IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model2

From OpenWetWare
Jump to navigationJump to search

Bounded Prey Growth

Introduction

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.

We start bridging the gap between the two models 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:
Sketch of Vector Field


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.

Retrieved from "https://openwetware.org/mediawiki/index.php?title=IGEM:IMPERIAL/2006/project/Oscillator/Theoretical_Analyses/Results/2D_Model2&oldid=83288"