IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3
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
- Preliminary Note on the Simulations:
As with the previous models, simulations using different initial conditions are assigned different colors (the open-end of the trajectories is the starting point). Finally in the phase diagrams , red dots symbolise a steady points.
- The typical behaviour of the system (convergence to stable steady point)
- The corresponding time variations of the prey and predator populations are dampened oscillations:
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.
Appendices
Related documents
- Stability analysis