Domains: Reaction-Diffusion in Irregular Domains
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Spot pattern in Gierer-Meinhardt model with irregular domain.
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Introduction
A 2D activator-inhibitor model (Gierer and Meinhardt, 1972), solved in a irregular domain that is loaded from an image file.

Description
This model uses an irregular Domain
with constant
boundary conditions. The domain is loaded from a TIFF image. See Space/Lattice/Domain
.
Reference
A. Gierer, H. Meinhardt: A Theory of Biological Pattern Formation. Kybernetik 12: 30-39, 1972.
Model
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Examples
→PDE
→ActivatorInhibitor_Domain.xml
or - Download:
ActivatorInhibitor_Domain.xml
.
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<MorpheusModel version="3">
<Description>
<Title>Example-ActivatorInhibitor-2D-Domain</Title>
<Details>Meinhardt-Gierer (activator-inhibitor) model solved in a nonregular domain with constant boundaries.</Details>
</Description>
<Global>
<Field symbol="a" value="rand_norm(0.5,0.1)" name="activator">
<Diffusion rate="0.02"/>
<BoundaryValue boundary="domain" value="0.01"/>
</Field>
<Field symbol="i" value="0.1" name="inhibitor">
<Diffusion rate="0.25"/>
<BoundaryValue boundary="domain" value="0"/>
</Field>
<System solver="runge-kutta" time-step="5" name="Meinhardt">
<Constant symbol="rho" value="0.001"/>
<Constant symbol="rho_a" value="0.001"/>
<Constant symbol="mu_i" value="0.02"/>
<Constant symbol="mu_a" value="0.04"/>
<Constant symbol="kappa" value="0.10"/>
<DiffEqn symbol-ref="a">
<Expression>(rho/i)*((a^2)/(1 + kappa*a^2)) - mu_a * a + rho_a</Expression>
</DiffEqn>
<DiffEqn symbol-ref="i">
<Expression>rho*((a^2)/(1+kappa*a^2)) - mu_i *i</Expression>
</DiffEqn>
</System>
</Global>
<Space>
<Lattice class="square">
<Size symbol="size" value="100 100 0"/>
<NodeLength value="1"/>
<Domain boundary-type="constant">
<Image path=":/examples/PDE/domain.tif"/>
</Domain>
<Neighborhood>
<Order>1</Order>
</Neighborhood>
</Lattice>
<SpaceSymbol symbol="space"/>
</Space>
<Time>
<StartTime value="0"/>
<StopTime value="10000"/>
<SaveInterval value="0"/>
<RandomSeed value="2"/>
<TimeSymbol symbol="time"/>
</Time>
<Analysis>
<Gnuplotter time-step="200" decorate="false">
<Terminal size="400 400 0" name="png"/>
<Plot>
<Field symbol-ref="a" min="0"/>
</Plot>
</Gnuplotter>
</Analysis>
</MorpheusModel>
Note: This model requires the external file domain.tiff.
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