On the Spatiotemporal Pattern Formation in Nonlinear Coupled Reaction–Diffusion Systems
Round 1
Reviewer 1 Report
The study delves into the pivotal role of nonlinear coupled reaction-diffusion (NCRD) systems in creating spatiotemporal patterns prevalent in a spectrum of scientific and technical arenas. To solve one and two-dimensional systems, the research adopts a mixed-type modal discontinuous Galerkin technique, adept at addressing the challenges posed by second-order derivatives in diffusion terms. Spatial solutions leverage hierarchical modal basis functions based on orthogonal scaled Legendre polynomials. The study also introduces an innovative method for handling reaction terms in NCRD systems, which ensures the accuracy of the discontinuous Galerkin scheme especially when faced with extreme nonlinearities. By translating the NCRD systems into a set of time-based ordinary differential equations, they are then tackled using an explicit third-order TVD Runge-Kutta method. The results align closely with existing literature, and the methodology offers potential scalability for complex multi-dimensional problems frequently encountered in advanced biological and chemical contexts.
Page 1, lines 20-23: among the studies of spatiotemporal pattern formation, you also have computational social sciences. See, for example, Dragicevic, A. (2018), Spacetime Discounted Value of Network Connectivity, Advances in Complex Systems, 21: 1850018-1–34.
Page 2: You also have convection-diffusion equations. See for example Dragicevic, A. and Gurtoo, A. (2022), Stochastic Control of Ecological Networks, Journal of Mathematical Biology, 85: 7. While both reaction-diffusion and convection-diffusion types of equations account for the diffusion process, convection-diffusion equations also incorporate the effects of bulk motion, whereas reaction-diffusion equations involve local reactions. In nonlinear coupled reaction-diffusion systems, convection could be introduced (−v⋅∇u) to capture a wider range of behaviors. What is your opinion on this matter?
Page, 2, line 41: Do you make any assumptions on the interplay between the chemical-species components u and v (stability, feedback…)?
Page 5, line 132: The statement "which can be thought of as the derivatives of unknown variables" is ambiguous. Which unknown variables or components? Where do they come from?
Page 9, lines 235-237: The major concern is that the validation of the spatiotemporal pattern is based solely on the computational results of Zegeling and Kok. Ideally, validation should be carried out against multiple studies or well-accepted benchmark problems to ensure the robustness and accuracy of the results. Can you provide such proof?
Pages 21-24: The conclusion tends to be verbose. While detailed explanations are valuable, conclusions typically summarize the main findings and implications succinctly. Here, there is a lot of repetition and over-explanation of the methods used. The statement "This innovative treatment showcases a fundamental characteristic of the novel DG scheme," is vague. It would be beneficial to specify what this fundamental characteristic is. The conclusion does a good job of pointing toward future work.
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Nonlinear coupled reaction-diffusion (NCRD) systems play a vital role in the emergence of spatiotemporal patterns across various scientific and engineering fields. In this study, the author examines NCRD systems that encompass a variety of models, including linear, Gray-Scott, Brusselator, isothermal chemical, and Schnakenberg models, all aimed at capturing the diverse spatiotemporal patterns they produce. These models represent intricate patterns observed in nature, such as spots, spot replication, stripes, hexagons, and more. To solve one and two-dimensional NCRD systems, the author employs a mixed-type modal discontinuous Galerkin approach. This approach introduces a mathematical formulation designed to handle second-order derivatives in diffusion terms. The article can be considered for publication after addressing the following comments:
1. Explain, why you considered The most general form of these reaction kinetics in (4), why this specific form?
2. Elaborate more on the Discretization process in the modal DG framework.
3. Mention the specifications of the machine used to depict the results.
4. Mention the platform used to generate all figures.
5. Comment on the results in Figure (3)
6. Define the tensor product used in the appendix.
7. Proofread the whole text for typos and Grammer Errors.
Minor editing of English language required.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
