The plots below show the number of predators and prey as predicted by the Lotka-Volterra predator-prey model: $$\begin{align} \frac{\textrm{dN}}{\textrm{dt}} &= a N - b N P \\ \frac{\textrm{dP}}{\textrm{dt}} &= c N P - d P. \end{align}$$ \(a\) and \(c\) are the birth rates with \(b\) and \(d\) the death rates of the prey and predators respectively.

The first plot shows the number of predators and prey as time progresses. Change the variables and see how the numbers change. Can you make either species die out?

The second plot shows the phase portrait of the system. The numbers of prey are plotted along the horizontal, with the predators along the vertical. Where is the equilibrium point?