2D Reaction-Diffusion: Activator-Inhibitor

A. Gierer, H. Meinhardt (Authors)

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Introduction

A 2D activator-inhibitor model (Gierer and Meinhardt, 1972).

Stripe pattern generated by 2D Gierer-Meinhardt model.

Description

This model uses a standard Lattice with square structure and periodic boundary conditions.

The PDE defined two species or Layers $A$ (activator) and $I$ (inhibitor) with resp. low and high Diffusion rates. The reaction part of the equations are defined in the System.

The results are visualized using the Gnuplotter.

Things to try

Alter the level of saturation of the activator by changing kappa (for example, using a ParamSweep). By changing this parameter, the model can generate stripes, nets and spots.

Reference

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

Model

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Download: ActivatorInhibitor_2D.xml

XML Preview

<MorpheusModel version="3">
    <Description>
        <Title>Example-ActivatorInhibitor-2D</Title>
        <Details></Details>
    </Description>
    <Global>
        <Field symbol="a" value="rand_norm(0.5,0.1)" name="activator">
            <Diffusion rate="0.02"/>
        </Field>
        <Field symbol="i" value="0.1" name="inhibitor">
            <Diffusion rate="1"/>
        </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="0.03"/>
            <Constant symbol="mu_a" value="0.02"/>
            <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="150 150 0"/>
            <NodeLength value="1"/>
            <BoundaryConditions>
                <Condition boundary="x" type="periodic"/>
                <Condition boundary="y" type="periodic"/>
            </BoundaryConditions>
            <Neighborhood>
                <Order>1</Order>
            </Neighborhood>
        </Lattice>
        <SpaceSymbol symbol="space"/>
    </Space>
    <Time>
        <StartTime value="0"/>
        <StopTime value="12500"/>
        <SaveInterval value="0"/>
        <RandomSeed value="2"/>
        <TimeSymbol symbol="time"/>
    </Time>
    <Analysis>
        <Gnuplotter time-step="500" decorate="false">
            <Terminal size="400 400 0" name="png"/>
            <Plot>
                <Field symbol-ref="a" min="0"/>
            </Plot>
        </Gnuplotter>
        <Logger time-step="250">
            <Input>
                <Symbol symbol-ref="a"/>
            </Input>
            <Output>
                <TextOutput file-format="csv"/>
            </Output>
            <Plots>
                <Plot time-step="0">
                    <Style style="lines" line-width="5.0"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="space.x"/>
                    </X-axis>
                    <Y-axis minimum="0" maximum="3">
                        <Symbol symbol-ref="a"/>
                    </Y-axis>
                    <Color-bar>
                        <Symbol symbol-ref="i"/>
                    </Color-bar>
                    <Range>
                        <Data/>
                        <Time mode="current"/>
                    </Range>
                </Plot>
            </Plots>
            <Restriction>
                <Slice value="size.y/2" axis="y"/>
            </Restriction>
        </Logger>
        <Logger time-step="100">
            <Input>
                <Symbol symbol-ref="a"/>
            </Input>
            <Output>
                <TextOutput file-format="matrix"/>
            </Output>
            <Plots>
                <SurfacePlot time-step="500">
                    <Color-bar>
                        <Symbol symbol-ref="a"/>
                    </Color-bar>
                    <Terminal terminal="png"/>
                </SurfacePlot>
            </Plots>
            <Restriction>
                <Slice value="size.y/2" axis="y"/>
            </Restriction>
        </Logger>
    </Analysis>
</MorpheusModel>

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Last updated on Mon, 13 May 2024