Partial differential equations as pattern forming systems

These are some early experiments inspired by the work of H Meinhardt and his book "The Algorithmic Beauty of Sea Shells".
In the book basic pattern forming systems are described as coupled nonlinear differential equations. Differential equations are good models for systems that evolve in time (as sea shells do). These simplified models try to approximate the processes that give rise to the patterns and pigmentation markings observed in some sea shells.
The examples here show very basic patterns that emerge from initial random fluctuations and simple repeated rules. Over time the random fluctuations and local interactions generate and maintain cycles and disruptions. At this stage of simplicity the pattern show primitive structures and features such as vertical and horizontal bands of pigment concentrations, propagating waves and wave interference, organic oscillations and zig-zagging boundaries, spots and blobs.