Comparison of Two Aspects of a PDE Model for Biological Network Formation
, Daniele Boffi 1,2, Jan Haskovec 1, Peter Markowich 1,3 and Giovanni Russo 1,4Round 1
Reviewer 1 Report
The manuscript presents the differences in the solutions of the two systems of partial differential equations, seen as two different interpretations of the same model that describes formation of complex biological networks.
To the best of my knowledge, the results in the manuscript are new. In addition, this paper is well organized, and the arguments are appropriate to obtain the present outcomes except for some typos.
In consequence, upon appropriate revision to the points listed below, the manuscript should be suitable for publication in Math. Comput. Appl..
1. P3, L54: The format of $C$ is different from others.
2. P3, L62-68: I think that one of the differences between the two systems comes from not only the metabolic exponent $\gamma$ but also the full metabolic term.
Precisely, since $\mathbb{C}=m\otimes m$, $|\mathbb{C}|^{\gamma-2}\mathbb{C}$ formally becomes $|m|^{2(\gamma-2)}m\otimes m$ rather than $|m|^{2(\gamma-1)}m$.
3. I don't sure that whether or not [3] should be replaced by (D. Hu, D. Cai, Commun. Math. Sci., 2019, 17(5)).
4. [11] should be updated.
Author Response
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Reviewer 2 Report
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Comments for author File: Comments.pdf
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Reviewer 3 Report
In this paper, the author used a system of PDEs of elliptic-parabolic type to model the formation of biological network. In particular, the authors compared the solutions of two different systems: one for the conductivity vector and one for the conductivity tensor exploring the dependence of
the solution on the parameters of the two models.
The authors present a numerical scheme based on finite differences scheme, with central differences for the space discretization and a symmetric-ADI method in time. This discretization allows to treat efficiently the stiffness induced by the components of the systems, and achieve second order accuracy and preserve spatial symmetry.
Comparisons of the solution of the two systems are provided and discussed in details, confirming the theoretical results of [7].
Overall the paper is very well discussed and presented, and I suggest pubblication after minor revision:
- Interestingly the comparison between the zero-diffusion and small diffusion for the C-model produces the same behavior. Indeed it seems that the sole reaction and the coupling with the pressure component can ingenerate the formation of ramification, without diffusion in the C-model. I wonder how much the numerical diffusion influences such comparison, and if the author can discuss better this point.
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