where L and T are whatever length and timescales we choose as the most relevant. The equations then become d n
= sp u, where we have omitted the * for algebraic convenience.
Relevant initial conditions for this system are u = 0, n = 1,p = 1 for r outside the initial wound boundary and n = 0 inside the wound. Since there is no ECM production we must assume that p (r, 0) = 1 inside the wound as well since we are in effect assuming the (fibrin) blood clot forms instantaneously relative to the contraction phase: this is not unreasonable (Clark 1985). Further, Murray et al. (1988) assumed that the mechanical properties of the wound matrix are the same as the surrounding matrix and remain so as it becomes modified by the cell biosynthesis.
As with all models the parameter values play an important role. In the case of this base model parameter sets must satisfy certain constraints. In particular the uniform steady state must be stable to perturbations in the dependent variables. With the number of stability analyses we have carried out in the book it is a now routine procedure to do a linear analysis on the model set of equations (10.5) about the uniform (unwounded) steady state n = 1,p = 1, u = 0. It is left as an exercise to show that the necessary and sufficient conditions for stability are:
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