Dynamics of a Lotka-Volterra map

Given the plane triangle with vertices (0, 0), (0, 4) and (4, 0) and the transformation F : (x, y) → (x(4 - x - y), xy) introduced by A. N. Sharkovskiǐ, we prove the existence of the following objects: a unique invariant curve of spiral type, a periodic trajectory of period 4 (given explicitly) and a periodic trajectory of period 5 (described approximately). Also, we give a decomposition of the triangle which helps to understand the global dynamics of this discrete system which is linked with the behavior of the Schrödinger equation.