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The Lotka-Volterra model

A more interesting, but still quite simple version of a predator-prey system is given with the following equations

\[\frac{dR}{dt}=a*R*(1-\frac R K)-b*R*N\] \[\frac{dN}{dt}=c*a*R*N-d*N\]

saying that \(R\), the prey, grows in respect to a limited carrying capacity \(K\) (the first part of this equation hence is a logistic equation) and declines in respect to its death-rate \(b\) and the strength of influence of the predator \(N\).

The predator \(N\) grows linearly in respect to its per capita consumption \(c\) and declines in respect to its death-rate \(d\).

The following interactive model lets you test some of the reactions of this model to parameter changes.

The small inset in the right upper corner shows a state space or phase space plot. It depicts the number of predator against the number of prey. Different to the abstracted version, this version of a predator-prey system always homes in on a steady state, which shows in the spiral of the state space curve.


The Michaelis-Menten function

In the model above it is assumed that each predator consumes proportionally to the number of prey and that the predator-population also grows proportionally to this number. With large numbers of prey this does not seem realistic. If one would want to consider that both, the consumption rate as well as the growth rate of the predator-population will be limited even if there is prey in abundance, one could use the so called Michaelis-Menten function:

\(f(x)=\frac{b*x}{1+\gamma*x}\) whith \(b, \gamma > 0\)




If \(b=\gamma\) the function approaches the limit of 1. In the plot to the right \(b=1\) and \(\gamma=1\).

modeling

A Lotka-Volterra-model with this function could look as follows:

\[\frac{dR}{dt}=a*R*(1-\frac R K)- \frac{b*R*N}{1+\gamma*R}\] \[\frac{dN}{dt}=\frac{c*R*N}{1+\gamma*R}-d*N\]

With \(\gamma = 0\) the equations equal the original version.

The following interactive model lets you test some reactions of this model to parameter changes.