The number of equilibrium points of perturbed nonlinear positive dynamical systems

Abstract

The number of equilibrium points of a dynamical system dictates important qualitative properties, such as the ability of the system to store different memory states, and may be significantly affected by state-dependent perturbations. In this paper, we develop a methodology based on tools from degree theory to determine whether the number of equilibrium points in a positive dynamical system changes due to structured state-dependent perturbations. Positive dynamical systems are particularly well suited to describe biological systems where the states are always positive. We prove two main theorems that utilize the determinant of the system’s Jacobian to find algebraic conditions on the parameters determining whether the number of equilibrium points is guaranteed either to change or to remain the same when a nominal system is compared to its perturbed counterpart. We demonstrate the application of the theoretical results to genetic networks where state-dependent perturbations arise due to disturbances in cellular resources. These disturbances constitute a major problem for predicting the behavior of genetic networks. Our results determine whether such perturbations change a genetic network’s number of steady states. The framework and results presented can be applied to a broad class of nonlinear dynamical systems beyond genetic networks.