  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.
