Stationary patterns in bistable reaction-diffusion cellular automata

Daniel Špale; Petr Stehlík; Daniel Špale; Petr Stehlík

doi:10.3934/mbe.2022283

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 6: 6072-6087. doi: 10.3934/mbe.2022283

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Stationary patterns in bistable reaction-diffusion cellular automata

Daniel Špale ,
Petr Stehlík ^,

Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8,306 14 Plzeň, Czech Republic

Received: 31 January 2022 Revised: 16 March 2022 Accepted: 05 April 2022 Published: 13 April 2022

In this paper, we study stationary patterns of bistable reaction-diffusion cellular automata, i.e., models with discrete time, space and state. We show the rich variability based on the interplay of the capacity and viability and the specific form of reaction functions. While stationary $ k $-periodic patterns occur naturally in many situations in large (exponential) numbers, there exist extreme situations for which there are no heterogeneous patterns. Moreover, nonmonotone dependence of the number of stationary patterns on the diffusion parameter is shown to be natural in the fully discrete setting.
- patterns,
- reaction-diffusion,
- cellular automata,
- discrete dynamics,
- difference equations
Citation: Daniel Špale, Petr Stehlík. Stationary patterns in bistable reaction-diffusion cellular automata[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 6072-6087. doi: 10.3934/mbe.2022283

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Abstract

In this paper, we study stationary patterns of bistable reaction-diffusion cellular automata, i.e., models with discrete time, space and state. We show the rich variability based on the interplay of the capacity and viability and the specific form of reaction functions. While stationary $ k $-periodic patterns occur naturally in many situations in large (exponential) numbers, there exist extreme situations for which there are no heterogeneous patterns. Moreover, nonmonotone dependence of the number of stationary patterns on the diffusion parameter is shown to be natural in the fully discrete setting.

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