scholarly journals Travelling Wave and Turing Patterns in Subdiffusive Autocatalytic Systems

2021 ◽  
Author(s):  
Uttam Kumar ◽  
Pushpavanam Subramanian
Keyword(s):  
Steady State ◽  
Stability Boundary ◽  
Mean Square Displacement ◽  
Travelling Wave ◽  
Turing Patterns ◽  
Logistic Growth ◽  
Stable Steady State ◽  
Phase Plane Analysis ◽  
Plane Analysis ◽  
The Stability

Abstract In this work, we analyse autocatalytic reactions in complex and disordered media which are governed by subdiffusion. The mean square displacement of molecules here scale as tγ where 0<γ<1. These systems are governed by fractional partial differential equations. Two systems are analysed i) in the first a logistic growth expression is used to represent the growth kinetics of bacteria. Here the system dynamics is governed by a single variable. ii) the second system is a two variable cubic autocatalytic system in a porous media. Here each reactant is involved in the autocatalytic generation of the other. These systems have multiple steady states. They exhibit traveling waves moving from an unstable steady state to a stable steady state. The minimum wave velocity has been obtained from phase plane analysis analytically for the first system. In addition, the two variable system also shows Turing patterns in selected regions of parameter space. The stability boundary for Turing patterns for subdiffusive system is found to be the same as that for regular diffusive systems obtained by Seshai et al. [1]. System behaviour as predicted by the stability analysis is verified using a robust implicit numerical method based on L1 scheme.

scholarly journals Suppression of marine ice sheet instability

2018 ◽  
Vol 857 ◽  
pp. 648-680 ◽  
Author(s):  
Samuel S. Pegler
Keyword(s):  
Steady State ◽  
Rapid Assessment ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Stable Steady State ◽  
Ice Shelf ◽  
West Antarctic Ice Sheet ◽  
West Antarctic ◽  
Grounding Line ◽  
The Stability

A long-standing open question in glaciology concerns the propensity for ice sheets that lie predominantly submerged in the ocean (marine ice sheets) to destabilise under buoyancy. This paper addresses the processes by which a buoyancy-driven mechanism for the retreat and ultimate collapse of such ice sheets – the marine ice sheet instability – is suppressed by lateral stresses acting on its floating component (the ice shelf). The key results are to demonstrate the transition between a mode of stable (easily reversible) retreat along a stable steady-state branch created by ice-shelf buttressing to tipped (almost irreversible) retreat across a critical parametric threshold. The conditions for triggering tipped retreat can be controlled by the calving position and other properties of the ice-shelf profile and can be largely independent of basal stress, in contrast to principles established from studies of unbuttressed grounding-line dynamics. The stability and recovery conditions introduced by lateral stresses are analysed by developing a method of constructing grounding-line stability (bifurcation) diagrams, which provide a rapid assessment of the steady-state positions, their natures and the conditions for secondary grounding, giving clear visualisations of global stabilisation conditions. A further result is to reveal the possibility of a third structural component of a marine ice sheet that lies intermediate to the fully grounded and floating components. The region forms an extended grounding area in which the ice sheet lies very close to flotation, and there is no clearly distinguished grounding line. The formation of this region generates an upsurge in buttressing that provides the most feasible mechanism for reversal of a tipped grounding line. The results of this paper provide conceptual insight into the phenomena controlling the stability of the West Antarctic Ice Sheet, the collapse of which has the potential to dominate future contributions to global sea-level rise.

scholarly journals THE DYNAMICS OF TACHYON FIELD WITH AN INVERSE SQUARE POTENTIAL IN LOOP QUANTUM COSMOLOGY

2013 ◽  
Vol 22 (06) ◽  
pp. 1350030 ◽  
Author(s):  
FEI HUANG ◽  
JIAN-YANG ZHU ◽  
KUI XIAO
Keyword(s):  
Quantum Cosmology ◽  
Dynamical Behavior ◽  
Loop Quantum Cosmology ◽  
Phase Plane Analysis ◽  
Late Time ◽  
Stable Node ◽  
Time Behavior ◽  
Tachyon Field ◽  
Plane Analysis ◽  
The Stability

