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Modelling pattern formation by numerical integration of systems of coupled convection-diffusion differential equations.

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pattern-formation

About

Project that models the spontaneous formation of patterns in reaction-diffusion system based on Alan Turing's paper of 1952 "The Chemical Basis of Morphogenesis" [1], where, the development of distinct biological features is modelled as a coupled reaction-diffusion process between two morphogens $u$ and $v$, i.e.,

$$ \partial_{t} u = \delta_{1} \Delta u + f(u,v), $$

$$ \partial_{t} v = \delta_{2} \Delta v + g(u,v). $$

Here we solved numerically the above system of differential equations using the Euler and the Crank-Nicolson integration rules and found a better performance in the latter case, as expected due to its implicit nature and a higher convergence order. The computational expense of an implicit integrator (solving a nonlinear fixed-point problem) is, in this case, compensated with by the property of unconditional stabillity. This allows integration with large time steps $\Delta t > 1$, which considerably reduces the computational time. The computational properties were studied using models from literature for abstract pattern generation [2], as well as formation of animal fur patterns [3-4], which were specified by the values of parameters $\delta_{1}$ and $\delta_{2}$ and functional dependencies $f(u,v)$ and $g(u,v)$.

For more details on the methods and results see the presentation slides.

Examples of patterns

me me

See other patterns in the figures folder and the presentation slides.

Credits

This was a coursework semester project in MSc Computational Sciences program at the Freie Universitaet Berlin in 2019. The project was initiated and supervised by Dr Gottfried Hastermann and performed by a group of 3 people: Cristina Melnic, Ece Sanin and Kevin Cyriac Edampurath.

References

[1]. A. M. Turing, “The chemical basis of morphogenesis", Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, vol. 237, pp. 37{72, Aug. 1952.}

[2]. J. E. Pearson, “Complex Patterns in a Simple System,” Science, vol. 261, pp. 189–192, July 1993.

[3]. A. J. Koch and H. Meinhardt “Biological Pattern Formation: from Basic Mechanisms to Complex Structures”, Rev. Modern Physics 66, 1481-1507 (1994)

[4]. R. T. Liu, S. S. Liaw, and P. K. Maini, "Two-stage Turing model for generating pigment patterns on the leopard and the jaguar," Physical Review E, vol. 74, p. 011914, July 2006.

Contents

  • solvers.py - Convection-diffusion solver classes that implement numerical integration of coupled convection-diffusion equations to get the solutions $u(t,x,y)$ and $v(t,x,y)$;
  • models.py - Classes that specify the type of parameters and functional dependencies between the 2 unknowns u and v;
  • performance.py - Benchmarking function.

TO DO:

  • Add description to classes and functions.
  • Update the test.

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Modelling pattern formation by numerical integration of systems of coupled convection-diffusion differential equations.

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