Stochastic simulations of the Schnakenberg model with spatial inhomogeneities using reactive multiparticle collision dynamics

Alireza Sayyidmousavi; Katrin Rohlf; Alireza Sayyidmousavi; Katrin Rohlf

doi:10.3934/math.2019.6.1805

AIMS Mathematics

2019, Volume 4, Issue 6: 1805-1823. doi: 10.3934/math.2019.6.1805

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Stochastic simulations of the Schnakenberg model with spatial inhomogeneities using reactive multiparticle collision dynamics

Alireza Sayyidmousavi ,
Katrin Rohlf ^,

Department of Mathematics, Ryerson University, 350 Victoria St., Toronto, ON, M5B 2K3, Canada

Received: 17 December 2018 Accepted: 19 August 2019 Published: 17 January 2019
MSC : 37M05, 65C35, 82C05, 92B05

A numerically efficient globally averaged number density approach is used to simulate a reaction-diffusion system using a particle-based stochastic simulation algorithm called reactive multiparticle collision (RMPC) dynamics. Constant diffusivity of the particles is achieved through a time-varying rotation angle (also called collision angle). Variation in the diffusion coefficient between two different chemical species is achieved in one of two ways: (ⅰ) using a different k_BT/m value for one species compared to the other, or (ⅱ) using the same k_BT/m value for both species, but using a different probability to free-stream for one species compared to another. For smaller diffusivities and larger spatial inhomogeneities, bath particles were necessary for the model to agree with the PDE solution. The latter approach was further used without a bath, and shown to be capable of producing Turing patterns after long simulation times. The significance of our work is that RMPC can serve as a feasible simulation tool for both short and long-term simulations, can handle spatial inhomogeneities, can model a fairly large range of diffusivities in a reaction-diffusion scenario, and is capable of producing Turing patterns. An advantage of this method includes more detailed system information in feasible simulation times.
- stochastic simulations,
- reaction-diffusion,
- reactive multiparticle collision dynamics,
- Schnakenberg model
Citation: Alireza Sayyidmousavi, Katrin Rohlf. Stochastic simulations of the Schnakenberg model with spatial inhomogeneities using reactive multiparticle collision dynamics[J]. AIMS Mathematics, 2019, 4(6): 1805-1823. doi: 10.3934/math.2019.6.1805

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Abstract

A numerically efficient globally averaged number density approach is used to simulate a reaction-diffusion system using a particle-based stochastic simulation algorithm called reactive multiparticle collision (RMPC) dynamics. Constant diffusivity of the particles is achieved through a time-varying rotation angle (also called collision angle). Variation in the diffusion coefficient between two different chemical species is achieved in one of two ways: (ⅰ) using a different k_BT/m value for one species compared to the other, or (ⅱ) using the same k_BT/m value for both species, but using a different probability to free-stream for one species compared to another. For smaller diffusivities and larger spatial inhomogeneities, bath particles were necessary for the model to agree with the PDE solution. The latter approach was further used without a bath, and shown to be capable of producing Turing patterns after long simulation times. The significance of our work is that RMPC can serve as a feasible simulation tool for both short and long-term simulations, can handle spatial inhomogeneities, can model a fairly large range of diffusivities in a reaction-diffusion scenario, and is capable of producing Turing patterns. An advantage of this method includes more detailed system information in feasible simulation times.

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