1D Reaction-Diffusion: Activator-Inhibitor
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Introduction
The first example models a 1D activator-inhibitor model (Gierer and Meinhardt, 1972).
Description
This 1D PDE model uses a Lattice
with linear
structure and periodic boundary conditions.
The PDE
defined two species called Layers
: $A$ (activator) and $I$ (inhibitor) with resp. low and high Diffusion
rates. The reaction part of the equations are defined in the System
. Similar to ODE model, the System
defines Constants
and DiffEqn
s.
The results are recorded and visualized using the Logger
and SpaceTimeLogger
.
Reference
A. Gierer, H. Meinhardt: A Theory of Biological Pattern Formation. Kybernetik 12: 30-39, 1972.
Model
Examples
→ PDE
→ ActivatorInhibitor_1D.xml
or
ActivatorInhibitor_1D.xml
XML Preview
<MorpheusModel version="3">
<Description>
<Title>Example-ActivatorInhibitor1D</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="1">
<DiffEqn symbol-ref="a">
<Expression>(rho*a^2) / i - mu_a * a + rho_a</Expression>
</DiffEqn>
<DiffEqn symbol-ref="i">
<Expression>(rho*a^2) - mu_i * i</Expression>
</DiffEqn>
<Constant symbol="rho_a" value="0.01"/>
<Constant symbol="mu_i" value="0.03"/>
<Constant symbol="mu_a" value="0.02"/>
<Constant symbol="rho" value="0.001"/>
</System>
</Global>
<Space>
<Lattice class="linear">
<Size symbol="size" value="100 0 0"/>
<BoundaryConditions>
<Condition boundary="x" type="periodic"/>
</BoundaryConditions>
<NodeLength value="0.25"/>
<Neighborhood>
<Order>1</Order>
</Neighborhood>
</Lattice>
<SpaceSymbol symbol="space"/>
</Space>
<Time>
<StartTime value="0"/>
<StopTime value="4000"/>
<SaveInterval value="0"/>
<!-- <Disabled>
<RandomSeed value="1"/>
</Disabled>
-->
<TimeSymbol symbol="time"/>
</Time>
<Analysis>
<Logger time-step="25">
<Input>
<Symbol symbol-ref="a"/>
<Symbol symbol-ref="i"/>
</Input>
<Output>
<TextOutput/>
</Output>
<Plots>
<Plot time-step="250" title="space plot">
<Style style="lines" line-width="3.0"/>
<Terminal terminal="png"/>
<X-axis>
<Symbol symbol-ref="space.x"/>
</X-axis>
<Y-axis minimum="0" maximum="3.5">
<Symbol symbol-ref="a"/>
<Symbol symbol-ref="i"/>
</Y-axis>
<Range>
<Time mode="current"/>
</Range>
</Plot>
<Plot time-step="-1" title="time-space plot">
<Style style="points"/>
<Terminal terminal="png"/>
<X-axis>
<Symbol symbol-ref="time"/>
</X-axis>
<Y-axis>
<Symbol symbol-ref="space.x"/>
</Y-axis>
<Color-bar>
<Symbol symbol-ref="a"/>
</Color-bar>
</Plot>
</Plots>
</Logger>
</Analysis>
</MorpheusModel>
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