scholarly journals Admissible wavefront speeds for a single species reaction-diffusion equation with delay

2008 ◽  
Vol 20 (2) ◽  
pp. 407-423 ◽  
Author(s):  
Elena Trofimchuk ◽  
◽  
Sergei Trofimchuk ◽  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Single Species ◽  
Reaction Diffusion Equation ◽  
Equation With Delay

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 Linear θ-Method and Compact θ-Method for Generalised Reaction-Diffusion Equation with Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Fengyan Wu ◽  
Qiong Wang ◽  
Xiujun Cheng ◽  
Xiaoli Chen
Keyword(s):  
Diffusion Equation ◽  
Necessary Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Asymptotically Stable ◽  
Sufficient And Necessary Conditions ◽  
Equation Solvability ◽  
Equation With Delay ◽  
Theoretical Results

This paper is concerned with the analysis of the linear θ-method and compact θ-method for solving delay reaction-diffusion equation. Solvability, consistence, stability, and convergence of the two methods are studied. When θ∈[0,1/2), sufficient and necessary conditions are given to show that the two methods are asymptotically stable. When θ∈[1/2,1], the two methods are proven to be unconditionally asymptotically stable. Finally, several examples are carried out to confirm the theoretical results.

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.

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