# Math Insight

### Applet: Lotka-Volterra model, with phase plane, functions of time and population display

Illustration of the solution to the predator-prey system \begin{align*} \diff{r}{t} &= \alpha r - \beta r p\\ \diff{p}{t} &= - \gamma p + \delta r p\\ r(t_0) &= r_0\\ p(t_0) &= p_0 \end{align*} for the population sizes $r(t)$ of prey and $p(t)$ of predators at time $t$. The left panel shows plots of the solutions $r(t)$ (blue curve) and $p(t)$ (red curve) as functions of time. The right panel shows a phase plane with a cyan curve illustrating the trajectory of the point $(r(t),p(t))$ as $t$ ranges from $t_0 \le t \le t_f$. The initial conditions $r_0$ and $p_0$ can be changed by dragging the blue and red points on the line $t=t_0$, by dragging the cyan point in the phase plane, or by typing values in the corresponding boxes. Green points on the blue, red, and cyan curves illustrate particular values of $r(t)$ and $p(t)$ for the value of $t$ shown on the green slider. You can change $t$ by dragging the green point on the slider, the green points on the blue and red curves, or by clicking the play button (triangle) in the lower left corner of one of the panels, which starts an animation where $t$ increases steadily. The values of $r(t)$ and $p(t)$ for the value of $t$ on the green slider are displayed at the top. If you click the “population display” checkbox, the population sizes are also illustrated by the number (capped at 10,000) of blue points (prey individuals) and red diamonds (predator individuals) overlaid on the phase plane. You can uncheck the “phase plane” checkbox to remove the phase plane from the right panel. The Greek letters $\alpha$, $\beta$, $\gamma$, and $\delta$ are parameters that can be changed by typing values in the boxes. The maximum values of $r$ and $p$ that are shown in the plots as well as the range of times $t_0 \le t \le t_f$ can be changed by typing values in their corresponding boxes.