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Analysis and Simulations of a Nonlocal Gray-Scott Model

Loic Cappanera, Gabriela Jaramillo, Cory Ward

Abstract

The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, $L^1$ convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.

Analysis and Simulations of a Nonlocal Gray-Scott Model

Abstract

The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.
Paper Structure (18 sections, 12 theorems, 124 equations, 2 figures, 2 tables)

This paper contains 18 sections, 12 theorems, 124 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model Equations
  3. Notation
  4. Preliminary Results
  5. Bilinear Forms
  6. Existence of Solutions
  7. The basis elements
  8. An auxiliary system of equations
  9. Convergence of Approximating Sequences
  10. Main Theorem
  11. Numerical Illustrations
  12. Time marching
  13. Space discretization
  14. Manufactured tests with nonlocal Dirichlet boundary conditions
  15. Manufactured tests with nonlocal Neumann boundary conditions
  16. ...and 3 more sections

Key Result

Theorem 1

\newlabelt:existence0 Let $u_0,v_0,\partial_x u_0,\partial_x v_0$ be in $L^2(\Omega)$. Then, there exists positive constants $C_1,C_2$ and $T$, such that if the system of equations has a unique weak solution $(u,v) \in [L^2(0,T, H^1(\Omega) )] \times [L^2(0,T, H^1(\Omega)]$ valid on the time interval $[0,T]$, and satisfying $u(x,0) = u_0$ and $v(x,0)=v_0$.

Figures (2)

  • Figure 1: Pulse solutions for the nonlocal Gray-Scott model for different values of the dispersive range $a=\{ 3,5,7,9\}$ (solid lines) and for the local Gray-Scott model $a= \infty$ (dashed line). Figures a) and b) depict the $u$ and $v$ profiles for the solution, respectively. Figures c) and d) zoom in into a neighborhood of the origin. Curves that are closer to dashed line correspond to larger values of $a$. Other parameters used are specified in the text.
  • Figure 2: Pulse solutions for the nonlocal Gray-Scott model with $a= 3$. Figure a) depicts the $u$ and $v$ profiles of the solution when $\Omega = [-40,40]$ (dashed-red line) and $\Omega = [-50,50]$ (solid-blue line). Figures b) zooms in into a neighborhood of the origin, where the solid-red line corresponds to $\Omega = [-40,40]$ and the blue circles to $\Omega= [-50,50]$.

Theorems & Definitions (27)

  • Theorem 1
  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proof 1
  • Proposition 2.4
  • Lemma 2.5
  • Proof 2: Sketch proof
  • Lemma 3.1
  • ...and 17 more