A new mechanochemical model: Coupled Ginzburg-Landau and Swift-Hohenberg equations in biological patterns of marine animals

- A mathematical model was proposed to describe some cellular interactions in three out layers of animal marine skin.
- A detailed stability analysis of the model was realized.
- The dynamics of the model shows a great diversity of patterns similar to the skin pigment of some marine animals.
- The model involves some biological behavior of the chromatophores cellular movement observed recently.

In this work the skin coating of some vertebrate marine animals is modeled considering only dermis, epidermis and basal layers. The biological process takes into account: cellular diffusion of the epidermis, diffusion inhibition and long-range spatial interaction (nonlocal effect on diffusive dispersal) for cells of dermal tissue. The chemical and physical interactions between dermis and epidermis are represented by coupling quadratic terms and nonlinear terms additional. The model presents an interesting property associated with their gradient form: a connection between some physical, chemical and biological systems. The model equations proposed are solved with numerical methods to study the spatially stable emergent configurations. The spatiotemporal dynamic obtained of the numerical solution of these equations, present similarity with biological behaviors that have been found recently in the cellular movement of chromatophores (as contact-dependent depolarization and repulsion movement between melanophores, xanthophores and iridophores). The numerical solution of the model shows a great variety of beautiful patterns that are robust to changes of boundary condition. The resultant patterns are very similar to the pigmentation of some fish.