You initialize a non-zero value for x, y. You take alternative steps to compute the gradient for x and y separately based on that equations. In each step, you update x (or y) and xy. You continue the program and plot x, y and xy for each iteration step.

Conceptually, both player will move in the opposite direction of each other because it is a min-max game. There is no damping effect on the learning rate and therefore the solution oscillates.