Critical threshold and stability of cluster solutions for large reaction-diffusion systems in \(\mathbb{R}^1\) (Q2782971)

The authors investigate the existence and stability of cluster solutions of a system of reaction-diffusion equations on \({\mathbb R}\). This system, introduced e.g. in [\textit{M. B. Cronhjort} and \textit{C. Blomberg}, Physica D 101, 289-298 (1997; Zbl 0888.92006)] to model prebiotic evolution, in non-dimensionalized variables has the form NEWLINE\[NEWLINE \partial_t X_i = \varepsilon^2 \partial_{xx} X_i - X_i +AM \sum_{i=1}^N k_{ij} X_i X_j, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \tau \partial_t M = \partial_{xx} M +1- M -M \sum_{i,j=1}^N k_{ij} X_i X_j, NEWLINE\]NEWLINE where \(A\), \(\tau\), \(\varepsilon\) are positive parameters with \(\varepsilon\) assumed small. The variables \(X_i\), \(i=1,\dots, N\) are interpreted as concentrations of different types of polymers, while \(M\) is the concentration of activated monomers, while the matrix \(k_{ij}\) measures the strength of \(X_i\) by \(X_j\). One looks for even stationary solutions, \(X_i(|x|) \in H^1({\mathbb R})\), \(1-M(|x|) \in H^1({\mathbb R})\). NEWLINENEWLINENEWLINEThe authors prove the existence of cluster steady state solutions that are defined by \(X_i= \xi_i X_0\), \(\xi_i > 0\), \(i=1,\dots, N\). To prove the existence of two such such solutions, ``large'' \((X_\varepsilon^l, M^l_\varepsilon)\) and ``small'' \((X_\varepsilon^s, M^s_\varepsilon)\), an assumption on the matrix \(k_{ij}\) is made; this is that the system of equations \(\sum_{j=1}^N k_{ij} \xi_j =1\), \(i=1,\dots, N\), has a unique solution \((\xi_1 \ldots \xi_N)\). NEWLINENEWLINENEWLINEThis assumption is satisfied in many important cases of prebiotic evolution (the hypercycle equation) and in generalizations of the Gray-Scott model. For the proof of the existence theorem the reader is referred to \textit{J. Wei} [Physica D 148, 20-48 (2001; Zbl 0981.35026)]. NEWLINENEWLINENEWLINEIt is in the discussion of stability of these solutions that the most interesting (threshold) result of the present paper is contained. The spectral stability results are proved under an additional assumption on the matrix \(k_{ij}\) (Assumption H2) and a genericity condition (Assumption H3). Under these conditions, and assuming \(\varepsilon\) sufficiently small, the authors prove, for example, the following result for the hypercycle (\(k_{ij} = k+k_0\delta_{i,j+1}\) if \(i>1\) and \(k_{1j} = k_0 \delta_{j,N}\)): The large cluster solution is always linearly unstable, while the small one \((X_\varepsilon^s,M^s_\varepsilon)\) is linearly stable for all \(\tau \in (0, \tau_0)\) with \(\tau_0\) independent of \(\varepsilon\) if \(N \leq 4\) and unstable otherwise. The proofs are by reduction, and careful analysis of a pair, a local and a nonlocal one, of scalar eigenvalue problems.