Tipping Cycles

TL;DR

This paper investigates how feedback loops and sign structures in ecological models influence system stability, revealing complex combinatorial patterns among destabilising cycles in various ecological interactions.

Contribution

It introduces a detailed analysis of destabilising feedback loops in ecological models, expanding understanding beyond traditional eigenvalue criteria.

Findings

Sign structure significantly affects destabilising feedback contributions.

Rich combinatorial patterns emerge among destabilising cycle sets.

Analysis applies to predator-prey, mutualistic, and competitive systems.

Abstract

Ecological systems are studied using many different approaches and mathematical tools. One approach, based on the Jacobian of Lotka-Volterra type models, has been a staple of mathematical ecology for years, leading to many ideas such as on questions of system stability. Instability in such methods is determined by the presence of an eigenvalue of the community matrix lying in the right half plane. The coefficients of the characteristic polynomial derived from community matrices contain information related to the specific matrix elements that play a greater destabilising role. Yet the destabilising circuits, or cycles, constructed by multiplying these elements together, form only a subset of all the feedback loops comprising a given system. This paper looks at the destabilising feedback loops in predator-prey, mutualistic and competitive systems in terms of sets of the matrix elements to explore how sign structure affects how the elements contribute to instability. This leads to quite rich combinatorial structure among the destabilising cycle sets as set size grows within the coefficients of the characteristic polynomial.