Bounded Prey Growth

Introduction

The assumptions made Lotka-Volterra equation are very remote to the ones made during the derivation of our model.

We start our analysis of our model 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

- (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:

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

Summary of Results

With the change in the equation, we could see the drastic change in the system behaviour
Hence it is important to carefully analysis each model to check whether we are able to get the required oscillation

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