# Spatial parameter sweep: Turing patterns

## Introduction

This model shows the pattern formation abilities of Turing’s linear activator-inhibitor model (Miyazawa et al., 2010). It shows how to vary parameters as a function of space.

## Model description

Instead of fixed parameters defined as Constants, this model uses Functions for two parameters. The parameters $C$ (activator production) and $A$ (rate of auto-activation) are defined as Function of space and varied over the $x$- and $y$-axes respectively. This requires the definition of a SpaceSymbol that can be used in expressions.

The results show the appearance of white spots (left), black spots (right) and labyrinthine patterns (middle).

## Model

In Morpheus GUI: Examples 🠒 PDE 🠒 TuringPatterns.xml.

<MorpheusModel version="3">
<Description>
<Title>Example-TuringPatterns</Title>
<Details>
Miyazawa, Okamoto and Kondo, Blending of animal colour patterns by hybridization, Nature Communications, 2010</Details>
</Description>
<Global>
<Field symbol="u" value="4.1+rand_uni(0,1)">
<Diffusion rate="1"/>
</Field>
<Field symbol="v" value="4.84+rand_uni(0,1)">
<Diffusion rate="20"/>
</Field>
<System solver="runge-kutta" time-step="0.25" name="Miyazawa">
<Function symbol="A">
<Expression>0.07 + ((0.07 * l.y)/ s.y)</Expression>
</Function>
<Constant symbol="B" value="0.08"/>
<Function symbol="C">
<Expression>-0.1 + ((0.5 * l.x)/ s.x)</Expression>
</Function>
<Constant symbol="D" value="0.03"/>
<Constant symbol="E" value="0.10"/>
<Constant symbol="F" value="0.12"/>
<Constant symbol="G" value="0.06"/>
<Constant symbol="R" value="20.0"/>
<Constant symbol="synU_max" value="0.23"/>
<Constant symbol="synV_max" value="0.50"/>
<Function symbol="s_u">
<Expression>max( 0, min( synU_max, A()*u-B*v+C()))</Expression>
</Function>
<Function symbol="s_v">
<Expression>max( 0, min( synV_max, E*u - F))</Expression>
</Function>
<DiffEqn symbol-ref="u">
<Expression>R*(s_u() - D*u)</Expression>
</DiffEqn>
<DiffEqn symbol-ref="v">
<Expression>R*(s_v() - G*v)</Expression>
</DiffEqn>
</System>
</Global>
<Space>
<Lattice class="square">
<Size symbol="s" value="512 512 0"/>
<NodeLength value="1"/>
<BoundaryConditions>
<Condition boundary="x" type="noflux"/>
<Condition boundary="y" type="noflux"/>
</BoundaryConditions>
<Neighborhood>
<Order>1</Order>
</Neighborhood>
</Lattice>
<SpaceSymbol symbol="l"/>
</Space>
<Time>
<StartTime value="0"/>
<StopTime value="30"/>
<SaveInterval value="0"/>
<RandomSeed value="1"/>
<TimeSymbol symbol="time"/>
</Time>
<Analysis>
<Gnuplotter time-step="2" decorate="false">
<Terminal persist="true" name="png"/>
<Plot>
<Field symbol-ref="u">
<ColorMap>
<Color value="1" color="black"/>
<Color value="0.0" color="white"/>
</ColorMap>
</Field>
</Plot>
</Gnuplotter>
<Logger time-step="0.0">
<Input>
<Symbol symbol-ref="u"/>
</Input>
<Output>
<TextOutput file-format="csv"/>
</Output>
<Restriction>
<Slice value="s.x/2" axis="x"/>
</Restriction>
<Plots>
<Plot title="slice at half of x extension" time-step="-1">
<Style style="lines" line-width="3.0"/>
<Terminal terminal="png"/>
<X-axis>
<Symbol symbol-ref="l.y"/>
</X-axis>
<Y-axis>
<Symbol symbol-ref="u"/>
</Y-axis>
<Color-bar>
<Symbol symbol-ref="v"/>
</Color-bar>
</Plot>
</Plots>
</Logger>
<Logger time-step="2">
<Input>
<Symbol symbol-ref="u"/>
</Input>
<Output>
<TextOutput file-format="csv"/>
</Output>
<Plots>
<SurfacePlot time-step="2">
<Color-bar>
<Symbol symbol-ref="u"/>
</Color-bar>
<Terminal terminal="png"/>
</SurfacePlot>
</Plots>
</Logger>
</Analysis>
</MorpheusModel>



## Reference

S. Miyazawa, M. Okamoto, S. Kondo: Blending of animal colour patterns by hybridization. Nature communications, 1(6), 66, 2010.