The dynamical behavior of tachyon field with an inverse potential is investigated in loop quantum cosmology. It reveals that the late-time behavior of tachyon field with this potential leads to a power-law expansion. In addition, an additional barotropic perfect fluid with the adiabatic index 0 < γ < 2 is added and the dynamical system is shown to be an autonomous one. The stability of this autonomous system is discussed using phase plane analysis. There exist up to five fixed points with only two of them possibly stable. The two stable node (attractor) solutions are specified and their cosmological indications are discussed. For the tachyon dominated solution, the further discussion is stretched to the possibility of considering tachyon field as a combination of two parts which respectively behave like dark matter and dark energy.

Steady State and Dynamic Analyses of Supercritical CO2 Natural Circulation Loop

2006 ◽  
Author(s):  
Prashant Jain ◽  
Rizwan Uddin
Keyword(s):  
Steady State ◽  
Natural Circulation ◽  
Stability Boundary ◽  
Operating Conditions ◽  
Circulation Loop ◽  
Steady State Flow ◽  
Natural Circulation Loop ◽  
Dynamic Analyses ◽  
The Stability ◽  
Flow Power

Numerical studies have been carried out to investigate supercritical flow instabilities in a CO2 natural circulation loop. For the steady state and dynamic analyses of the loop under supercritical conditions, a single-channel, one-dimensional model is developed. In this model, equations for the conservation of mass, momentum and energy are discretized using an implicit finite difference scheme. A computer code called FIASCO (Flow Instability Analysis under SuperCritical Operating conditions) is developed in FORTRAN90 to simulate the dynamics of natural circulation loops with supercritical fluid. Results obtained for the stability boundary substantially deviate from the results reported by previous investigators, and thus contradict some of the reported findings. The disagreement in results is most likely due to the undesirable dissipative and dispersive effects produced from the large time steps used in previous studies, thereby leading to a larger stable region than those found using smaller time step. Results presented here suggest that the stability boundary of a natural circulation loop with supercritical fluid, is not confined to the near-peak region of the (steady state) flow-power curve. Additional and more extensive experimental data are needed to resolve the differences between results obtained here and those reported earlier. However, results obtained for the range of parameter values used in this investigation always predict the stability threshold to be in the positive slope region of the (steady state) flow-power curve. Parametric studies for different operating conditions reveal the similarity of stability characteristics under supercritical conditions with those in two-phase flows.

The travelling wave fronts of nonlinear reaction–diffusion systems via Friedlin's stochastic approaches

1994 ◽  
Vol 124 (2) ◽  
pp. 273-299 ◽  
Author(s):  
Huaizhong Zhao
Keyword(s):  
Steady State ◽  
Large Deviation ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Stable Steady State ◽  
Reaction Diffusion Systems ◽  
Nonlinear Reaction ◽  
Diffusion Systems ◽  
Special Case ◽  
Deviation Theory

In this paper we study the asymptotic behaviour of reaction–diffusion systems with a small parameter by using then-dimensional Feynman–Kac formula and large deviation theory. The generalised solutions are introduced in Section 2. We obtain the travelling wave joining an unstable steady state and an asymptotically stable steady state of a diffusionless dynamical system in a reaction–diffusion system with nonlinear ergodic interactions, and a special case with nonlinear reducible interactions.

