February  2009, 25(1): 363-398. doi: 10.3934/dcds.2009.25.363

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

Received  May 2007 Revised  January 2008 Published  June 2009

We consider the following Gierer-Meinhardt system with a precursor $ \mu (x)$ for the activator $A$ in $\mathbb{R}^1$:

$A_t=$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.

Citation: Juncheng Wei, Matthias Winter. On the Gierer-Meinhardt system with precursors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 363-398. doi: 10.3934/dcds.2009.25.363
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