On a hypercycle system with nonlinear rate (Q1848677)

The authors study the following nonlinear \((N+1)\)-hypercycle of polymers \(X_i\) and activated monomers \(M\) with diffusion: \[ \begin{aligned} \partial X_i / \partial t &= D_X \partial^2 X_i / \partial x^2 - g_X X_i + M \sum_{j=1}^{N} k_{ij} X_i X_j^n,\\ \partial M / \partial t &= D_M \partial^2 M / \partial x^2 + k_M - g_M M - L M \sum_{j=1}^{N} k_{ij} X_i X_j^n, \end{aligned} \] where \(i=1,\dots,N\), \(x\in \mathbb{R}\), and \(n>0\). They consider stationary cluster-like solutions (corresponding to high concentration \(\sum_{j=1}^{N} X_i\) of polymers and low concentration \(M\) of the activated monomer). Generalizing their previous results in [Nonlinearity 13, No. 6, 2005-2032 (2000; Zbl 0979.35075)], the authors show that, for a given range of parameters that guarantee existence of solutions, the system is stable for \(N \leq N_0\) and unstable for \(N > N_0\), and that \(N_0 = 5\) for \(n \geq n_0 \sim 3.35\) and \(N_0 = 4\) for \(n \leq n_0 \sim 3.35\).