Photobioreactor Stability by Phase Plane Technique Applied to Spirulina Maxima Growth

2003 ◽  
Vol 1 (1) ◽  
Author(s):  
Khaled Belkacemi ◽  
Safia Hamoudi
Keyword(s):  
Steady State ◽  
Phase Plane ◽  
Biologically Active ◽  
Raw Material ◽  
System Stability ◽  
Stable Steady State ◽  
Monod Kinetics ◽  
Spirulina Maxima ◽  
Plane Technique ◽  
The Stability

Spirulina maxima is a worthy multicellular filamentous micro-algae used as a food supplement and raw material for fine chemicals and biologically active compounds production. Intensive approach consisting of cultivating pure strains of this photoautotroph microorganism in photobioreactors is more desirable than extensive ones, largely incontrollable with regard to production stability. Determining the best reaction conditions to reach a steady state in the runway events is often needed in biological systems. For a biochemical engineer, knowing the system stability for an optimal bioreactor configuration is crucial to estimate the rate at which dependent variables grow or decay with the time reaction. The stability analysis becomes important in recycle processes in which possibility that these systems influence themselves exists. The aim of this work deals with the 1) description of the growth kinetics by a logistic and unstructured model based on Monod kinetics taking into account the maintenance in life of viable cells; 2) establishment of a dynamic growth model for Spirulina maxima cultivated in continuous lamellar photobioreactors using industrial manures as macro-nutrients; 3) determination of optimal culture conditions sustaining a stable growth of S. maxima in a system of two bioreactors in series; and 4) investigation of the dynamic stability of this multivariable system with nonlinear dynamics using phase plane technique (PPT). Although good mixing of the culture is essential for ensuring adequate supply of nutrients and prevention of the accumulation of toxic metabolites. Excessive agitation causes mechanical damage to Spirulina cells. An air flow rate of 2.5 L/min for airlift agitation represented a balance between the need to provide good mixing and to avoid cell damage. A stable steady state was achieved corresponding to a productivity of 10.8 g. m2/day when the system was supplied with 0.2 g N/L of minerals, at a dilution rate of 0.1 1/day, temperature of 30 °C under light intensity of 18 Klux. PPT as a powerful procedure successfully predicts the stability of such a complex system very well.

scholarly journals Hopf Bifurcation and Stability of Periodic Solutions for Delay Differential Model of HIV Infection of CD4+T-cells

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
P. Balasubramaniam ◽  
M. Prakash ◽  
Fathalla A. Rihan ◽  
S. Lakshmanan
Keyword(s):  
T Cells ◽  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Hiv Infection ◽  
Logistic Growth ◽  
Differential Model ◽  
Discrete Time Delay ◽  
The Stability ◽  
Delay Differential

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection ofCD4+T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results.

scholarly journals Revisiting the Fisher–Kolmogorov–Petrovsky–Piskunov equation to interpret the spreading–extinction dichotomy

2019 ◽  
Vol 475 (2229) ◽  
pp. 20190378 ◽  
Author(s):  
Maud El-Hachem ◽  
Scott W. McCue ◽  
Wang Jin ◽  
Yihong Du ◽  
Matthew J. Simpson
Keyword(s):  
Moving Boundary ◽  
Travelling Wave ◽  
Phase Plane Analysis ◽  
Travelling Wave Solutions ◽  
Wave Speeds ◽  
Population Extinction ◽  
Plane Analysis ◽  
Wave Solutions ◽  
Stefan Condition ◽  
Approximate Expressions

The Fisher–Kolmogorov–Petrovsky–Piskunov model, also known as the Fisher–KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher–KPP model cannot replicate, such as the extinction of invasive populations. The Fisher–Stefan model is an adaptation of the Fisher–KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher–Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher–Stefan model is that it is able to simulate population extinction, giving rise to a spreading–extinction dichotomy . In this work, we revisit travelling wave solutions of the Fisher–KPP model and show that these results provide new insight into travelling wave solutions of the Fisher–Stefan model and the spreading–extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher–Stefan model and often-neglected travelling wave solutions of the Fisher–KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher–Stefan model in the limit of slow travelling wave speeds, c ≪1.

scholarly journals Importance of various antioxidant enzymes for cell stability. Confrontation between theoretical and experimental data

