Most of this work is in collaboration with Sana Jahedi and Tim Sauer. The final part on Lyapunov functions is joint with Naghmen Akhaven. It applies the work with Jahedi and Sauer.
Systems of $M$ 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.
More specifically, a structured system of equations $F(x)=c$ where $F:\mathbb R^N\to\mathbb R^M$ is a system of $M$ equations in which it is specified which of the $N$ variables are allowed to appear in each equation. Our goal in this article is to describe the properties that will hold for almost every $F$ that satisfies the structure, and in particular the global properties of solutions for structured systems of $C^\infty$ functions.
An important question when modeling natural systems is whether solutions are robust to small changes in the model. To address this question, we say $F$ is "flat" if there exists an integer $k$ such that for almost every $x_0$, the set $F^{-1}F(x_0)$ of solutions $x$ of $F(x)=F(x_0)$ is a $k$-dimensional manifold. When $k=0$, the solutions are isolated points. Systems of polynomials are examples of flat functions, but different polynomials may have different properties. There are $C^\infty$ functions $F$ that are not flat, even in one dimension, when $N=M=1$. The set of $F$ that have a given structure is a vector space. We state conditions on vector spaces of $C^\infty$ functions that imply that "almost every" (in the sense of prevalence) $F$ in the vector space is $k$-flat for some $k$, and we show that if $k=M-N$ almost every $F$ has the property that almost every $x$ is a robust solution of $F(x)=c$ for some $c$. Then $x$ is robust to small changes in $c$ and $F$.
We also describe the "bottlenecks" that are the obstructions to robustness in structured systems. In particular for fragile (i.e., non- robust) structured systems, we characterize the unique "minimax bottleneck", and describe how the model structure can be changed to make it robust.
Lyapunov functions. Motivated by the above theory of bottlenecks, we consider an $n$-dimension competitive Lotka-Volterra system of form \begin{equation*} x_i' = \frac{d}{dt}{x_i} = x_i(c_i +\sum_{j=1}^{d} s_{ij}x_j),\quad \text{where } i=1,\ldots,d. \end{equation*} Let $S$ be the $d\times d$ matrix $(s_{ij})$. When the system satisfies our "trophic" condition, all solutions are bounded. When there is a bottleneck indicating that $k$ species must die out, we construct $k$ distinct Lyapunov functions, each of which shows that a different species must die out exponentially fast. Each Lyapunov function is based on a different null vector of $S$. The $k$ species that die out are in the bottleneck.