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