Consider the Lotka-Volterra predator
(*y*) prey (*x*) model:

Solution curves for various initial populations are plotted below:

(*x*(0), *y*(0)) = (5.25, 7.5), (5.5, 7), (5.75, 6.5), (6, 6), (6.25, 5.5), (6.5, 5), (6.75, 4.5)

Another way to visualize the solution
curves is plotting the trajectories in the phase plane. This is
called the *phase portrait*.

Notice the *critical point* (5,8).
The constant functions *x*(*t*) = 5, *y*(*t*) = 8
form a solution to the system.

Images generated by sage.