(11C) Rabbits VS Foxes Simulation

[Thomas Klemm]

(07-29-2018 06:46 AM)Gamo Wrote:  The system can be approximated by a pair of nonlinear, first-order differential equations.

From the code I assume they are:
\[ \begin{eqnarray}
\dot{r}&=2r-\alpha fr \\
\dot{f}&=\alpha fr-f
\end{eqnarray} \]

This looks very much like the Lotka–Volterra equations:
\[ \begin{aligned}
\frac {dx}{dt}&=\alpha x-\beta xy\\
\frac {dy}{dt}&=\delta xy-\gamma y
\end{aligned} \]

What is reason for the peculiar factor 2 in \(\dot{r}=2r-\alpha fr\)?

At a stationary point the derivative vanishes: \(\dot{r}=\dot{f}=0\).
This leads to:
\begin{aligned}
r(2-\alpha f)=0&\Rightarrow &f=\frac{2}{\alpha}\\
f(\alpha r-1)=0&\Rightarrow &r=\frac{1}{\alpha}
\end{aligned}

Thus for the value given \(\alpha=0.01\) we end up with:
\begin{aligned}
r=100\\
f=200
\end{aligned}

This looks like a lot of foxes compared to the rabbits.

Cheers
Thomas