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.
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1. Polish Academy of Sciences, IPPT, Poland
2. University of Notre Dame