IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results: Difference between revisions
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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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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
- 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
- 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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