Modelling under uncertainties in data generates special definitions of stability, for instance, D-stability of the Jacobi matrix for a system of nonlinear ODEs, or other, stronger-than-Lyapunov, notions of matrix stability, which were suggested in a number of fields of application including theoretical population biology, mathematical ecology, macroeconomic models, chemical kinetics, etc.). Few more examples are: total stability, diagonal stability (dissipativeness in Volterra sense), diagonal quasi-dominance, sign stability. Strongly motivated by applications, each of those formal notions is pretty interpretable in the meaningful terms, yet quite few of them (excluding D-stability) posses mathematical characterization in a sufficiently general or visible form.
Therefore, given a variety of particular results the matrix theory can propose for a particular category of stability, a visible representation for the general 'topology' of mutual relations among the subsets of stable matrices in the space Mnxn(R), - what is called the Matrix Flower - serves both a framework to combine the findings available and a heuristic tool to promote further research. It stimulates studying general topological properties of particular subsets ('petals' of the Flower), such as to be open or connected; it helps searching for unknown characterizations, defining new notions of matrix stability that correspond to particular model requirements, and formulating sound conjectures on logical relations between particular 'petals'.
This approach will be illustrated by several stories related to models of multi-species community dynamics and biogeochemical cycles.