IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model4

2D Model 4: Bounded Predator and Prey Growth with Regulated killing of Preys

Introduction

After proving that bounding both growth and enzymatic degradation of the preys grants the system the capacity to oscillate, we now reintroduce the bounded predator growth and check that is does not disturb the balance to the extent that oscillations are impossible again. From a physical point of view the equations are those of the molecular predation system in the absence of chemostat (or equivalently with zero washout).

The ODE (Ordinary Differential Equation) system is:

Since we now have 7 variables , we normalise the system by rescaling the X,Y axes and changing the time scale as we did with model 3, which leaves us with 4 variables only:

We will use this dimension-less version of the Dynamic System for the rest of the study.

Basic Results on the Steady Points

• The origin (0,0) remains a steady point of the system and a saddle point
• The other steady points of the system are associated to a quadratic equation
• If BC/D<1 the equation yields two admissible steady points

- One of them is a saddle point

- The nature of the other point can be adjusted (from stable to unstable via centre)

• If BC/D= 1 the quadratic degenerates into a linear equation

- And yields one steady point

- The nature of this point can be adjusted (from stable to unstable via centre)

• Finally if BC/D> 1 the quadratic yields one steady point, whose nature can be adjusted (from stable to unstable via centre)

Behaviour at Infinity

We could not prove simply that the trajectories remained bounded as time goes to infinity. Simulations quickly showed this was a hopeless effort: for a combination of parameters trajectories are not bounded!!!

• When BC/D < 1 (the case when the system has 3 steady points)

- trajectories are not bounded

- the predator population converges to a steady value D/C

- this predator population is too low to repress the prey population

- the prey population diverges to infinity

• Conversely when BC/D>=1 (the system has 2 steady points)

- trajectories remain bounded and spin around the second steady point

- we can apply Poincare-Bendixson

- consequently

- when the point is stable, the system stabilises itself at the steady point

- else we have oscillations around the 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.

It follows from the results stated above that the system has three different modes

- Prey Explosion

- Stability

- Oscillations

First Case : Prey Explosion

Phase Diagram

Time Diagram

Second case: Oscillations

Phase Diagram

Time Diagram

Last Case: Stability

Phase Diagram

Time Diagram

Conclusion

Bounding the predator growth has not removed the capacity of the system to oscillate on a unique limit-cycle. However, it has introduced a new dangerous (and highly unrealistic) mode where the prey population explodes.

Fortunately a final modification still needs introducing: the use of a chemostat in our final construct. The chemostat is used to keep the ratio of cells constant primarily, but also to get rid of any excess in the system. It is hoped that this regularising behaviour will help stabilise the system enough to eliminate the 'explosion' mode.

Appendices

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