Diffusion, Cross-diffusion and Competitive Interaction

Iida, Masato; Mimura, Mamoru; Ninomiya, Hirokazu

doi:10.1007/s00285-006-0013-2

Cited by 112 publications

(104 citation statements)

References 15 publications

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“…This form is slightly different from the conventional Fickian diffusion coefficient, which is studied in mathematical ecology under a different context [14,15]. For example, if we apply the molecule in liquid to the extracellular matrix region, which is similar to the situation in Fig.…”

Section: /Hour)mentioning

confidence: 99%

In vitroVasculogenesis Models Revisited - Measurement of VEGF Diffusion in Matrigel

Miura

Tanaka

2009

Math. Model. Nat. Phenom.

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Abstract. The circulatory system is one of the first to function during development. The earliest event in the system's development is vasculogenesis, whereby vascular progeniter cells form clusters called blood islands, which later fuse to form capillary networks. There exists a very good in vitro system that mimics this process. When HUVECs (Human Umbilical Vein Endothelial Cells) are cultured on Matrigel, they spontaneously form a capillary network structure. Two theoretical models have been proposed to explain the pattern formation of this in vitro system. Both models utilize chemotaxis to generate spatial instability, and one model specifies VEGF as the chemoattractant. However, there are several unknown factors concerning the experimental model. First, the pattern formation process occurs at the interface between the liquid medium and Matrigel, and it is unclear whether diffusion in the liquid or gel is critical. Second, the diffusion coefficient of VEGF, which determines the spatial scale of the capillary structure, has not been properly measured. In the present study, we modified the experimental system to clarify the effect of diffusion in Matrigel, and experimentally measured the diffusion coefficient of VEGF in this system. The relationship with the spatial scale of the pattern generated is discussed.

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Section: /Hour)mentioning

confidence: 99%

In vitroVasculogenesis Models Revisited - Measurement of VEGF Diffusion in Matrigel

Miura

Tanaka

2009

Math. Model. Nat. Phenom.

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“…In developmental biology, recent experimental findings demonstrate that cross-diffusion can be quite significant in generating spatial structure [10]. The effects of cross-diffusion on models for pattern formation (i.e., reaction-diffusion type) have been studied in many theoretical papers [13][14][15][16][17][18][19][20][21][22][23][24]. Recently, in [25] we showed that introducing cross-diffusion to a system of reaction-diffusion equations results in further relaxation of the conditions necessary for the emergency of patterns.…”

Section: Introductionmentioning

confidence: 99%

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

Madzvamuse

Barreira

2014

Phys. Rev. E

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Article (Published Version) http://sro.sussex.ac.uk Madzvamuse, A and Barreira, R (2014) Exhibiting cross-diffusion-induced patterns for reactiondiffusion systems on evolving domains and surfaces. Physical Review E, 90. 043307-1-043307-14. ISSN 1539-3755 This version is available from Sussex Research Online: http://sro.sussex.ac.uk/51334/ This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the URL above for details on accessing the published version. Copyright and reuse:Sussex Research Online is a digital repository of the research output of the University.Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available.Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.PHYSICAL REVIEW E 90, 043307 (2014) Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion system with cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in...

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“…This is why they are called cross-diffusions. The subject of cross-diffusions is also very specific and, to gain an insight, some papers are [4,5,9,12]. A strong property is existence of an entropy under the condition for the nonlinearities a i in (8. and the right hand side is negative under the condition (8.29).…”

Section: Cross-diffusionmentioning

confidence: 99%

The Fokker-Planck Equation

Perthame

2015

Lecture Notes on Mathematical Modelling in the Life Sciences

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Diffusion, Cross-diffusion and Competitive Interaction

Cited by 112 publications

References 15 publications

In vitroVasculogenesis Models Revisited - Measurement of VEGF Diffusion in Matrigel

In vitroVasculogenesis Models Revisited - Measurement of VEGF Diffusion in Matrigel

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

The Fokker-Planck Equation

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