Variational discretization for the planar Lotka–Volterra equations in the Birkhoffian sense

In this paper, we derive the variational characterization of the planar Lotka–Volterra equations in the Birkhoffian sense and ulteriorly construct variational integrators for the group of equations. The planar Lotka–Volterra equations turn out to admit a Birkhoffian representation and consequently can be discretized according to the discrete Birkhoffian equations. By means of the transformation theory of the Birkhoffian equations if necessary, efficient variational integrators of the Lotka–Volterra equations can be obtained. These variational integrators, compared with traditional difference schemes as well as Poisson integrators, have better numerical performance in terms of stability, accuracy and preservation of conserved quantities, demonstrated by numerical results.