The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

This paper deals with an arbitrary-order autocatalysis model with delayed feedback subject to Neumann boundary conditions. We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. By analyzing the distribution of eigenvalues, the existence of Hopf bifurcation is obtained. Then we derive an algorithm for determining the direction and stability of the bifurcation by computing the normal form on the center manifold. Moreover, some numerical simulations are given to illustrate the analytical results. Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the spatial inhomogeneous periodic solutions. In addition, the delayed feedback also makes the unstable equilibrium become stable under certain conditions

Gene expression time delays & Turing pattern formation systems

The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this paper, we investigate the effects of incorporating gene expression time delays into a one-dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. While oscillatory behaviour is rare, we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning

Design of novel chemical oscillators

Designing new chemical and/or electrochemical oscillatory systems is an important area in nonlinear chemical dynamics. We successfully designed two new chemical oscillators, the pyrocatechol-bromate-sulfuric acid and aminophenol-bromate-sulfuric acid systems. Both chemical systems exhibit a very rich oscillatory behavior and we obtained their phase diagrams in uncatalyzed and ferroin-catalyzed systems. Phase diagrams in the bromate - pyrocatechol - sulfuric acid concentration space illustrate that the observed chemical oscillations strongly depend on the ratio of [bromate]/[pyrocatechol] rather than their actual concentrations. Also, in both uncatalyzed and catalyzed systems kinetics and mechanisms have been investigated. In mechanistic studies, we have tried to determine intermediate species with various analytical techniques such as: FTIR., 1H NMR, 13C NMR, Mass spectroscopy, TLC, Elemental Analysis, etc. The aminophenol system is found to be a photo-mediated oscillatory system which does not exhibit spontaneous oscillations in the absence of light. Investigation of the role of illumination, in particular the wavelength of light responsible for the oscillatory behaviour, in the aminophenol-acidic bromate system has been carried out. Study shows that the long induction time in this photochemical oscillator has an exponential dependence on the light intensity. On the other hand, the pyrocatechol system is a photosensitive oscillatory system which light is capable of quenching and inducing oscillation in the system. Furthermore, chemical wave activities in the ferroin-catalyzed pyrocatechol system have been investigated, in 2-dimensional (2-D) beads and homogeneous systems, and in a 1-dimensional (1-D) medium. In the 1-D pyrocatechol system, we observed various types of pulse instabilities such as: breathing, propagation failure, merging pulses, and packing phenomena. In the homogeneous 2-D medium, the pyrocatechol system exhibited two stages of wave activity. Spontaneous transitions to complex spatiotemporal patterns, as a result of anomalous dispersions, have also been observed. In the beads pyrocatechol system, variation of wave propagation speeds and spiral tip trajectories versus four different factors including concentrations of bromate, acid, and ferroin concentration and beads mass have been characterized. Wave studies in the 1-D aminophenol system showed different types of pulse instabilities as well, where global breathing phenomena lasted for more than 48 hours in most cases. In 2-D reaction diffusion media in bead, the ferroin-catalyzed aminophenol system is capable of supporting slow waves even in the absence of light

Criticality control of insulin release: a novel alternative for detailed modelling of insulin secretion

Representation of the dynamic components of the glucose-insulin system has posed a major challenge in the field of physiological modelling for the past six decades. Early stages in development focused on a descriptive approach where mathematical complexity was usually compromised, causing misrepresentation of the system. The use of such models allowed the study of the system under extremely specific circumstances, and although it aided in the development of devices to manage type 1 Diabetes, it allowed little opportunity to study the system under normal circumstances. In the early 1990's, evidence of insulin oscillations in the system motivated a more detailed approach, with the analysis and representation of the molecular interactions involved. This work presents a novel modelling methodology which aligns with the aforementioned focus. The methodology focuses on representing a network of cells represented by systems which are inherently non-linear. In order to develop it, a review of reaction-diffusion systems was conducted, where four models were chosen as candidates to represent the building blocks for the resulting model. These were chosen due to their biological relevance and capability of generating a wide range of dynamics using a relatively simple formulation. The resulting model is comprised of a set of sixteen coupled oscillators organized into four clusters. It successfully incorporates characteristics that have been observed in the glucose-insulin system, such as nonlinear dynamics, coupling, and response to external influences. In order to tune the system and achieve multiple stable states, a biologically inspired control method (Rate Control of Chaos) was implemented. The overall structure will allow the study of the mechanisms that keep the system from reaching a chaotic state (diabetes), based on the property of self-organized criticality. The results show that the chosen candidate models are capable of representing the desired structure whilst maintaining the desired dynamics; achieved through the variation of system parameters and initial conditions. They are responsive to the controller and are tolerant to modifications in the system such as the increment of the control signal and coupling strength. The behaviour observed differs among the models and was instrumental in assessing biological relevance