Biochemical and environmental modeling utilizes reaction networks to represent complex transformations, yet the standard Linkage Class Decomposition (LCD), which is based purely on visual connectivity, frequently fails to capture the algebraic properties that govern long-term system dynamics. This work introduces a structural classification based on the finest decompositions of chemical...
Polynomial dynamical systems arise in many applications (e.g., biochemistry, population dynamics) but are hard to analyze because they can display multistability, oscillations, and chaos. Mass-action systems, and in particular complex-balanced (toric) systems, are remarkably stable: they admit a unique attracting equilibrium and rule out oscillations and chaos. We study the set of rate...
I plan to give a rigorous proof that the Willamowski-Roessler reaction network $$X \leftrightarrow 2X, \quad X+Y \leftrightarrow 2Y, \quad Y \leftrightarrow 0, \quad X+Z \leftrightarrow 0, \quad Z \leftrightarrow 2Z$$ has chaotic dynamics.
Recent work has revealed that biochemical networks with bifunctional enzymes can display remarkably rich dynamics, including ultrasensitivity, switch-like responses, concentration robustness, and even species exhaustion. This talk presents a dynamical systems analysis of such networks, shedding light on the subtle architectural differences that produce these vast differences in functional...
I'll outline some recent results on the geometry of the positive equilibrium sets of mass action networks. We obtain useful parameterisations of equilibria and bounds on the number of (positive, nondegenerate) equilibria a mass action network can admit on any stoichiometric class. The techniques also lead to new approaches to studying bifurcations in mass action networks. Sometimes, via...
I will discuss two types of mathematical models of biochemical reaction networks: (i) deterministic models described by reaction-rate equations, i.e., ordinary differential equations (ODEs) for the concentrations of the involved biochemical species~\cite{ref1,ref2}, and (ii) stochastic models described by the Gillespie stochastic simulation algorithm, which provides more detailed information...
Structural identifiability answers a theoretical question about which parameter combinations of a mathematical model are uniquely recovered from ideal, noise-free input–output data. Identifiability has been long studied on linear compartment models, and is closely related to the algebraic structure of the input-output equation obtained from the model. In this work, we investigate how...
Systems of $N$ first-order autonomous ordinary-differential equations with polynomials of at most degree $n$ on the right-hand side, called $N$-dimensional $n$-degree polynomial dynamical systems (DSs), can display a rich set of solutions whose complexity increases with $N$ and $n$. When $N \ge 3$ and $n \ge 2$, polynomial DSs can exhibit chaos — aperiodic long-term behavior that is sensitive...
Biochemical reaction networks provide a powerful and flexible mathematical framework for modelling the dynamics of complex biochemical systems. Such models, which tend to be nonlinear and have unknown parameter values, can exhibit diverse and interesting behavior, including bistability, oscillations, and even chaos. Existing methods for analyzing such models draw on not only dynamical systems...
