Effects on global coupling of an interface problem in an activator-inhibitor system. (Q1421239)

The authors study the following activator-inhilator system \[ \begin{gathered} \sigma\varepsilon u_t= \varepsilon^2 u_{xx}+ H(u- a)- u-v,\\ v_t= v_{xx}+ \mu u-v,\quad t> 0,\quad x\in\mathbb{R},\end{gathered}\tag{1} \] where \(\sigma\) and \(\varepsilon\) are positive constants and \(H\) is a Heaviside function. The goal of the paper is to explore the dynamics of interfaces (1) with global coupling \[ a= a_0+ \alpha\Biggl(\int_{\mathbb{R}} (u+ v)\,dx- s_0\Biggr),\quad 0<a_0<{1\over 2}, \] \(\alpha> 0\), in order to investigate the existence of a time periodic solution as the global coupling intensity and the bifurcation parameter vary.