Exercise: Predator-prey modelΒΆ

Suppose the populations of rabbits (denoted by $r(t)$) and foxes (denoted by $x(t)$) at time $t$ in a jungle are modeled by the ODE system $$ \begin{aligned} \frac{dr}{dt} & = \alpha r - \beta r x \\ \frac{dx}{dt} & = \delta r x - \gamma x \end{aligned} $$ where $\alpha = 1.1$, $\beta=0.4$, $\delta= 0.1$, and $\gamma=0.1$.

Task 1: Given initial conditions $r(0) = 5$ and $x(0) = 2,$ solve for $r(t)$ and $x(t)$ and plot the solution for $0 \le t \le 70$.

Task 2: The phase plot of the solution consists of points $(x(t), r(t))$ for various $t$ values. Prepare a figure (phase portrait) with phase plots of, say, 10 solutions, one each for randomly chosen initial values $r(0)$ and $x(0)$ between 1 and 9.

Task 3: This system has two equilibria. Solve for them and mark them in your phase portrait.