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{{Template:IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analysis}} | |||
=='''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: | |||
: | <br><br> | ||
:* '''2D Model 1: Lotka – Volterra''' | :*<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 | |||
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model1| Detailed Analysis for Lotka-volterra]]</b> | |||
:* '''2D Model 2: Bounded Prey Growth''' | <br><br> | ||
:*<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 | ::*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/Results/2D Model2| Detailed Analysis for Model with Bounded Prey Growth]]</b> | ||
<br><br> | |||
:* '''2D Model 3: Bounded Predator and Prey Growth''' | :*<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. 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/Results/2D Model3| Detailed Analysis for Model with Bounded Growths]]</b> | ||
<br><br> | |||
:* '''2D Model | [[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]] | ||
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3a| | ::*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. | |||
::*<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]] | ||
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model4| | ::* 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/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 also exhibits some very unrealistic properties. | |||
::*Model 4 can be made to oscillate | ::* 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> | |||
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Latest revision as of 07:40, 2 November 2006
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
- 2D Model 2: Bounded Prey Growth
-
- 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.
- 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
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