IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3a

Model 3 bis: Bounded Predator and Prey Growth

Introduction

Generalities

• The analysis of the model with bounded growth terms for both preys and predators has hinted that in order to get oscillations it was more important to balance the respective growth and decay terms of a population than to balance the growth terms within the whole system (the idea being to achieve some form of control of the ratios of the populations).
• We now test the validity of this interpretation by bounding both growth and decay of the preys while leaving the terms relating to the predators identical to those in the Lotka-Volterra model.

Physical interpretation of the equations

• All the arguments that bounded the growth of the preys still apply (production is limited by the number of promoters).
• The decay term of the preys is due to an enzymatic reaction, typically modelled with a Michaelis-Menten kinetics.
• For all the predator terms of the system (second differential equation) we make the same assumption as with Lotka-Volterra that is:

- The predator’s growth is the direct result of interacting the preys.

- In the present case it is proportional to the population of both preys and predators.

- The population of the predators is controlled by its natural death (proportional to the population)

Basic Results on the Steady Points

The dynamic analysis of the system yields results that differ from the previous system in a significant way:

• The system has two stationary points
• The first stationary point (0, 0) is a saddle point
• The second stationary point ( Xs=D/C, Ys=A(Bo+Xs)/(B(Ao+Xs)) ) is not always stable
• The nature of the second steady point depends on the respective values of Ao and Bo:

- if Ao>Bo, the point is stable (it attracts all the trajectories in its vicinity)

- If Ao=Bo, the point is a centre (the trajectories in its vicinity spin around it)

- if Ao<Bo, the point is unstable (it attracts all the trajectories in its vicinity)

Behaviour at Infinity

The trajectories remain bounded as time goes to infinity regardless of the choice of parameters.

• We can therefore apply Poincare-Bendixson's theorem.
• Due to the saddle point nature of the origin (0,0)

- any trajectory starting on the Y-axis will be attracted to the origin

- any trajectory starting on the X-axis will be repelled to infinity

• For initial conditions (Xo,Yo) both non-null the trajectories will converge

- to the stable steady point if Ao>Bo

- else (Ao<=Bo) to a Limit Cycle: We have Oscillations

Typical Simulations

First Case: Stable System

The first typical behaviour of the system (convergence to stable steady point) can be illustrated by the following phase diagram, and the corresponding time variations of the prey and predator populations (dampened oscillations)

Phase Diagram

Time Diagram

Second Case: Oscillations

The second typical behaviour of the system (Oscillations around a limit cycle) can be illustrated by the following phase diagram , and the corresponding time variations of the prey and predator populations (oscillations)

Phase Diagram

Time Diagram

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). In the phase diagrams , red dots symbolise a steady points.

Control of the Oscillations

We obtained results that are very similar to the case of Lotka-Volterra:

- By playing on the ratios D/C and B/A we can push the second steady point far away from the origin, which has the desired effect of increasing the size of the limit cycle - and its amplitude

- Once a cycle of desired amplitude its frequency can be changed by multiplying A,B,C and D by a positive constant.

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 and decay of the preys has succeeded in yielding oscillations.
• Furthermore we have total control over the amplitude and frequency of the oscillations.