IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3: Difference between revisions

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Bounding the growth term for the preys moves us further from our goal of creating oscillations as it stabilises the system too much. We now seek to counteract this by also bounding the growth of the predators.
Bounding the growth term for the preys moves us further from our goal of creating oscillations as it stabilises the system too much. We now seek to counteract this by also bounding the growth of the predators.


 
<b>Physical interpretation of the equations</b>
All the arguments that bounded the growth of the preys apply to the predators ( production is limited by the number of promoters as well). This is modeled by Michaelis-Menten kinetics.  
* All the arguments that bounded the growth of the preys apply to the predators (production is limited by the number of promoters as well). This results in a Michaelis-Menten-like kinetics.  
 
* For all the degradation terms of the system the assumptions made for Lotka-Volterra remain.
For all the degradation terms of the system the assumptions made for Lotka-Volterra remain.


<b>ODE (Ordinary Differential Equation)</b>
<b>ODE (Ordinary Differential Equation)</b>


<b>Physical interpretation of the equations</b>
* The prey (AHL) production is limited by the limited number of promoters, which can be modelled by a Michaelis-Menten-like kinetics. For very small values of the prey-population (X) the growth can still be assumed proportional to the population (as with Lotka-Volterra). However, for larger populations it becomes constant as the system reaches saturation due to the limited number of promoters.
*
<b>ODE (Ordinary Differential Equation)</b>
:::[[Image: Theoretical_Analysis_6h.PNG]]
:::[[Image: Theoretical_Analysis_6h.PNG]]




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The complexity of system has increased notably (6 independent parameters) so much so that direct symbolic computation with Matlab is not available. We therefore normalised the system before analysing it - thus reducing the number of free parameters. Normalisation was achieved by changing the scale reference on the X and Y axes as well as in time. Therefore the behaviour of a system is unchanged by normalisation.   
The complexity of system has increased notably (6 independent parameters) so much so that direct symbolic computation with Matlab is not available. We therefore normalised the system before analysing it - thus reducing the number of free parameters. Normalisation was achieved by changing the scale reference on the X and Y axes as well as in time. Therefore the behaviour of a system is unchanged by normalisation.   
After normalisation the system only has 3 free parameters
After normalisation the system only has 3 free parameters
:::[[Image:2d model 3a.PNG]]


= '''Basic Results on the Steady Points and Vector Field''' =
= '''Basic Results on the Steady Points and Vector Field''' =

Revision as of 12:28, 28 October 2006

2D Model 3: Bounded Predator and Prey Growth

Introduction

Bounding the growth term for the preys moves us further from our goal of creating oscillations as it stabilises the system too much. We now seek to counteract this by also bounding the growth of the predators.

Physical interpretation of the equations

  • All the arguments that bounded the growth of the preys apply to the predators (production is limited by the number of promoters as well). This results in a Michaelis-Menten-like kinetics.
  • For all the degradation terms of the system the assumptions made for Lotka-Volterra remain.

ODE (Ordinary Differential Equation)


Dimensionless Version of the System

The complexity of system has increased notably (6 independent parameters) so much so that direct symbolic computation with Matlab is not available. We therefore normalised the system before analysing it - thus reducing the number of free parameters. Normalisation was achieved by changing the scale reference on the X and Y axes as well as in time. Therefore the behaviour of a system is unchanged by normalisation. After normalisation the system only has 3 free parameters

Basic Results on the Steady Points and Vector Field

Steady Points

  • After analysis, there are 3 stationary points, but only 2 in the first quadrant (i.e. X&Y>0)
  • The existence of the 2 stationary points is independent of the parameters
  • After Jacobian analysis, the stationary point (0, 0) is a saddle point, while the second one is always stable

Limit cycles

  • Clearly, the second stable stationary point will ensure the system is stable.
  • Trajectories will be remain bounded for all the different parameters

Results of Dynamic Analysis

  • 2 stationary points, (0, 0) – saddle, the second point– stable
  • Thus we are expected to get a damped oscillation with this model

Conclusion

Summary of Results

  • The bounded production will result in stability of the system. This is expected because the rate of production is not longer fast enough; the exponential decay of the prey in the equation will bring the system into equilibrium.

Appendices

Related documents

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