819 research outputs found
Global dynamics of a novel delayed logistic equation arising from cell biology
The delayed logistic equation (also known as Hutchinson's equation or
Wright's equation) was originally introduced to explain oscillatory phenomena
in ecological dynamics. While it motivated the development of a large number of
mathematical tools in the study of nonlinear delay differential equations, it
also received criticism from modellers because of the lack of a mechanistic
biological derivation and interpretation. Here we propose a new delayed
logistic equation, which has clear biological underpinning coming from cell
population modelling. This nonlinear differential equation includes terms with
discrete and distributed delays. The global dynamics is completely described,
and it is proven that all feasible nontrivial solutions converge to the
positive equilibrium. The main tools of the proof rely on persistence theory,
comparison principles and an -perturbation technique. Using local
invariant manifolds, a unique heteroclinic orbit is constructed that connects
the unstable zero and the stable positive equilibrium, and we show that these
three complete orbits constitute the global attractor of the system. Despite
global attractivity, the dynamics is not trivial as we can observe long-lasting
transient oscillatory patterns of various shapes. We also discuss the
biological implications of these findings and their relations to other logistic
type models of growth with delays
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
Local thermal non-equilibrium effects on thermal convection in a rotating anisotropic porous layer
Effects of local thermal non-equilibrium (LTNE) on thermal convection in a rotating fluid-saturated anisotropic porous layer are investigated. The analysis has been carried out by constructing a simplified model consisting of six coupled nonlinear ordinary differential equations. The study reveals the equivalence of linear and nonlinear stability boundaries indicating the linearized instability theory captures completely the physics of the onset of convection. Results show that the presence of rotation is to introduce oscillatory convection once the Taylor number exceeds a threshold value. The preferred mode of instability is found to be influenced significantly by the mechanical anisotropy parameter as well and it is demonstrated that it has both stabilizing and destabilizing effects on the steady onset in the presence of rotation. Besides, asymptotic analyses for small and large values of the interphase heat transfer coefficient are presented. Heat transport is calculated in terms of Nusselt number. Also, the coupled nonlinear ordinary differential equations are solved numerically using Runge-Kutta method and the transient behavior of Nusselt number is demonstrated for various values of physical parameters. © 2015 Elsevier Inc
Optimal linear stability condition for scalar differential equations with distributed delay
Linear scalar differential equations with distributed delays appear in the
study of the local stability of nonlinear differential equations with feedback,
which are common in biology and physics. Negative feedback loops tend to
promote oscillations around steady states, and their stability depends on the
particular shape of the delay distribution. Since in applications the mean
delay is often the only reliable information available about the distribution,
it is desirable to find conditions for stability that are independent from the
shape of the distribution. We show here that for a given mean delay, the linear
equation with distributed delay is asymptotically stable if the associated
differential equation with a discrete delay is asymptotically stable. We
illustrate this criterion on a compartment model of hematopoietic cell dynamics
to obtain sufficient conditions for stability
Dynamics of Patterns
This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects
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