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

TL;DR

This paper proves that solutions of the classical Lotka-Volterra system are algebraically independent over complex numbers when certain parameters are linearly independent over rationals, using model-theoretic methods.

Contribution

It establishes algebraic independence of solutions for the classical Lotka-Volterra system under specific parameter conditions, employing differential algebra and model theory.

Findings

Solutions are algebraically independent over a9 when parameters are linearly independent over a9.

The solution set is strongly minimal and geometrically trivial in universal differential fields.

Partial results are obtained for generalized Lotka-Volterra systems.

Abstract

Let $(x_1,y_1),\ldots,(x_n,y_n)$ be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system \begin{equation}\notag \begin{split} x'&= axy + bx\\ y'&= cxy + dy, \end{split} \end{equation} where $a,b,c,d\in\mathbb{C}\setminus\{0\}$. We show that if $d$ and $b$ are linearly independent over $\mathbb{Q}$, then the solutions are algebraically independent over $\mathbb{C}$, that is $tr.deg_{\mathbb{C}}\mathbb{C}(x_1,y_1,\ldots,x_n,y_n)=2n$. 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 $\mathbb{Q}$, 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.