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Turing Pattern Simulations

Solutions Approximated by Explicit Forward Euler + Finite Differences

General System of Reaction Diffusion Equations

\[ \begin{aligned} \frac{\partial u_1}{\partial t} &= \sigma_1 \cdot \nabla^2 u_1 + R_1(u_1, u_2, .... , u_n)\\ \\ \frac{\partial u_2}{\partial t} &= \sigma_2 \cdot \nabla^2 u_2 + R_2(u_1, u_2, .... , u_n)\\ \\ \vdots \\ \\ \frac{\partial u_n}{\partial t} &= \sigma_n \cdot \nabla^2 u_n + R_n(u_1, u_2, .... , u_n)\\ \end{aligned} \] Here is a brief summary of the variables used in the equations:

\(u_n\): The concentration of the i-th species/chemical/etc.

\(t\): Time variable.

\(\sigma_i\): Diffusivity constant of the i-th species/chemical/etc.

\(\nabla^2\): Laplacian operator, representing diffusion in space.

\(R_i\): Reaction term for the i-th species/chemical/etc., which describes how the concentration changes due to reactions with other species/chemicals/etc.

Turing Pattern System

\[ \begin{aligned} \frac{\partial u}{\partial t} &= \sigma_u \cdot \nabla^2 u + \alpha(u-h) + \beta(v-k)\\ \\ \frac{\partial v}{\partial t} &= \sigma_v \cdot \nabla^2 v + \omega(u-h) + \lambda(v-k) \end{aligned} \]
Here we have two chemical concentrations, \(u\) and \(v\), which are governed by the reaction-diffusion equations. The parameters \(\sigma_u\), \(\sigma_v\), \(\alpha\), \(\beta\), \(\omega\), and \(\lambda\) control the diffusion and reaction rates of the chemicals. The constants \(h\) and \(k\) represent the threshold concentrations for the chemicals to react.

Red represents the concrentration of \(u\) and blue represents the concentration of \(v\). I represent the higher concentration as the pixel color of the corresponding chemical color.
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