05C50 Online Seminar: Pauline van den Driessche

Biological systems often give rise to systems of first order ordinary differential equations (ODEs). Linearization then yields a system x' = Ax, where A is the community matrix. By contrast, mechanical systems often give rise to a second order ODE system x'' = Ax' + Bx, which is equivalent to a first order system with coefficient matrix with block form A B I 0. In cases for which the signs rather than the magnitudes of matrix entries are known, the matrices become sign patterns with entries { +, -, 0 }. What can be determined about the behaviour of a system governed by such sign pattern matrices? This general question is addressed by developing results on sign patterns and their associated digraphs. Some answers in special cases are given that determine stability and inertia properties important for the underlying systems.