The distance-to-bifurcation problem in non-negative dynamical systems with kinetic realizations

Consider a dynamical system ẋ = f(x, k₀) whose vector field is parameterized by real parameters k₀. The distance-to-bifurcation problem seeks the smallest parameter variation γ = |k - k₀| that results in a bifurcation of the original system´s phase portrait. Prior work on this problem used numerical methods to search for the minimum γ, but these methods were computationally demanding and only guaranteed locally optimal solutions. This paper recasts the minimum distance-to-bifurcation problem as a sum-of-squares (SOS) relaxation for non-negative dynamical systems that have kinetic realizations. The class of systems with kinetic realizations is large enough to characterize a wide range of real world applications and the use of such realizations allows one to find explicit parameterizations of the system Jacobian as a rational function of system parameters. This parameterization of the Jacobian was originally proposed for chemical reaction networks and its value is that it greatly simplifies the distance-to-bifurcation problem by removing the need to keep the system´s equilibria as decision variables in the distance-to-bifurcation problem. The proposed approach is illustrated on an food web in aquatic eco-systems. The example demonstrates that our approach is able to identify how a coordinated set of parameter variations may result in a smaller distance-to-bifurcation than predicted by competing computational tools.