5 research outputs found
Characteristic, completion or matching timescales? An analysis of temporary boundaries in enzyme kinetics
Scaling analysis exploiting timescale separation has been one of the most
important techniques in the quantitative analysis of nonlinear dynamical
systems in mathematical and theoretical biology. In the case of enzyme
catalyzed reactions, it is often overlooked that the characteristic timescales
used for the scaling the rate equations are not ideal for determining when
concentrations and reaction rates reach their maximum values. In this work, we
first illustrate this point by considering the classic example of the
single-enzyme, single-substrate Michaelis--Menten reaction mechanism. We then
extend this analysis to a more complicated reaction mechanism, the auxiliary
enzyme reaction, in which a substrate is converted to product in two sequential
enzyme-catalyzed reactions. In this case, depending on the ordering of the
relevant timescales, several dynamic regimes can emerge. In addition to the
characteristic timescales for these regimes, we derive matching timescales that
determine (approximately) when the transitions from initial fast transient to
steady-state kinetics occurs. The approach presented here is applicable to a
wide range of singular perturbation problems in nonlinear dynamical systems.Comment: 35 pages, 11 figure
Complex oscillatory patterns near singular Hopf bifurcation in a two time-scale ecosystem
We consider an ecological model consisting of two species of predators
competing for their common prey with explicit interference competition. With a
proper rescaling, the model is portrayed as a singularly perturbed system with
one-fast (prey dynamics) and two-slow variables (dynamics of the predators).
The model exhibits variety of rich and interesting dynamics, including, but not
limited to mixed mode oscillations (MMOs), featuring concatenation of small and
large amplitude oscillations, relaxation oscillations and bistability between a
semi-trivial equilibrium state and a coexistence oscillatory state. Existence
of co-dimenison two bifurcations such as fold-Hopf and generalized Hopf
bifurcations make the system further intriguing. More interestingly, in a
neighborhood of {\emph{singular Hopf}} bifurcation, long lasting transient
dynamics in form of chaotic MMOs or relaxation oscillations are observed as the
system approaches the periodic attractor born out of supercritical Hopf
bifurcation or a semi-trivial equilibrium state respectively. The transient
dynamics could persist for hundreds or thousands of generations before the
ecosystem experiences a regime shift. The time series of population cycles with
different types of irregular oscillations arising in this model stem from a
biological realistic feature, namely, by the variation in the intraspecific
competition amongst the predators. To explain these oscillations, we use
bifurcation analysis and methods from {\emph{geometric singular perturbation
theory}}
Spike-adding in a canonical three time scale model: superslow explosion & folded-saddle canards
International audienceWe examine the origin of complex bursting oscillations in a phenomenological ordinary differential equation model with three time scales. We show that bursting solutions in this model arise from a Hopf bifurcation followed by a sequence of spike-adding transitions, in a manner reminiscent of spike- adding transitions previously observed in systems with two time scales. However, the details of the process can be much more complex in this three-time-scale context than in two-time-scale systems. In particular, we find that spike-adding can involve canard explosions occurring on two different time scales and is associated with passage near a folded-saddle singularity. We show that the character of the bursting and the form of spike-adding transitions that occur depend on the geometry of certain singular limit systems, specifically the relative positions of the critical and superslow manifolds. We also show that, unlike the case of spike-adding in two-time-scale systems, the onset of a new spike in our model is not typically associated with a local maximum in the period of the bursting oscillation