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Notes: Abstract The initiation of reaction-diffusion travelling waves in two regions coupled together by the linear diffusive interchange of an autocatalytic species is considered. In one region a purely autocatalytic production process (either quadratic or cubic) is assumed, while in the other region there are both autocatalytic and decay processes (either linear or quadratic). A perturbation analysis based on small initial inputs of the autocatalyst is presented. This indicates conditions under which travelling wave formation is possible as well as identifying two special cases which need further consideration, namely cubic autocatalysis in both regions with quadratic or linear decay in one region. The former case gives rise to a zero eigenvalue and the perturbation method has to be extended to include the higher order terms to resolve this case. The latter case requires a threshold on the initial input of autocatalyst and further information about this threshold is gained from a solution for strong coupling.

Notes: Abstract The formation of stable patterns is considered in a reaction-diffusion system based on the cubic autocatalator, A+2B → 3B, B → C, with the reaction taking place within a closed region, the reactant A being replenished by the slow decay of precursor P via the reaction P → A. The linear stability of the spatially uniform Steady state % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaabmGabaGaamyyaiaacYcacaWGIbaacaGLOaGaayzkaaGaeyyp% a0ZaaeWaceaacqaH8oqBdaahaaWcbeqaaiabgkHiTiaaigdaaaGcca% GGSaGaeqiVd0gacaGLOaGaayzkaaaaaa!48C3!\[\left( {a,b} \right) = \left( {\mu ^{ - 1} ,\mu } \right)\], where a and b are the dimensionless concentrations of reactant A and autocatalyst B and μ is a parameter representing the initial concentration of the precursor P, is discussed first. It is shown that a necessary condition for the bifurcation of this steady state to stable, spatially non-uniform, solutions (patterns) is that the parameter % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadseacqGH8aapcaaIZaGaeyOeI0IaaGOmamaakaaabaGaaGOm% aaWcbeaaaaa!4139!\[D 〈 3 - 2\sqrt 2 \] where D=D b/Da (D a and D b are the diffusion coefficients of chemical species A and B respectively). The values of μ at which these bifurcations occur are derived in terms of D and λ (a parameter reflecting the size of the system). Further information about the nature of the spatially non-uniform solutions close to their bifurcation points is obtained from a weakly nonlinear analysis. This reveals that both supercritical and subcritical bifurcations are possible. The bifurcation branches are then followed numerically using a path-following method, with μ as the bifurcation parameter, for representative values of D and λ. It is found that the stable patterns can lose stability through supercritical Hopf bifurcations and these stable, temporally periodic, spatially non-uniform solutions are also followed numerically.