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Mathematical analysis of a model of morphogenesis
On the Gierer-Meinhardt system with precursors
| 1. | Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong |
| 2. | Brunel University, Department of Mathematical Sciences, Uxbridge UB8 3PH, United Kingdom |
$A_t=$ε2$A^{''}- \mu (x) A+\frac{A^2}{H} \mbox{ in } (-1, 1),$
$\tau H_t=D H^{''}-H+ A^2 \mbox{ in } (-1, 1),$
$ A' (-1)= A' (1)= H' (-1) = H' (1) =0.$
Such an equation
exhibits a typical Turing bifurcation of the
second kind, i.e., homogeneous uniform
steady states do not exist in the system.
We establish the existence and stability of
$N-$peaked steady-states in terms of the
precursor $\mu(x)$ and the diffusion
coefficient $D$. It is shown that $\mu (x)$
plays an essential role for both existence and
stability of spiky patterns. In particular, we
show that precursors can give rise to
instability. This is a new effect which
is not present in the homogeneous case.
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