Heteroclinic solutions for a model of skin morphogenesis
B. Kazmierczak1,2 K. Piechor1
In the early stages of embryonic growth the skin of vertebrates consists of two layers: the epidermis with its cells densely packed and the dermis with its cells more loosely packed. The epidermis and the dermis are separated with a thin fibrous basal lamina. One of the models of skin morphogenesis was proposed by Cruywagen and Murray (1992). In this model the epithelial layer is treated as a visco-elastic medium with a small Reynolds number, whereas the motion of the dermis cells is described by reaction-diffusion-chemotaxis equations.
The above model is considered from the point of view of travelling wave solutions. These waves connect two characteristic states of the dermis cell densities (with zero dilation and the same epithelial densities).
The system is analyzed under two simplifying conditions:
· traction produced in the epithelial layer by the morphogen secreted in the epidermis is much weaker than the elastic forces and the force exerted by the basal lamina
· The force exerted by the basal lamina is much bigger than the other forces acting on the epithelial.
In the second case one the equations of the model becomes singularly perturbed.
The existence of travelling wave solutions is proved as well as its positivity.
1. Polish Academy of Sciences, IPPT, Poland
2. University of Notre Dame