Stronger-than-Lyapunov notions of matrix stability, or how “flowers” help solve problems in mathematical ecology

Persistent patterns of interactions in a multi-component system (e.g., intra- and inter-species relations in a community of n interacting species) may imply a number of formalizations as special, stronger-than-Lyapunov, notions of matrix stability, like D-stability, qualitative stability, Volterra-Lyapunov stability, and others. A variety of these notions, each having a certain motivation with regard to uncertainties inherent in model applications, constitute a hierarchical topology, sometimes very intricate and not yet well-understood, in a formal space of real n × n-matrices. As visible forms of this hierarchy, Matrix Flowers are suggested where 'petals' correspond to subsets of particular stability kinds, whose visible inclusion/intersection represent logical implication/junction. The Flowers are constructed under a few simple conventions, and, in the absence of ready characterizations, to draw a 'petal' often poses a challenging mathematical problem, whose solution may reveal a new biological knowledge of a general nature. Particular 'petals' concern the topics of strong and weak interactions in a community, 'key' species in its structure, diffusion instability in its spatial dynamics, where the flower plays a heuristic role in formulating a new problem or/and stimulating a new application.