Optimal spatial patterns in feeding, fishing, and pollution

<p style='text-indent:20px;'>Infinite time horizon spatially distributed optimal control problems may show so–called optimal diffusion induced instabilities, 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: optimal feeding, optimal fishing, and optimal pollution. The (numerical) analysis uses the continuation and bifurcation package <inline-formula><tex-math id="M1">\begin{document}$\mathtt{pde2path} $\end{document}</tex-math></inline-formula> to first compute bifurcation diagrams of canonical steady states, and then time–dependent optimal controls to control the systems from some initial states to a target steady state as <inline-formula><tex-math id="M2">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>. 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.</p>