http://ptrinh.com/ 2026-05-14T21:56:22+00:00 http://ptrinh.com/ http://ptrinh.com/ttp://ptrinh.com/lib/tpl/white/images/favicon.ico text/html 2021-09-06T01:53:43+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:about?rev=1630893223&do=diff About this project Rationale Pattern formation is a beautiful subject that has strong links to both applied mathematics and the natural sciences. The purpose of this project is to use a multimedia web environment to visualize the quantitative principles underlying reaction-diffusion models of pattern formation. text/html 2021-09-06T01:06:06+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:navigation?rev=1630890366&do=diff <WRAP noprint> </WRAP> text/html 2021-09-06T01:53:17+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:numerics?rev=1630893197&do=diff A primer in numerical methods Beyond visualizing the mathematical concepts in pattern formation, you may be curious to learn about how the animations were actually implemented using MATLAB. In this section I want introduce some of the key ideas behind the code that was used to generate the animated gifs, guide you through some of the code, and make it accessible for you to make similar animations as well. \begin{equation} \label{heatEqn} \frac{\partial u}{\partial t}\, - \,D_u \frac{\partial … text/html 2021-09-06T01:06:06+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:nummethods?rev=1630890366&do=diff Numerical Methods for solving PDE's As a amateur in applied mathematics, I find that my biggest frustration is not knowing how to implement the models presented in papers I read. The pretty figures in the papers seemed so compelling but at the same time unreachable. In this section I want to try and step through some of the key ideas behind the code that was used to generate the gifs scattered around the website. This is an extremely basic introduction to the numerical methods to solve PDE's… text/html 2021-09-06T01:51:55+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:pattern?rev=1630893115&do=diff Pattern Formation in Nature Have you ever wondered why tigers have stripes, giraffes have spots, and humans don't have either? If you have, how did you go about answering that question? Since leopards, tigers, and humans are all different species, it might make sense that our appearance would be different. But what exactly makes us look different? Maybe, you might say, it is all genetically programmed. Maybe nature has a library with volumes of recipe book: one for humans, one for zebras, on… text/html 2021-09-06T01:52:13+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:reactdiffuse1?rev=1630893133&do=diff Reaction-Diffusion Equations In this chapter, we study a class of partial differential equations (PDE's) called Reaction-Diffusion Equations, which are frequently used in modeling, and describe the diffusion (spreading out) and reaction of one or several chemical species.$u(x,t)$$x$$t$\begin{equation} \frac{\partial u}{\partial t} = \underbrace{D_u \frac{\partial^2 u} {\partial x^2}}_{\text{Diffusion}} + \overbrace{f(u)}^{\text{Reaction}}, \label{eqn:reactdiffuse1D} \end{equation}$D_u$$u$$f(u… text/html 2021-09-06T01:52:29+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:reactdiffuse2?rev=1630893149&do=diff Reaction-Diffusion Equations 2 Why do we need Diffusion AND Reactions? What benefit do diffusion-reaction models have over just reaction or just diffusion models? As discussed in the introduction, one of the big questions in developmental biology is how complex organisms emerge from a single fertilized egg. The answer has long been thought to be chemical gradients. If you have different chemicals at different concentrations throughout the embryo, and you apply a threshold function so tha… text/html 2021-09-06T01:52:46+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:turinginst1?rev=1630893166&do=diff Turing Instabilities: Part 1 Before we go into pattern formation let's quickly review the key concepts of Reaction-Diffusion equations: * Diffusion: It can be understood as arising from Brownian motion, and it tends to have a “smoothing-out”\begin{alignat}{3} \frac{\partial u}{\partial t} &= D_u\frac{\partial^2u}{\partial x^2} &&+ f(u,v), \label{difrxn1} \\ \frac{\partial v}{\partial t} &= D_v\frac{\partial^2 v}{\partial x^2} &&+ g(u,v). \label{difrxn2} \end{alignat}$D_u$$D_v$$u$$v$$u(… text/html 2021-09-06T01:52:56+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:turinginst2?rev=1630893176&do=diff Turing Instabilities Part 2: Gierer-Meinhardt Model The previous section derived the conditions that are needed for a Turing instability to exist. Now, let us step through an example in order to see how this works in practice. We will consider the Gierer-Meinhardt model, which is a reaction diffusion system that describes an activator-inhibitor interaction. It is one of the equations that have been used to model morphogenesis and patterns in development, though experimental evidence is sti… text/html 2021-09-06T01:53:08+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:turinginst3?rev=1630893188&do=diff Turing Instabilities and Tiger stripes All of the sections up until now have been building up the mathematical foundations that we need to answer the question I asked at the very beginning: Where do tigers get their stripes? The systems that we have considered so far are all one-dimensional, but it is not difficult to extend our analysis into two dimensions. We can revisit the idea of spatial dependence of patterns with the 2D Gierer Meinhardt model on a thin domain and on a square domain.\… text/html 2021-09-06T01:53:32+00:00 Anonymous (anonymous@undisclosed.example.com) http://ptrinh.com/dynsys:vids?rev=1630893212&do=diff Mathematics of Patterns: Video Series To watch all of the videos on a Youtube playlist go here * Home * Pattern formation in nature * Reaction-diffusion I * Reaction-diffusion II * Turing I * Turing II * Turing III * Numerical methods and Matlab * Videos * About