Instability and Oscillations in Reaction Networks

Abstract: In the past two decades, much mathematical research on reaction networks has focused on multistationarity, both under mass-action and more general kinetics. Multistationarity typically arises in parameter space when a stable steady state loses stability through a real-zero eigenvalue crossing. More recently, attention has turned also to stability loss via purely imaginary eigenvalues (Hopf bifurcation), which is associated with periodic oscillations.

In this talk, based on established results from dynamical systems theory developed in late 1970s and 1980s, I will first clarify that for monostationary networks, in essence, any loss of stability necessarily leads to oscillations. I will then present sufficient conditions for oscillations expressed as algebraic criteria (for mass-action kinetics) and as minimal network motifs (for general kinetics). In turn, excluding Hopf bifurcations in monostationary networks with a stable steady state is equivalent to 'universal’ stability, meaning that the unique steady state is locally stable for all parameter choices. I will conclude with a conjecture characterizing universal stability for reaction networks with general kinetics. This is ongoing joint work with Peter F. Stadler and Alex Blokhuis.

chemical biologychemical kineticsalgebraic geometrydynamical systemsprobability

Audience: researchers in the topic

( video )

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Seminar on the Mathematics of Reaction Networks

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This seminar series focuses on progress in mathematical theory for the study of reaction networks, mainly in biology and chemistry. The scope is broad and accommodates works arising from dynamical systems, stochastics, algebra, topology and beyond.

We aim at providing a common forum for sharing knowledge and encouraging discussion across subfields. In particular we aim at facilitating interactions between junior and established researchers. These considerations will be represented in the choice of invited speakers and we will strive to create an excellent, exciting and diverse schedule.

The seminar runs twice a month, typically on the 2nd and 4th Thursday of the month, at 17:00 Brussels time (observe that this webpage shows the schedule in your current time zone). Each session consists of two 25-minute talks followed by 5-minute questions. After the two talks, longer discussions will take place for those interested. To this end, we will use breakout rooms. For this to work well, you need to have the latest version of Zoom installed (version 5.3.0 or higher), and use the desktop client or mobile app (not supported on ChromeOS).

We look forward hearing about new work and meeting many of you over zoom! Many of the talks are recorded; to see the recording, from Past Talks, open details of the listed talk for a video link.

The organizers.

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