IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model2: Difference between revisions
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:- (D/C, AC/(B(AoC+D)) which is always stable | :- (D/C, AC/(B(AoC+D)) which is always stable | ||
* The resulting '''Vector Field''' is a slightly distorted version of the Vector Field of Lotka-Volterra: | |||
:::[[Image:Theoretical_Analysis_6c.PNG]] | :::[[Image:Theoretical_Analysis_6c.PNG]] |
Revision as of 10:29, 28 October 2006
Bounded Prey Growth
Introduction
The assumptions made Lotka-Volterra equation are very remote to the ones made during the derivation of our model.
We start our analysis of our model by modifying the growth term of the prey population in Lotka-volterra:
ODE (Ordinary Differential Equation)
Physical interpretation of the equations
- 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.
Basic Results on the Steady Points and Vector Field
Steady Points
- The Dynamical System has two stationary points
- - (0, 0), which is always a saddle point
- - (D/C, AC/(B(AoC+D)) which is always stable
- The resulting Vector Field is a slightly distorted version of the Vector Field of Lotka-Volterra:
Behaviour at Infinity
It is easy to show that the trajectories remain bounded as time goes to infinity regardless of he 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:
the corresponding time variations of the prey and predator populations are dampened oscillations:
Conclusion
The bounding of the predator growth has a dramatic effect on the system. The balance between the growth term of the preys and predators and their decay/degradations terms is broken. We cannot have any sustained oscillations anymore.