Persistence, permanence and global stability for an n-dimensional Nicholson system

TL;DR

This paper investigates the global stability and persistence of solutions in an n-dimensional Nicholson system with delays, providing explicit bounds and criteria for extinction, persistence, and stability, including cases with reducible community matrices.

Contribution

It offers new criteria for global stability and persistence in delayed Nicholson systems, including non-cooperative and reducible cases, with explicit bounds and improved stability conditions.

Findings

Population becomes extinct if spectral bound is non-positive.

Total population persists if spectral bound is positive.

Existence and global stability of a unique positive equilibrium.

Abstract

For a Nicholson's blowflies system with patch structure and multiple discrete delays, we analyze several features of the global asymptotic behavior of its solutions. It is shown that if the spectral bound of the community matrix is non-positive, then the population becomes extinct on each patch, whereas the total population uniformly persists if the spectral bound is positive. Explicit uniform lower and upper bounds for the asymptotic behavior of solutions are also given. When the population uniformly persists, the existence of a unique positive equilibrium is established, as well as a sharp criterion for its absolute global asymptotic stability, improving results in the recent literature. While our system is not cooperative, several sharp threshold-type results about its dynamics are proven, even when the community matrix is reducible, a case usually not treated in the literature.