The equations of nature and the nature of equations

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 ${\mathcal{F}(G)}$ of systems that respect ${G}$. We define a solution ${X}$ of ${F(X) = 0}$ to be robust if for each continuous ${F^*}$ sufficiently close to ${F}$, a solution ${X^*}$ 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" ${F\in\mathcal{F}(G)}$ in the sense of prevalence, every solution is robust. Conversely, when ${G}$ fails to be cycle-coverable, each system ${F\in\mathcal{F}(G)}$ 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

When the Best Pandemic Models are the Simplest

As the coronavirus pandemic spreads across the globe, people are debating policies to mitigate its severity. Many complex, highly detailed models have been developed to help policy setters make better decisions. However, the basis of these models is unlikely to be understood by non-experts. We describe the advantages of simple models for COVID-19. We say a model is “simple” if its only parameter is the rate of contact between people in the population. This contact rate can vary over time, depending on choices by policy setters. Such models can be understood by a broad audience, and thus can be helpful in explaining the policy decisions to the public. They can be used to evaluate the outcomes of different policies. However, simple models have a disadvantage when dealing with inhomogeneous populations. To augment the power of a simple model to evaluate complicated situations, we add what we call “satellite” equations that do not change the original model. For example, with the help of a satellite equation, one could know what his/her chance is of remaining uninfected through the end of an epidemic. Satellite equations can model the effects of the epidemic on high-risk individuals, death rates, and nursing homes and other isolated populations. To compare simple models with complex models, we introduce our “slightly complex” Model J. We find the conclusions of simple and complex models can be quite similar. However, for each added complexity, a modeler may have to choose additional parameter values describing who will infect whom under what conditions, choices for which there is often little rationale but that can have big impacts on predictions. Our simulations suggest that the added complexity offers little predictive advantage

Robust steady states in ecosystems with symmetries

Steady states of dynamical systems, whether stable or unstable, are critical for understanding future evolution. Robust steady states, ones that persist under small changes in the model parameters, are desired when modelling ecological systems, where it is common for accurate and detailed information on functional form and parameters to be unavailable. Previous work by Jahedi et al. [Robustness of solutions of almost every system of equations, SIAM J. Appl. Math. 82(5) (2022), pp. 1791–1807; Structured systems of nonlinear equations, SIAM J. Appl. Math. 83(4) (2023), pp. 1696–1716.] has established criteria to imply the prevalence of robust steady states for systems with minimal predetermined structure, including conventional structured systems. We review that work and extend it by allowing symmetries in the system structure, which present added obstructions to robustness