scholarly journals Asymptotic and transient behaviour for a nonlocal problem arising in population genetics

2018 ◽  
Vol 31 (1) ◽  
pp. 84-110
Author(s):  
J.-B. BURIE ◽  
R. DJIDJOU-DEMASSE ◽  
A. DUCROT
Keyword(s):  
Asymptotic Behaviour ◽  
Large Time ◽  
Nonlocal Problem ◽  
Genetic Adaptation ◽  
System Of Equations ◽  
Pathogen Population ◽  
Transient Behaviour ◽  
Selection Processes ◽  
The One ◽  
Mutation And Selection

This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. First, we study the asymptotic behaviour of the system and prove that it eventually converges to a stationary state. Next, we more closely investigate the behaviour of the system in the presence of multiple EAs. Under suitable assumptions and based on a small mutation variance asymptotic, we describe the existence of a long transient regime during which the pathogen population remains far from its asymptotic behaviour and highly concentrated around some phenotypic value that is different from the one described by its asymptotic behaviour. In that setting, the time needed for the system to reach its large time configuration is very long and multiple evolutionary attractors may act as a barrier of evolution that can be very long to bypass.

Wavelets in the Solution of Nongray Radiative Heat Transfer Equation

1998 ◽  
Vol 120 (1) ◽  
pp. 133-139 ◽  
Author(s):  
Y. Bayazitoglu ◽  
B. Y. Wang
Keyword(s):  
Transfer Equation ◽  
Solution Method ◽  
Radiative Heat ◽  
Radiative Transfer Equation ◽  
Basis Functions ◽  
Wavelet Basis ◽  
System Of Equations ◽  
Heat Transfer Equation ◽  
One Dimensional ◽  
The One

The wavelet basis functions are introduced into the radiative transfer equation in the frequency domain. The intensity of radiation is expanded in terms of Daubechies’ wrapped-around wavelet functions. It is shown that the wavelet basis approach to modeling nongrayness can be incorporated into any solution method for the equation of transfer. In this paper the resulting system of equations is solved for the one-dimensional radiative equilibrium problem using the P-N approximation.

On the minimizers of the Ginzburg–Landau energy for high kappa: the one-dimensional case

1997 ◽  
Vol 8 (4) ◽  
pp. 331-345 ◽  
Author(s):  
AMANDINE AFTALION
Keyword(s):  
Magnetic Field ◽  
Asymptotic Behaviour ◽  
Dimensional Case ◽  
Numerical Computations ◽  
Ginzburg Landau ◽  
Landau Model ◽  
One Dimensional ◽  
Landau Parameter ◽  
Convergence Results ◽  
The One

The Ginzburg–Landau model for superconductivity is examined in the one-dimensional case. First, putting the Ginzburg–Landau parameter κ formally equal to infinity, the existence of a minimizer of this reduced Ginzburg–Landau energy is proved. Then asymptotic behaviour for large κ of minimizers of the full Ginzburg–Landau energy is analysed and different convergence results are obtained, according to the exterior magnetic field. Numerical computations illustrate the various behaviours.

scholarly journals Transitions and Instabilities in Imperfect Ion-Selective Membranes

2020 ◽  
Vol 21 (18) ◽  
pp. 6526
Author(s):  
Jarrod Schiffbauer ◽  
Evgeny Demekhin ◽  
Georgy Ganchenko
Keyword(s):  
Electrolyte Solutions ◽  
Darcy Number ◽  
Porous Membrane ◽  
Quiescent State ◽  
System Of Equations ◽  
High Concentration ◽  
Electrokinetic Instability ◽  
The One ◽  
Overlimiting Current ◽  
Selective Membranes

Numerical investigation of the underlimiting, limiting, and overlimiting current modes and their transitions in imperfect ion-selective membranes with fluid flow through permitted through the membrane is presented. The system is treated as a three layer composite system of electrolyte-porous membrane-electrolyte where the Nernst–Planck–Poisson–Stokes system of equations is used in the electrolyte, and the Darcy–Brinkman approach is employed in the nanoporous membrane. In order to resolve thin Debye and Darcy layers, quasi-spectral methods are applied using Chebyshev polynomials for their accumulation of zeros and, hence, best resolution in the layers. The boundary between underlimiting and overlimiting current regimes is subject of linear stability analysis, where the transition to overlimiting current is assumed due to the electrokinetic instability of the one-dimensional quiescent state. However, the well-developed overlimiting current is inherently a problem of nonlinear stability and is subject of the direct numerical simulation of the full system of equations. Both high and low fixed charge density membranes (low- and high concentration electrolyte solutions), acting respectively as (nearly) perfect or imperfect membranes, are considered. The perfect membrane is adequately described by a one-layer model while the imperfect membrane has a more sophisticated response. In particular, the direct transition from underlimiting to overlimiting currents, bypassing the limiting currents, is found to be possible for imperfect membranes (high-concentration electrolyte). The transition to the overlimiting currents for the low-concentration electrolyte solutions is monotonic, while for the high-concentration solutions it is oscillatory. Despite the fact that velocities in the porous membrane are much smaller than in the electrolyte region, it is further demonstrated that they can dramatically influence the nature and transition to the overlimiting regimes. A map of the bifurcations, transitions, and regimes is constructed in coordinates of the fixed membrane charge and the Darcy number.

