Algebraic independence of the solutions of the classical Lotka-Volterra system

Abstract

Let $(x_1,y_1),\dots ,(x_n,y_n)$ be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system$x^{\prime }=axy+bx{y}^{\prime }=cxy+dy$ where $a,b,c,d\in C\setminus \left\{0\right\}$. We show that if d and b are linearly independent over $Q$, then the solutions are algebraically independent over $C$, that is $tr.de{g}_{C}C(x_1,y_1,\dots ,x_n,y_n)=2n$. As a main part of the proof, we show that the set defined by the system in universal differential fields, with d and b linearly independent over $Q$, is strongly minimal and geometrically trivial. Our techniques also allows us to obtain partial results for some of the more general 2d-Lotka-Volterra system.