Abstract for:Robust Stability Measures in Ecology via Pseudospectra

When try to assess behavior of the dynamical systems, like empirical food webs, modelled by generalized Lotka-Volterra equations, it is important to discuss the corresponding amplification and timescales of transient instability. Commonly, to derive stability, eigenvalues are being used, since they provide answers to (local) asymptotic stability. However, ecological stability is not the same as mathematical asymptotic stability, so, considering eigenvalues only, can be misleading! In order to find a robust measure, that incorporates uncertainty level of empirical data and amplification-timescale frame in the stability analysis, one should use the pseudospectra, instead. As a result, more realistic notion of stability will be achieved.

In this presentation, after a short preliminaries, we will focus on linear time invariant dynamical systems and certain matrix characteristics that can explain properties of the evolution function (amplification envelope), and thus explain global properties of linear system or local properties of a nonlinear one: resilience (by matrix eigenvalues, i.e. matrix spectrum), transient behavior (by matrix pseudospectrum) and reactivity (by logarithmic matrix norm, or matrix measure). Corresponding stability indicators will be illustrated on the example of a soil food web, and an efficient numerical algorithm for computation of stability indicators, which allows possible applications to high resolution food webs, will be presented.