5,870 research outputs found
A delay differential model of ENSO variability: Parametric instability and the distribution of extremes
We consider a delay differential equation (DDE) model for El-Nino Southern
Oscillation (ENSO) variability. The model combines two key mechanisms that
participate in ENSO dynamics: delayed negative feedback and seasonal forcing.
We perform stability analyses of the model in the three-dimensional space of
its physically relevant parameters. Our results illustrate the role of these
three parameters: strength of seasonal forcing , atmosphere-ocean coupling
, and propagation period of oceanic waves across the Tropical
Pacific. Two regimes of variability, stable and unstable, are separated by a
sharp neutral curve in the plane at constant . The detailed
structure of the neutral curve becomes very irregular and possibly fractal,
while individual trajectories within the unstable region become highly complex
and possibly chaotic, as the atmosphere-ocean coupling increases. In
the unstable regime, spontaneous transitions occur in the mean ``temperature''
({\it i.e.}, thermocline depth), period, and extreme annual values, for purely
periodic, seasonal forcing. The model reproduces the Devil's bleachers
characterizing other ENSO models, such as nonlinear, coupled systems of partial
differential equations; some of the features of this behavior have been
documented in general circulation models, as well as in observations. We
expect, therefore, similar behavior in much more detailed and realistic models,
where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure
Hopf bifurcation in a gene regulatory network model: Molecular movement causes oscillations
Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper we analyse a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatio-temporal interactions between the protein and its mRNA in a 1-dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behaviour and this is the case for our model, shown initially via computational simulations. In order to investigate this behaviour more deeply, we next solve our system using Greens functions and then undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. This has implications for transcription factors such as p53, NF-B and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer
A delay recruitment model of the cardiovascular control system.
Copyright will be owned by Springer. We develop a nonlinear delay-differential equation for the human cardiovascular control system, and use it to explore blood pressure and heart rate variability under short-term baroreflex control. The model incorporates an intrinsically stable heart rate in the absence of nervous control, and features baroreflex influence on both heart rate and peripheral resistance. Analytical simplifications of the model allow a general investigation of the rôles played by gain and delay, and the effects of ageing.
Global Stability and Periodicity in a Glucose-Insulin Regulation Model with a Single Delay
A two-dimensional system of differential equations with delay modelling the
glucose-insulin interaction processes in the human body is considered.
Sufficient conditions are derived for the unique positive equilibrium in the
system to be globally asymptotically stable. They are given in terms of the
global attractivity of the fixed point in a limiting interval map. The
existence of slowly oscillating periodic solutions is shown in the case when
the equilibrium is unstable. The mathematical results are supported by
extensive numerical simulations. It is shown that typical behaviour in the
system is the convergence to either a stable periodic solution or to the unique
stable equilibrium. The coexistence of several periodic solutions together with
the stable equilibrium is demonstrated as a possibility.Comment: Accepted to Communications in Nonlinear Science and Numerical
Simulatio
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