# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2009, 25 (1) : 363-398. doi: 10.3934/dcds.2009.25.363
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