On How the Generation of Lift Can Be Explained in a Closed Form Based on the Fundamental Conservation Equations

2020 ◽  
Author(s):  
Philipp Epple ◽  
Holger Babinsky ◽  
Michael Steppert ◽  
Manuel Fritsche
Keyword(s):  
Fluid Mechanics ◽  
Closed Form ◽  
Fundamental Problem ◽  
System Of Equations ◽  
Bernoulli Equation ◽  
Conservation Equations ◽  
Research Institutes ◽  
Conservation Principles ◽  
Newton’S Laws ◽  
The One

Abstract The generation of lift is a fundamental problem in aerodynamics and in general in fluid mechanics. The explanations on how lift is generated are often very incomplete or even not correct. Perhaps the most popular explanation of lift is the one with the Bernoulli equation and with the longer path over an airfoil as compared to the path below the airfoil, assuming the flow arrives at the same time at the trailing edge on both paths. This is an intuitive assumption, but no equation is derived from this assumption. In some explanations the Bernoulli equation is also complemented with Newton’s laws of motion. In other explanations Newton’s law is said to be the only explanation. Other explanations mention the Venturi suction effect to explain the generation of lift. In books of aerodynamics and on the homepage of well-known research institutes the explanations are, although better and partially correct, still very often incomplete. In this contribution the generation of lift is explained in a scientific way based on the conservation principles of mass, momentum and energy and how they have to be applied to close the system of equations in order to explain the generation of lift. The most common incomplete or incorrect explanations of lift are also analysed and it is explained why they are incomplete or wrong. In this work the generation of lift is explained based on the conservation equations. It is shown how and when they apply to the problem of lift generation and how the system of equations has to be closed.

scholarly journals Ground state solutions for fractional Schrödinger equation with variable potential and Berestycki–Lions type nonlinearity

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jing Chen ◽  
Zu Gao
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlocal Problem ◽  
Minimization Method ◽  
Fractional Schrödinger Equation ◽  
Ground State Solutions ◽  
The One ◽  
Least Energy Solutions ◽  
Fractional Schrodinger Equation

Abstract We consider the following nonlinear fractional Schrödinger equation: $$ (-\triangle )^{s} u+V(x)u=g(u) \quad \text{in } \mathbb{R} ^{N}, $$ ( − △ ) s u + V ( x ) u = g ( u ) in  R N , where $s\in (0, 1)$ s ∈ ( 0 , 1 ) , $N>2s$ N > 2 s , $V(x)$ V ( x ) is differentiable, and $g\in C ^{1}(\mathbb{R} , \mathbb{R} )$ g ∈ C 1 ( R , R ) . By exploiting the minimization method with a constraint over Pohoz̆aev manifold, we obtain the existence of ground state solutions. With the help of Pohoz̆aev identity we also process the existence of the least energy solutions for the above equation. Our results improve the existing study on this nonlocal problem with Berestycki–Lions type nonlinearity to the one that does not need the oddness assumption.

Solitary waves in excitable systems with cross-diffusion

2005 ◽  
Vol 461 (2064) ◽  
pp. 3711-3730 ◽  
Author(s):  
Vadim N Biktashev ◽  
Mikhail A Tsyganov
Keyword(s):  
Asymptotic Behaviour ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
System Of Equations ◽  
Soliton Interaction ◽  
Predator Prey ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Excitable Systems

We consider a FitzHugh–Nagumo system of equations where the traditional diffusion terms are replaced with linear cross-diffusion of components. This system describes solitary waves that have unusual form and are capable of quasi-soliton interaction. This is different from the classical FitzHugh–Nagumo system with self-diffusion, but similar to a predator–prey model with taxis of populations on each other's gradient which we considered earlier. We study these waves by numerical simulations and also present an analytical theory, based on the asymptotic behaviour which arises when the local dynamics of the inhibitor field are much slower than those of the activator field.

scholarly journals Asymptotics for a parabolic double obstacle problem

1994 ◽  
Vol 444 (1922) ◽  
pp. 429-445 ◽  
Keyword(s):  
Asymptotic Behaviour ◽  
Obstacle Problem ◽  
Thin Strip ◽  
Normal Velocity ◽  
One Dimensional ◽  
Cahn Equation ◽  
Small Constant ◽  
The One ◽  
Short Time ◽  
Double Well Potential

We consider a parabolic double obstacle problem which is a version of the Allen-Cahn equation u t = Δ u — ϵ -2 ψ '( u ) in Ω x (0, ∞), where Ω is a bounded domain, ϵ is a small constant, and ψ is a double well potential; here we take ψ such that ψ ( u ) = (1 — u 2 ) when | u | ≤ 1 and ψ ( u ) = ∞ when | u | > 1. We study the asymptotic behaviour, as ϵ → 0, of the solution of the double obstacle problem. Under some natural restrictions on the initial data, we show that after a short time (of order ϵ 2 |ln ϵ |), the solution takes value 1 in a region Ω + t and value — 1 in Ω - t , where the region Ω ( Ω + t U Ω - t ) is a thin strip and is contained in either a O ( ϵ |ln ϵ |) or O ( ϵ ) neighbourhood of a hypersurface Γ t which moves with normal velocity equal to its mean curvature. We also study the asymptotic behaviour, as t → ∞, of the solution in the one-dimensional case. In particular, we prove that the ω -limit set consists of a singleton.

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