Alan Turing-ek morfogenesiaren inguruan egindako ikerketaren analisi matematikoa

Abstract

Artikulu honetan, Alan Turing matematikariak 1952an morfogenesiaren inguruan aurkeztu zuen The Chemical Basis of Morphogenesis, [19], lanaren zati bat aurkeztu eta matematikoki garatuko dugu. Horretarako, beharrezkoak diren matematikako kontzeptuak eta baliabideak azalduko ditugu. Konkretuki, Alan Turingek [19] artikuluko "Reactions and Diffusion in a Ring of Cells" (eraztun diskretuko eremua) eta "Continuous Ring of Tissue" (eraztun jarraituko eremua) ataletan gehiegi sakondu edo zehaztu gabe erabili zituen ekuazio diferentzialak, erreakzio-difusio ekuazioak, Fourieren serieak eta funtzioen linealizazioa azalduko ditugu, eta bi eremu horietan planteatutako ekuazio-sistemen soluzioak bilatzeko erabiliko ditugu. Artikulu hau UPV/EHUko Idoia Marauri ikasle ohiaren Gradu Amaierako Lanean oinarrituta dago.; In this article, we show and mathematically develop a part of the work that the mathematician Alan Turing did on Morphogenesis which he published in 1952 in his article The Chemical Basis of Morphogenesis. We will explain the mathematical concepts and resources needed to do so: differential equations, reaction-diffusion equations, Fourier series and function linearization. Specifically, we will show and explain all these mathematical tools that Alan Turing used but did not develop nor delve into a lot in his aforementioned article’s two sections: “Reactions and Diffusion in a Ring of Cells" (for the discrete ring region) and “Continuous Ring of Tissue" (for the continuous ring region). This article is based on the UPV/EHU former student Idoia Marauri’s Final Degree Project.