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.

Appendix A

Let $\mathcal {A}_n$ be an n-by-n sign pattern.

Theorem 3

(Bodine et al. 2012, Theorem 2.3) If $\mathcal {A}_4$ is sign nonsingular, requires a negative trace and allows ${\mathbb {H}}_4$, then $\mathcal {A}_4$ requires ${\mathbb {H}}_4$.

Theorem 4

(Bodine et al. 2012, Theorem 3.3) If $\mathcal {A}_n$ allows ${\mathbb {H}}_n$ and has all diagonal entries negative, then any superpattern of $\mathcal {A}_n$ allows ${\mathbb {H}}_n$.

Theorem 5

(Bodine et al. 2012, Corollary 3.4) If $\mathcal {A}_n$ allows ${\mathbb {H}}_n$ and has all diagonal entries negative, then every $(n+m)$-by-$(n+m)$ superpattern of $\mathcal {A}_n \oplus -\mathcal {I}_m$ allows ${\mathbb {H}}_{n+m}$.

Theorem 6

(Bodine et al. 2012, Corollary 3.6) For $n\ge 3$, if every entry of $\mathcal {A}_n$ is negative, then $\mathcal {A}_n$ allows ${\mathbb {H}}_n$.