# Sign of the solution to a non-cooperative system

## TL;DR

This paper extends previous work to determine the sign of solutions in non-cooperative systems of arbitrary size near the principal eigenvalue, combining recent theoretical results and methods from a PhD thesis.

## Contribution

It generalizes the sign determination of solutions from 2x2 systems to n x n systems near the principal eigenvalue, using a novel combination of existing results and new methods.

## Key findings

- Sign of solutions can be characterized near the lowest principal eigenvalue.
- Extension from 2x2 to n x n systems achieved.
- Method provides a way to analyze solution signs in complex systems.

## Abstract

Combining the results of a recent paper by Fleckinger-Hernandez-deTh{\'e}lin [14] for a non cooperative $2\times2$ system with the method of PhD Thesis of MH Lecureux we compute the sign of the solutions of a $n\times n$ non-cooperative systems when the parameter varies near the lowest principal eigenvalue of the system.