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.
Highlights
Reaction-diffusion models are used in the design and study of many detecting, measuring, and sensing devices
Immunosensors, which are studied here as an example, represent a kind of them. Such spatial-temporal models are described by the systems of partial or lattice differential equations
A reaction-diffusion model of a hexagonal immunosensor was considered. It was described by the system of lattice delayed differential equations on the hexagonal grid
Summary
Reaction-diffusion models are used in the design and study of many detecting, measuring, and sensing devices. Immunosensors, which are studied here as an example, represent a kind of them Such spatial-temporal models are described by the systems of partial or lattice differential equations. In [1], it was shown that the model describing the chemical reaction of two morphogens (reactants), one of them diffusing within two compartments, resulted in “bi-chaotic” behavior. The origin of such chaotic phenomena (They called it “spiral turbulence” in [2]) were explained with the help of the statistics of topological defects [2]. The lattice differential equations describe systems with a discrete spatial structure, which is more consistent with pixel devices. The aim of this work is to construct one hexagonal model of the reaction-diffusion type, which describes the hexagonal lattice of immunopixels and to study its qualitative behavior when changing parameters
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