State transition graph and stability of singular equilibria for piecewise linear biological models

Many biological networks such as gene regulatory networks and neural networks can be modeled by piecewise linear switching systems of differential equations. For these biological network models, it is very difficult to describe the dynamics of the governing system near singular equilibria as they belong to the discontinuity set of the system. In this paper, using the concept of Filippov solutions, with the help of the state transition graph, and employing a generalized Lyapunov method, we establish some stability results for singular equilibria of a class of piecewise linear biological models. Our results generalize many existing ones. In particular, we present an affirmative answer to Conjecture 5.4 proposed in [R. Casey, H. de Jong, J.-L. Gouzé, Piecewise-linear models of genetic regulatory networks: Equilibria and their stability, Journal of Mathematical Biology 52 (1) (2006) 27–56].