scholarly journals On Nonlinear Reaction-Diffusion Model with Time Delay on Hexagonal Lattice

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 758 ◽  
Author(s):  
Vasyl Martsenyuk ◽  
Olga Veselska
Keyword(s):  
Time Delay ◽  
Diffusion Model ◽  
Numerical Study ◽  
Hexagonal Lattice ◽  
Deterministic Chaos ◽  
Reaction Diffusion ◽  
Qualitative Behavior ◽  
Nonlinear Reaction ◽  
Reaction Diffusion Model ◽  
Spatial Operator

In the work, a nonlinear reaction-diffusion model in a class of delayed differential equations on the hexagonal lattice is considered. The system includes a spatial operator of diffusion between hexagonal pixels. The main results deal with the qualitative investigation of the model. The conditions of global asymptotic stability, which are based on the Lyapunov function construction, are obtained. An estimate of the upper bound of time delay, which enables stability, is presented. The numerical study is executed with the help of the bifurcation diagram, phase trajectories, and hexagonal tile portraits. It shows the changes in qualitative behavior with respect to the growth of time delay; namely, starting from the stable focus at small delay values, then through Hopf bifurcation to limit cycles, and finally, through period doublings to deterministic chaos.

Effect of delay on pattern formation of a Rosenzweig–MacArthur type reaction–diffusion model with spatiotemporal delay

2021 ◽  
Author(s):  
Gao-Xiang Yang ◽  
Xiao-Yu Li
Keyword(s):  
Time Delay ◽  
Diffusion Model ◽  
Reaction Diffusion ◽  
Multiple Scale ◽  
Predator Prey ◽  
Turing Bifurcation ◽  
Reaction Diffusion Model ◽  
Scale Perturbation ◽  
Spatiotemporal Delay ◽  
Theoretical Results

In this paper, a predator–prey reaction–diffusion model with Rosenzweig–MacArthur type functional response and spatiotemporal delay is investigated through using the tool of Turing bifurcation theories. First, by taking the average time delay as a bifurcation parameter, conditions of occurrence of Turing bifurcation are obtained through employing the Routh–Hurwitz criteria. Second, as the average time delay varies the amplitude equations of Turing bifurcation patterns including spots pattern and stripes pattern are also obtained through the multiple scale perturbation method. Finally, the two kinds of spatiotemporal evolution distributions of species such as spots pattern and stripes pattern are shown to illustrate theoretical results.

Traveling wavefronts of a nonlinear reaction–diffusion model of tumor growth under the acid environment

2014 ◽  
Vol 07 (03) ◽  
pp. 1450034
Author(s):  
Huiyan Zhu ◽  
Yang Luo ◽  
Xiufang Wang
Keyword(s):  
Diffusion Model ◽  
Perturbation Methods ◽  
Tumor Tissue ◽  
Reaction Diffusion ◽  
Traveling Wave Solution ◽  
Ion Concentration ◽  
Banach Fixed Point Theorem ◽  
Nonlinear Reaction ◽  
Reaction Diffusion Model ◽  
Acid Environment

In this paper, a reaction–diffusion model describing temporal development of tumor tissue, normal tissue and excess H+ ion concentration is considered. Based on a combination of perturbation methods, the Fredholm theory and Banach fixed point theorem, we theoretically justify the existence of the traveling wave solution for this model.

scholarly journals A solvable nonlinear reaction-diffusion model

1995 ◽  
Vol 198 (2) ◽  
pp. 100-104 ◽  
Author(s):  
Max-Olivier Hongler ◽  
Ricardo Lima
Keyword(s):  
Diffusion Model ◽  
Reaction Diffusion ◽  
Nonlinear Reaction ◽  
Reaction Diffusion Model
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