Discretization of the Lotka–Volterra System and Asymptotic Focal and Prefocal Sets

We revisit the Kahan–Hirota–Kimura discretization of a quadratic vector field. The corresponding discrete system is generated by successive iterations of a birational map [Formula: see text]. We include a proof of a formula for the Jacobian of this map. In the following, we essentially focus on the case of the Lotka–Volterra system. We discuss the notion of focal points and prefocal lines of the map [Formula: see text] and of its inverse [Formula: see text]. We show that the map [Formula: see text] is the product of two involutions. The nature of the fixed points of [Formula: see text] is studied. We introduce the notion of asymptotic focal and prefocal sets. We further provide a new proof of the theorem of Sanz-Serna. We show that the mapping [Formula: see text] is integrable for [Formula: see text] and that it preserves a pencil of conics (generic hyperbolas). To conclude, we provide several numerical simulations for [Formula: see text].