1992 ◽  
Vol 286 (1) ◽  
pp. 41-46 ◽  
Author(s):  
J Remacle ◽  
D Lambert ◽  
M Raes ◽  
E Pigeolet ◽  
C Michiels ◽  
...  
Keyword(s):  
Experimental Data ◽  
Antioxidant Enzymes ◽  
Steady State ◽  
Qualitative Description ◽  
Rate Equations ◽  
Stable Steady State ◽  
General View ◽  
Complex Situation ◽  
Cell Stability ◽  
The Stability

A theoretical model was developed taking into account the production and destruction of oxygen-derived free radicals. The steady state of the system was derived by using the rate equations of these reactions, and the stability of the system was tested. In the simplified model, only one stable steady state was found. However, we know that glutathione peroxidase can be inhibited by hydroperoxides, and, when incorporated into the model, this effect led to a complex situation with the presence of some stable and some unstable domains according to the concentration of either the enzyme or the hydroperoxide. This qualitative description of the system was compared with experimental data on the protection given by three antioxidant enzymes, and concordance of data was found which allows some quantification of the system. A general view of the efficiency of the three antioxidant enzymes and of the stability of the system according to their concentrations could be produced.

Analytical Solutions for an Avian Influenza Epidemic Model incorporating Spatial Spread as a Diffusive Process

2015 ◽  
Vol 5 (2) ◽  
pp. 150-159 ◽  
Author(s):  
Phontita Thiuthad ◽  
Valipuram S. Manoranjan ◽  
Yongwimon Lenbury
Keyword(s):  
Avian Influenza ◽  
Analytical Solutions ◽  
Epidemic Model ◽  
Travelling Wave ◽  
Phase Plane Analysis ◽  
Influenza Epidemic ◽  
Spatial Spread ◽  
Diffusive Process ◽  
Plane Analysis ◽  
Spatial Transmission

AbstractWe consider a theoretical model for the spread of avian influenza in a poultry population. An avian influenza epidemic model incorporating spatial spread as a diffusive process is discussed, where the infected individuals are restricted from moving to prevent spatial transmission but infection occurs when susceptible individuals come into contact with infected individuals or the virus is contracted from the contaminated environment (e.g. through water or food). The infection is assumed to spread radially and isotropically. After a stability and phase plane analysis of the equivalent system of ordinary differential equations, it is shown that an analytical solution can be obtained in the form of a travelling wave. We outline the methodology for finding such analytical solutions using a travelling wave coordinate when the wave is assumed to move at constant speed. Numerical simulations also produce the travelling wave solution, and a comparison is made with some predictions based on empirical data reported in the literature.

scholarly journals INDUCED PHANTOM AND 5D ATTRACTOR SOLUTION IN SPACETIME–MATTER THEORY

2005 ◽  
Vol 20 (26) ◽  
pp. 1973-1981 ◽  
Author(s):  
HONGYA LIU ◽  
HUANYING LIU ◽  
BAORONG CHANG ◽  
LIXIN XU
Keyword(s):  
Phase Plane ◽  
Phase Plane Analysis ◽  
Late Time ◽  
Plane Analysis ◽  
Spacetime Singularity ◽  
Attractor Solution ◽  
Matter Theory ◽  
Cosmological Solutions ◽  
The Stability ◽  
Induced Matter

In Spacetime–Matter theory we assume that the 4D induced matter of the 5D Ricci-flat bouncing cosmological solutions contains a perfect fluid as well as an induced scalar field. Then we show that the conventional 4D quintessence and phantom models of dark energy could be recovered from the 5D cosmological solutions. By using the phase-plane analysis to study the stability of evolution of the 5D models, we find that the conventional 4D late-time attractor solution is also recovered. This attractor solution shows that the scale factors of the phantom dominated universes in both the 4D and 5D theories will reach infinity in a finite time and the universes will be ended at a new kind of spacetime singularity at which everything will be annihilated. We also find that the repulsive force of the phantom may provide us with a mechanics to explain the bounce.

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