cristina-v-melnic / pattern-formation

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

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

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.