Team:Evry/Auxin diffusion
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Using PDE instead of ODE allows one to take into account the space dimensions. Where in the ODE model, the concentration of auxin in a compartment was considered homogeneous, here we can represent the variations internal to the compartment. When taking into account the space dimensions, new problems arise: finding a coherent geometry, 2D vs 3D model, precision, etc. We chose to model a slice of a tadpole's tail based on images from [ADD BOOK REF]. Putting multiple such slices one after the other allows us to approximate a 3 dimensions model.  Using PDE instead of ODE allows one to take into account the space dimensions. Where in the ODE model, the concentration of auxin in a compartment was considered homogeneous, here we can represent the variations internal to the compartment. When taking into account the space dimensions, new problems arise: finding a coherent geometry, 2D vs 3D model, precision, etc. We chose to model a slice of a tadpole's tail based on images from [ADD BOOK REF]. Putting multiple such slices one after the other allows us to approximate a 3 dimensions model.  
Revision as of 17:17, 23 September 2012
Model using Partial Differential Equations(PDE)
Overview

Using PDE instead of ODE allows one to take into account the space dimensions. Where in the ODE model, the concentration of auxin in a compartment was considered homogeneous, here we can represent the variations internal to the compartment. When taking into account the space dimensions, new problems arise: finding a coherent geometry, 2D vs 3D model, precision, etc. We chose to model a slice of a tadpole's tail based on images from [ADD BOOK REF]. Putting multiple such slices one after the other allows us to approximate a 3 dimensions model.
 The quantity of auxins is homogeneous in the blood
 x = (x_{1}, x_{2}) is a 2 dimensional vector
 c is a function representing the auxin concentration
 D is the diffusion constant
 Δ is the Laplacian operator
 Atlas of xenopus development,G. Bernardini, Springer, 1999.
Hypothesis
There are the different hypothesis we were constrained to make in order to model the system:Model description
The PDE model is similar to the ODE one except that it takes into account the geometry. This allows us to model more complex phenomenon such as diffusion and transport.Equations
The diffusion equation is used to model the repartition of auxin's molecule when not subject to any flow (in the skin for example). The equation is states as follows:Geometry
Limit conditions
Calibration
Results
Conclusion
References
References: