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

Abstract

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.where we consider a lattice Λ ⊂ R n , which can be presented as a discrete subset of R n , consisting of either a finite or infinite number of points, which are located in accordance with some regular spatial structure. The vectors u ξ , ξ ∈ Λ are the values of the state u = u ξ ξ∈Λ , calculated at the the points of the lattice, and g ξ are the right sides of the equations with the properties enabling us to find the existence of the solution.As a rule, without loss of generality, these consider Λ = Z n , which is the integer lattice in R n . The methods developed can be easily applied to a different type of lattice, namely the planar rectangular and hexagonal lattice, the crystallographic lattices in R 3 .These pay attention to the notion of delay in lattice differential equations, the so-called delayed lattice differential equations. One of the applications dealing with them is the investigation of traveling wave fronts and their stability [5]. The main results are applied to the delayed and discretely diffusive models for the population (see, e.g., [6,7]).Lattice differential equations are used as models in many applications, for example cellular neural networks, image processing, chemical kinetics, materials science, in particular metallurgy, and biology [5,8]. Lattice models are extremely attractive from the viewpoint of population dynamics, especially in the case of spatially-separated populations [5,6,[8][9][10][11].There are a few reasons for requiring the consideration of the hexagonal grid instead of rectangular one (primarily in image and vision computing); namely, the equal distances between neighboring pixels for hexagonal coordinate systems [12]; hexagonal points are packed more densely [13]; since "hexagons are 'rounder' than squares", the presentation of curves is more consistent with the help of hexagonal systems [13]; hence, mathematical operations of edge detection and shape extraction are more successful when applying hexagonal lattices [14].With the purpose of indexing hexagonal pixels, as a rule, two-(This is the so-called "skewed-axis" coordinate system) or three-element (It is also known as the "cube hex coordinate system") coordinate system...

Nonlinear Analytics for Electrochemical Biosensor Design Using Enzyme Aggregates and Delayed Mass Action

“…The dynamics of such type systems in a general case is studied in [ 16 ]. Some of our previous results were also related to their behavior [ 17 , 18 , 19 ]. In the next section, we will consider one simple case allowing us to develop the method of parameter identification for the model such as (24).…”

Комп'ютерне Моделювання Співіснування Вірусних Штамів: Непередбачуваність Через Нелінійні Явища

“…For this purpose, linearization methods and the direct method of lyapunay were used. At the same time, the model of interaction of subpopulations in the case of the spread of several virus strains and the question of nonlinear analysis of such models are of considerable interest [5][6][7][8].…”

Global Asymptotic Stability and Nonlinear Analysis of the Model of the Square Immunopixels Array Based on Delay Lattice Differential Equations

“…It was shown that they depend on the value of delay τ. In [10], the corresponding model was constructed in the case of a three-dimensional array (or hexagonal lattice) of immunopixels, for which the global asymptotic stability was investigated. In both cases, we evidenced the dependence of phase portraits on τ.…”

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