On stability of a reaction diffusion system described by difference equations (Q6561464)

The authors study the discrete Gierer-Meinhardt reaction-diffusion system \N\[ \left\{\begin{aligned} \Delta u_i(t)&=\frac{a_1}{k^2} \Delta^2 u_{i-1}(t)+\sigma-\mu u_i(t)+\frac{u_i^p(t)}{v_i^q(t)} ,\\\N\Delta v_i(t)&=\frac{a_2}{k^2} \Delta^2 v_{i-1}(t)-\nu v_i(t)+\frac{u_i^r(t)}{v_i^s(t)}, \end{aligned}\right. \] \Nthat is widely used for modeling pattern formation. It describes the change in the variables \(u\) and \(v\) at position \(i\) due to diffusion. Here \(a_1\) and \(a_2\) are the diffusion coefficients, \(k\) is the spatial step size, and \(\mu,\nu,\sigma,p, q, r\), and \(s\) are positive parameters with \(p>1\).\N\NUsing linearization techniques, they obtain sufficient criteria for the asymptotic stability of the unique positive equilibrium both in absence and presence of diffusion. By applying Lyapunov's direct method they are also able to prove a result on global asymptotic stability.\N\NNumerical examples are also presented to support the theoretical results.