Domains: Reaction-Diffusion in Irregular Domains

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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.

Spot pattern in Gierer-Meinhardt model with irregular domain.
Spot pattern in Gierer-Meinhardt model with irregular domain.

Description

This model uses an irregular Domain with constant BoundaryConditions. The domain is loaded from a TIFF image. See Space:Lattice:Domain.

Reference

This model reproduces a published result, originally obtained with a different simulator:

A. Gierer, H. Meinhardt: A Theory of Biological Pattern Formation. Kybernetik 12: 30-39, 1972.

Model

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  • Morpheus GUI: ExamplesPDEActivatorInhibitor_Domain.xml or
  •  Download: ActivatorInhibitor_Domain.xml
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    <MorpheusModel version="4">
        <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" name="activator" value="rand_norm(0.5,0.1)">
                <Diffusion rate="0.02"/>
                <BoundaryValue boundary="domain" value="0.01"/>
            </Field>
            <Field symbol="i" name="inhibitor" value="0.1">
                <Diffusion rate="0.25"/>
                <BoundaryValue boundary="domain" value="0"/>
            </Field>
            <System time-step="5" name="Meinhardt" solver="Runge-Kutta [fixed, O(4)]">
                <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="M0013_model_domain.tiff"/>
                </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>
            <ModelGraph format="dot" reduced="false" include-tags="#untagged"/>
        </Analysis>
    </MorpheusModel>
    
    

    This model requires the external file M0013_model_domain.tiff, which must be inside the same directory as ActivatorInhibitor_Domain.xml ( | ).
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