On the growth of filamentary structures in planar media

dc.contributor.authorAndreucci, D.
dc.contributor.authorHerrero, Miguel A.
dc.contributor.authorVelázquez, J.J. L.
dc.description.abstractWe analyse a mathematical model for the growth of thin filaments into a two dimensional medium. More exactly, we focus on a certain reaction/diffusion system, describing the interaction between three chemicals (an activator, an inhibitor and a growth factor), and including a fourth cell variable characterising irreversible incorporation to a filament. Such a model has been shown numerically to generate structures shaped like nets. We perform an asymptotical analysis of the behaviour of solutions, in the case when the system has parameters very large and very small, thereby allowing the onset of different time and space scales. In particular, we describe the motion of the tip of a filament, and the changes in the relevant chemical species nearby.
dc.description.departmentDepto. de Análisis Matemático y Matemática Aplicada
dc.description.facultyFac. de Ciencias Matemáticas
dc.identifier.citationMeinhardt H. Morphogenesis of lines and nets. Differentiation 1976; 6:117–123. Meinhardt H. Biological pattern formation as a complex dynamic phenomenon. International Journal of Bifurcation and Chaos 1997; 7(1):1–26. Gierer A, Meinhardt H. A theory of biological pattern formation. Kibernetik 1972; 12:30–39. Bellettini G, Fusco G. Stable dynamics of spikes in solutions to a system of reaction-diffusion equations. Asymptotic Analysis 2001; 26:307–357. Kolokolnikov T, Ward M. Reduced Green's functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model. European Journal of Applied Mathematics 2003; 14:513–545. Ni WM, Takagi I. Locating the peaks of least-energy solutions to a semilinear Neuman problem. Duke Mathematical Journal 1993; 70:247–281. Ni WM, Takagi I. Point condensation generated by a reaction-diffusion system in axially symmetric domains. Japan Journal of Industrial and Applied Mathematics 1995; 12:327–365. Ward M. An asymptotic analysis of localised solutions for some reaction-diffusion models in multidimensional domains. Studies in Applied Mathematics 1996; 2:103–126. Wei J. On the interior spike layer solutions for some singular perturbation problems. Proceedings of the Royal Society of Edinburgh, Section A 1996; 128:849–874. Wei J. On single interior spike solutions of the Gierer-Meinhardt system: Uniqueness and spectrum estimates. European Journal of Applied Mathematics 1999; 10:353–378. Wei J, Winter M. On the two-dimensional Gierer-Meinhardt system with strong coupling. SIAM Journal on Mathematics Analysis 1999; 30:1241–1263. Ni WM. Diffusion, cross-diffusion and their spike-layer steady states. Notices of the AMS 1998; 451:9–18. Levine HA, Sleeman BD, Nilsen-Hamilton M. Mathematical modeling of the onset of capillary formation initiating angiogenesis. Journal of Mathematical Biology 2001; 42:195–238. Levine HA, Pamuk S, Sleeman BD, Nilsen-Hamilton M. Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bulletin of Mathematical Biology 2001; 63:801–863. Meinhardt H. Models of Biological Pattern Formation. Academic Press: NY, 1982. del Pino M, Felmer PL. Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting. Indiana University Mathematics Journal 1999; 48(3):883–898. Rothe F. Global solutions of reaction-diffusion systems. Lecture Notes in Mathematics, vol. 1072. Springer: Berlin, 1984. Berestycki H, Lions PL, Peletier LA. An ODE approach to the existence of positive solutions to semilinear problems in RN. Indiana University Mathematics Journal 1981: 30(1):141–157. Gidas B, Spruck J. Global and local behaviour of positive solutions of nonlinear elliptic equations. Communications on Pure Applied Mathematics 1981; 34:525–598. Gidas B, Ni WM, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Mathematical Analysis and Applications, Part A (Adv. Math. Suppl. Stud.), vol. 7. Academic Press: NJ, 1981; 369–402. Kwong MK. Uniqueness of the positive solutions of Δu−u+up = 0 in Rn. Archive for Rational Mechanics and Analysis 1989; 105:243–266. Bender CM, Orzag SA. Advanced Mathematical Methods for Scientists and Engineers. Springer: Berlin, 1999. Abramowitz M, Stegun IA. Handbook of Mathematical Functions. Dover: New York, 1970. Lebedev NN. Special Functions And Their Applications. Dover: NY, 1970. Ikebe T. Eigenfunction expansions associated with the Schrödinger operators and their applications to scattering theory. Archive for Rational Mechanics and Analysis 1960; 5:1–34. Kato T. Perturbation Theory for Linear Operators. Springer: Berlin, 1970. Gilbarg D, Trudinger N. Elliptic Partial Differential Equations of Second Order. Springer: Berlin, 2000. Iron D, Ward MI. A metastable spike solution for a nonlocal reaction-diffusion model. SIAM Journal on Applied Mathematics 2000; 60(3):778–802. Reed M, Simon B. Functional Analysis. Academic Press: NY, 1980.
dc.journal.titleMathematical Methods in the Applied Sciences
dc.publisherJohn Wiley and Sons
dc.rights.accessRightsrestricted access
dc.subject.keywordBiological pattern-formation
dc.subject.keywordGierer-Meinhardt system
dc.subject.keywordpositive solutions
dc.subject.keywordcapillary formation
dc.subject.keywordreaction-diffusion systems
dc.subject.keywordasymptotic behaviour of solutions
dc.subject.keywordsingular perturbation techniques
dc.subject.keywordmathematical biology
dc.subject.ucmEcuaciones diferenciales
dc.subject.unesco2404 Biomatemáticas
dc.subject.unesco1202.07 Ecuaciones en Diferencias
dc.titleOn the growth of filamentary structures in planar media
dc.typejournal article
Original bundle
Now showing 1 - 1 of 1
No Thumbnail Available
260.55 KB
Adobe Portable Document Format