A Method to Solve the Reaction-Diffusion-Chemotaxis System

2019 ◽  
Vol 20 (6) ◽  
pp. 633-650
Author(s):  
Yao-Hsin Hwang ◽  
Jui-Ling Yu ◽  
Chin-Kun Hu
Keyword(s):  
Numerical Analysis ◽  
Control Volume ◽  
Reaction Diffusion ◽  
Two Dimensional ◽  
Rectangular Grid ◽  
Simulation Accuracy ◽  
Grid Systems ◽  
The Stability ◽  
Chemotaxis System ◽  
Solution Gradient

AbstractThe objective of this article is to use a recent developed scheme to simulate reaction-diffusion-chemotaxis equations. The solution gradient required for an accurate discretization is computed directly as an additional variable rather than interpolated from solution values around neighboring computational nodes. To achieve this goal, a supplementary equation and its associated control volume are introduced to retain a compact and accurate discretization. Scheme essentials are exposed by the numerical analysis on two-dimensional chemotaxis problems to reveal its formal accuracy. Due to its highly comprehensible and practical features, this formulation can be easily extended to solve problems for other two-dimensional rectangular grid systems. One- and two-dimensional problems are solved to verify its simulation accuracy and to study the possible formation of bacteria bands. We further perform the linearization technique to the reaction term to increase the stability of the current scheme. From the numerical analysis and computational results, it is found that the present formulation is a useful tool to solve reaction-diffusion-chemotaxis equations.

The Free Energies of Partially Open Magnetic Fields and Coronal Mass Ejections

1994 ◽  
Vol 144 ◽  
pp. 97-100
Author(s):  
D. F. Smith ◽  
B. C. Low
Keyword(s):  
Numerical Analysis ◽  
Magnetic Fields ◽  
Closed Field ◽  
Two Dimensional ◽  
Free Energies ◽  
Magnetic Bubble ◽  
Stabilizing Effect ◽  
Field Lines ◽  
Line Tying ◽  
The Stability

AbstractThe results of Low and Smith (1993) on how to circumvent Aly’s limitation on the energy of force-free magnetic fields are briefly reviewed. Two non-force-free configurations with energies in excess of the energy of the corresponding completely open field are discussed: 1. A magnetic bubble configuration with completely detached field lines. 2. A combined configuration consisting of closed field lines with plasma compressed against the star and a superimposed helmet streamer with all field lines tied to the surface. The results of Low and Smith are extended by considering the stability of these two-dimensional equilibria. It is shown that the bubble configuration is unstable. The stability of the combined configuration can only be answered by a numerical analysis because of the stabilizing effect of line-tying.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

scholarly journals The influence of density in population dynamics with strong and weak Allee effect

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kamrun Nahar Keya ◽  
Md. Kamrujjaman ◽  
Md. Shafiqul Islam
Keyword(s):  
Population Dynamics ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Logistic Growth ◽  
Allee Effects ◽  
Reaction Diffusion Model ◽  
Positive Steady States ◽  
The Stability ◽  
The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

Sign in / Sign up

Export Citation Format

Share Document