2D model and prove that the new dissipative term confers it some very interesting characteristics.
  • Detailed Analysis of the complete 2D Model

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