Stronger-than-Lyapunov notions of matrix stability, or how «flowers» help solve problems in mathematical ecology amg

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.