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

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=2D Model 3: Bounded Predator and Prey Growth=
<br><br>
<font size="6"><center>'''Model 3: Bounded Predator and Prey Growth'''</center></font size="6">
<br><br><br>
 
= '''Introduction''' =
= '''Introduction''' =
<font size="4">'''Generalities''' </font size="4">
:*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.
[[Image:LotkaV3.png|center]]


<br><br>
<font size="4">'''Physical interpretation of the equations'''</font size="4">
:* 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.


<b>Relevance of the Model</b>
*The predator production is limited by the limited number of promoters as well. This is modeled by Michaelis-Menten kinetics.
*However, the LuxR in the predator is the direct product after the promoter is activated. Hence LuxR cannot be treated saturated to a constant.
*The rate of production becomes linear when number of promoters is saturated.


<font size="4">'''Dimensionless Version of the System'''</font size="4">
:* 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


The assumptions made Lotka-Volterra equation are very remote to the ones made during the derivation of our model. Consequently our model is very different from the pure Lotka-Volterra.  
[[Image:LotkaV3-norm.png|center]]
<br><br>


We start bridging the gap between the two models by modifying the growth term of the prey population in Lotka-volterra:
= '''Basic Results on the Steady Points and Vector Field''' =


<b>ODE (Ordinary Differential Equation)</b>
* The dynamic analysis of the system yields results that are '''very similar''' to the previous system where only the growth of preys was bounded.
:* The system has two stationary points
:* The first stationary point (0, 0) is a saddle point
:* The second stationary point (coordinates) is '''always stable'''


* The Vector Field is again reminiscent of Lotka-Volterra (if only even more distorted).
<br><br>


<b>Physical interpretation of the equations</b>
= '''Behaviour at Infinity''' =
* 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.
* For all the other terms of the system the assumptions made for Lotka-Volterra remain.
<b>ODE (Ordinary Differential Equation)</b>
:::[[Image: Theoretical_Analysis_6h.PNG]]


* We get again similar results to those obtained for the bounded prey growth.


*The trajectories remain bounded as time goes to infinity regardless of the choice of parameters.
::- We can therefore apply Poincare-Bendixson's theorem.
::- Because the system only has 2 steady points , one a saddle point the other a stable point we are sure the system will not oscillate.
::- For initial conditions (Xo,Yo) both non-null the trajectories will converge to the stable steady point
<br><br>


='''Typical Simulations''' =


<b>Dimensionless Version of the System</b>
: The typical behaviour of the system (convergence to stable steady point) can be illustrated by the following simulation, and the corresponding time variations of the prey and predator populations are dampened oscillations


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. 
[[Image:2d model 3b.PNG|thumb|380px|left|Phase Diagram]]
After normalisation the system only has 3 free parameters
[[Image:2d model 3c.PNG|thumb|380px|right|Time Diagram]]
 
<br style="clear:both;"/>
:::[[Image:2d model 3a.PNG]]
 
= '''Basic Results on the Steady Points and Vector Field''' =
<b>Steady Points</b>
*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''' =
:Simulations using different initial conditions are assigned different colors (the open-end of the trajectories is the starting point). In the phase diagrams , red dots symbolise a steady points.
*Clearly, the second stable stationary point will ensure the system is stable.
<br><br>
*Trajectories will be remain bounded for all the different parameters


='''Results of Dynamic Analysis''' =
*2 stationary points, (0, 0) – saddle, the second point– stable
:::[[Image:2d model 3b.PNG]]
:::[[Image:2d model 3c.PNG]]
*Thus we are expected to get a damped oscillation with this model
='''Conclusion'''=
='''Conclusion'''=
<b>Summary of Results</b>
*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'''=
*The bounding of the predator growth had a dramatic effect on the system and broke the balance between the growth terms of the preys and predators and their decay/degradations terms.
<b>Related documents</b>
*Our attempt to re-balance the system by bounding the growth of the predators has '''failed to yield any oscillation'''.
*Stability [http://openwetware.org/images/1/17/MM_model_analysis_3.doc analysis]
*This suggests that a '''better way to re-balance the system''' is to '''balance the growth and decay terms of the preys''' and/or predators instead.

Latest revision as of 07:37, 31 October 2006



Model 3: Bounded Predator and Prey Growth




Introduction

Generalities

  • 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.


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

  • The dynamic analysis of the system yields results that are very similar to the previous system where only the growth of preys was bounded.
  • The system has two stationary points
  • The first stationary point (0, 0) is a saddle point
  • The second stationary point (coordinates) is always stable
  • The Vector Field is again reminiscent of Lotka-Volterra (if only even more distorted).



Behaviour at Infinity

  • We get again similar results to those obtained for the bounded prey growth.
  • The trajectories remain bounded as time goes to infinity regardless of the choice of parameters.
- We can therefore apply Poincare-Bendixson's theorem.
- Because the system only has 2 steady points , one a saddle point the other a stable point we are sure the system will not oscillate.
- For initial conditions (Xo,Yo) both non-null the trajectories will converge to the stable steady point



Typical Simulations

The typical behaviour of the system (convergence to stable steady point) can be illustrated by the following simulation, and the corresponding time variations of the prey and predator populations are dampened oscillations
Phase Diagram
Time Diagram


Simulations using different initial conditions are assigned different colors (the open-end of the trajectories is the starting point). In the phase diagrams , red dots symbolise a steady points.



Conclusion

  • The bounding of the predator growth had a dramatic effect on the system and broke the balance between the growth terms of the preys and predators and their decay/degradations terms.
  • Our attempt to re-balance the system by bounding the growth of the predators has failed to yield any oscillation.
  • This suggests that a better way to re-balance the system is to balance the growth and decay terms of the preys and/or predators instead.
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