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Analytical analysis of spatially extended autocatalytic and hypercyclicsystems is presented. It is shown that spatially explicit systems in the formof reaction-diffusion equations with global regulation possess the same majorqualitative features as the corresponding local models. In particular, usingthe introduced notion of the stability in the mean integral sense we prove thecompetitive exclusion principle for the autocatalytic system and the permanencefor the hypercycle system. Existence and stability of stationary solutions arestudied. For some parameter values it is proved that stable spatiallynon-uniform solutions appear.

Existence and stability of stationary solutions to spatially extended  autocatalytic and hypercyclic systems under global regulation and with  nonlinear growth rates