scholarly journals A stage-structured delayed advection reaction-diffusion model for single species

2020 ◽  
Vol 10 (6) ◽  
pp. 6260
Author(s):  
Raed Ali Alkhasawneh
Keyword(s):  
Diffusion Equation ◽  
Time Delays ◽  
Reaction Diffusion ◽  
Single Species ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Equation ◽  
Reaction Diffusion Model ◽  
Advection Term ◽  
Species Population ◽  
Homogeneous Dirichlet Boundary Conditions

In this paper, we derived a delay advection reaction-diffusion equation with linear advection term from a stage-structured model, then the derived equation is used under the homogeneous Dirichlet boundary conditions u_m (0,t)=0, u_m (L,t)=0, and the initial condition u_m (x,0)=u_m^0 (x)>0,x∈[-τ,0] with u_m^0 (0)>0 in order to find the minimum value of domain L that prevents extinction of the species under the effect of advection reaction diffusion equation. Finally, for the measurement the time lengths from birth to the development of the species population, time delays are integrated.

scholarly journals A Numerical Method for Time-Fractional Reaction-Diffusion and Integro Reaction-Diffusion Equation Based on Quasi-Wavelet

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Sachin Kumar ◽  
Jinde Cao ◽  
Xiaodi Li
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Numerical Solutions ◽  
Exact Analytical Solution ◽  
Research Work ◽  
Reaction Diffusion ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Equation ◽  
Fractional Reaction

In this research work, we focused on finding the numerical solution of time-fractional reaction-diffusion and another class of integro-differential equation known as the integro reaction-diffusion equation. For this, we developed a numerical scheme with the help of quasi-wavelets. The fractional term in the time direction is approximated by using the Crank–Nicolson scheme. The spatial term and the integral term present in integro reaction-diffusion are discretized and approximated with the help of quasi-wavelets. We study this model with Dirichlet boundary conditions. The discretization of these initial and boundary conditions is done with a different approach by the quasi-wavelet-based numerical method. The validity of this proposed method is tested by taking some numerical examples having an exact analytical solution. The accuracy of this method can be seen by error tables which we have drawn between the exact solution and the approximate solution. The effectiveness and validity can be seen by the graphs of the exact and numerical solutions. We conclude that this method has the desired accuracy and has a distinctive local property.

Dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity

2010 ◽  
Vol 140 (5) ◽  
pp. 1081-1109 ◽  
Author(s):  
Zhi-Cheng Wang ◽  
Wan-Tong Li
Keyword(s):  
Diffusion Equation ◽  
Global Attractivity ◽  
Reaction Diffusion ◽  
Single Species ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Positive Equilibrium ◽  
Cauchy Type ◽  
Delayed Reaction ◽  
Non Local

AbstractThis paper is concerned with the dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity on an infinite n-dimensional domain, which can be derived from the growth of a stage-structured single-species population. We first prove that solutions of the Cauchy-type problem are positively preserving and bounded if the initial value is non-negative and bounded. Then, by establishing a comparison theorem and a series of comparison arguments, we prove the global attractivity of the positive equilibrium. When there exist no positive equilibria, we prove that the zero equilibrium is globally attractive. In particular, these results are still valid for the non-local delayed reaction–diffusion equation on a bounded domain with the Neumann boundary condition. Finally, we establish the existence of new entire solutions by using the travelling-wave solutions of two auxiliary equations and the global attractivity of the positive equilibrium.

scholarly journals Comparison principles for impulsive parabolic equations with applications to models of single species growth

1991 ◽  
Vol 32 (4) ◽  
pp. 382-400 ◽  
Author(s):  
L. H. Erbe ◽  
H. I. Freedman ◽  
X. Z. Liu ◽  
J. H. Wu
Keyword(s):  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Single Species ◽  
Reaction Diffusion Equation ◽  
Equation Modeling ◽  
Positive Equilibrium ◽  
Comparison Principles ◽  
Nonlinear Parabolic ◽  
Species Population ◽  
Single Species Population

AbstractThis paper establishes some maximum and comparison principles relative to lower and upper solutions of nonlinear parabolic partial differential equations with impulsive effects. These principles are applied to obtain some sufficient conditions for the global asymptotic stability of a unique positive equilibrium in a reaction-diffusion equation modeling the growth of a single-species population subject to abrupt changes of certain important system parameters.

scholarly journals An Accelerating Numerical Computation of the Diffusion Term in a Nonlocal Reaction-Diffusion Equation

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2111
Author(s):  
Mitică CRAUS ◽  
Silviu-Dumitru PAVĂL
Keyword(s):  
Diffusion Equation ◽  
Numerical Approximation ◽  
Approximation Scheme ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Diffusion Term ◽  
Reaction Diffusion Model ◽  
Computationally Intensive ◽  
Nonlocal Reaction ◽  
Nonlocal Formulation

In this paper we propose and compare two methods to optimize the numerical computations for the diffusion term in a nonlocal formulation for a reaction-diffusion equation. The diffusion term is particularly computationally intensive due to the integral formulation, and thus finding a better way of computing its numerical approximation could be of interest, given that the numerical analysis usually takes place on large input domains having more than one dimension. After introducing the general reaction-diffusion model, we discuss a numerical approximation scheme for the diffusion term, based on a finite difference method. In the next sections we propose two algorithms to solve the numerical approximation scheme, focusing on finding a way to improve the time performance. While the first algorithm (sequential) is used as a baseline for performance measurement, the second algorithm (parallel) is implemented using two different memory-sharing parallelization technologies: Open Multi-Processing (OpenMP) and CUDA. All the results were obtained by using the model in image processing applications such as image restoration and segmentation.

A Nonstandard Finite Difference Scheme for the Reaction Diffusion Equation

2012 ◽  
Vol 166-169 ◽  
pp. 3265-3268
Author(s):  
Ming Ding Liu
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Reaction Diffusion ◽  
Discrete Models ◽  
Reaction Diffusion Equation ◽  
Nonstandard Finite Difference Scheme ◽  
Advection Term ◽  
Nonstandard Finite Difference

We use the nonstandard finite difference (NSFD) method to construct discrete models of the reaction diffusion equation. A nonstandard finite difference scheme for the reaction-diffusion equation is given. We demonstrated that the space denominator function can be based on the use of a transformation from the simple expression (Δx)2to an 4C(sin[(1/C)1/2((Δx)/2)])2.which is clearly valid for sufficiently small Δx. Another important class for which this method keeps the equation solutions are positivity and can be applied is those PDE's without advection term.

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