IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results
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=='''Our Results'''==  =='''Our Results'''==  
  
:During the run of the summer 2006, we had time to study six 2dimensional 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 2dimensional 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]]  
::*LotkaVolterra is the first (and most famous) model for preypredator interactions and is notoriously endowed with some very appealing properties. LotkaVolterra also was a major inspiration for the design of the molecular predation oscillator.  ::*LotkaVolterra is the first (and most famous) model for preypredator interactions and is notoriously endowed with some very appealing properties. LotkaVolterra also was a major inspiration for the design of the molecular predation oscillator.  
  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model1  +  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model1 Detailed Analysis for Lotkavolterra]]</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]]  
::*LotkaVolterra is far too simple to yield essential results on the complex 2D model.  ::*LotkaVolterra 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  +  ::*<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  +  ::*<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">  +  [[Image:blockdiagram.jpgthumb600pxcenterThe path from LotkaVolterra 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  +  ::*We have studied this model in parallel with Model 3. 
  ::*Instead of bounding the production of the predator, we bound the degradation of  +  ::*Instead of bounding the production of the predator, we bound the degradation of preys 
  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3a  +  ::* 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/Results/2D Model3a Detailed Analysis for Model with bounded prey growth and degradation]]</b>  
  :* '''2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys'''  +  <br><br> 
+  :*<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. 
  ::*  +  ::* 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  +  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model4 Detailed Analysis for Model 4]]</b> 
  +  <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  +  ::*Model 4 can be made to oscillate but also exhibits some very unrealistic properties. 
  ::*We investigate the properties of this final 2D model and prove that the new dissipative term confers it some very interesting characteristics  +  ::* 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.  
  +  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model5 Detailed Analysis of the complete 2D Model]]</b>  
  +  
  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model5  +  <html> 
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Current revision
Analysis of the Model of the Molecular Predation Oscillator
 Introduction
 Our Approach
 Model Simplication
 Our Results
 Conclusion
 Appendix
Our Results
 During the run of the summer 2006, we had time to study six 2dimensional 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
 2D Model 2: Bounded Prey Growth

 LotkaVolterra 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.
 Detailed Analysis for Model with Bounded Prey Growth
 2D Model 3: Bounded Predator and Prey Growth

 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
 2D Model 3bis: Bounded Prey Growth and Prey Killing

 We have studied this model in parallel with Model 3.
 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.
 Detailed Analysis for Model with bounded prey growth and degradation
 2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys

 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

 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