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Title: Optimal spatial patterns in feeding, fishing, and pollution

Abstract: Infinite time horizon spatially distributed optimal control problems may show so-called optimal diffusion induced instabilities (Brock&Xepapadeas, JEDC 2008), which may lead to patterned optimal steady states, although the problem itself is completely homogeneous. Here we show that this can be considered as a generic phenomenon, in problems with scalar distributed states, by computing optimal spatial patterns and their canonical paths in three examples, from (Uecker, DCDS-S, 2021): optimal feeding, optimal fishing, and optimal pollution. The (numerical) analysis uses the continuation and bifurcation package pde2path to first compute bifurcation diagrams of canonical steady states, and then time-dependent optimal controls to control the systems from some initial states to a target steady state as time goes to infinity. We consider two setups: The case of discrete patches in space, which allows to gain intuition and to compute domains of attraction of canonical steady states, and the spatially continuous (PDE) case.