Systems of N equations in N unknowns are ubiquitous in mathematical
modeling. These systems, often nonlinear, are used to identify equilibria of
dynamical systems in ecology, genomics, control, and many other areas.
Structured systems, where the variables that are allowed to appear in each
equation are pre-specified, are especially common. For modeling purposes, there
is a great interest in determining circumstances under which physical solutions
exist, even if the coefficients in the model equations are only approximately
known.
The structure of a system of equations can be described by a directed graph
G that reflects the dependence of one variable on another, and we can
consider the family F(G) of systems that respect G.
We define a solution X of F(X)=0 to be robust if for each
continuous Fβ sufficiently close to F, a solution Xβ exists.
Robust solutions are those that are expected to be found in real systems. There
is a useful concept in graph theory called "cycle-coverable". We show that if
G is cycle-coverable, then for "almost every" FβF(G) in the
sense of prevalence, every solution is robust. Conversely, when G fails to
be cycle-coverable, each system FβF(G) has no robust solutions.
Failure to be cycle-coverable happens precisely when there is a configuration
of nodes that we call a "bottleneck," a criterion that can be verified from the
graph. A "bottleneck" is a direct extension of what ecologists call the
Competitive Exclusion Principle, but we apply it to all structured systems