IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3a
2D Model 3: Bounded Predator and Prey Growth
Introduction
The analysis of the model with bounded growth terms for both preys and predators has shown such a has hinted that oscillations are more the result of the balance between growth and decay of populations
We now test this hypothesis 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.
ODE (Ordinary Differential Equation)
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:
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:
- The corresponding time variations of the prey and predator populations are dampened oscillations:
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.
Appendices
Related documents
- Stability analysis
