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


 * 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.


 *  Detailed Analysis for Lotka-volterra


 * 2D Model 2: Bounded Prey Growth
 * [[Image:Model2.PNG]]
 * 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
 * [[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


 * 2D Model 3bis: Bounded Prey Growth and Prey Killing 
 * [[Image:Model3a.PNG]]
 * 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
 * [[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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