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

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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

  • 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

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

  • 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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