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

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.