title: Using sign patterns to detect the possibility of periodicity in biological systems.
abstract: Many models in the physical and life sciences, formulated as dynamical systems, exhibit a positive steady state, with its local qualitative behavior determined by the eigenvalues of its Jacobian matrix. Our interest lies in detecting if this steady state is linearly stable or if the system has periodic solutions arising from a Hopf bifurcation. We address this by considering the sign pattern of the Jacobian matrix and its set of allowed refined inertias. The refined inertia of a matrix, which is an extension of the classical matrix inertia, is a property of its eigenvalues. A Hopf bifurcation, leading to periodic solutions, may be possible if the sign pattern of the Jacobian matrix allows a specific set of refined inertias. For most systems, we also need to consider magnitude restrictions on the entries of the Jacobian matrix that are a consequence of the particular biological model. The usefulness of sign pattern analysis to detect linear stability or the possibility of periodicity is illustrated with several biological examples, including metabolic-genetic circuits, biochemical reaction networks, predator-prey and competition systems.
authors.affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 2Y2, Canada. garrettc@uvic.ca.
authors.forename: G J
authors.initials: GJ
authors.lastname: Culos
authors.affiliation: Department of Computer Science, University of Victoria, Victoria, BC, V8W 2Y2, Canada. dolesky@cs.uvic.ca.
authors.forename: D D
authors.initials: DD
authors.lastname: Olesky
authors.affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 2Y2, Canada. pvdd@math.uvic.ca.
authors.forename: P
authors.initials: P
authors.lastname: van den Driessche
mesh_terms: D000818:Animals; D053898:Biosynthetic Pathways; D053263:Gene Regulatory Networks; D016014:Linear Models; D055641:Mathematical Concepts; D053858:Metabolic Networks and Pathways; D008954:Models, Biological; D010507:Periodicity; D011235:Predatory Behavior
keywords: Goodwin model; Lotka–Volterra system; Metabolic-genetic circuit; Periodic solution; Refined inertia; Sign pattern