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doi: 10.3934/dcdss.2021099
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## Optimal spatial patterns in feeding, fishing, and pollution

 Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, D-26129 Oldenburg, Germany

* Corresponding author

Received  February 2021 Revised  July 2021 Early access August 2021

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 $\mathtt{pde2path}$ 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 $t\to\infty$. 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.

Citation: Hannes Uecker. Optimal spatial patterns in feeding, fishing, and pollution. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021099
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1D sample canonical paths going to a PCSS (top: control, bottom: states). (a) FEED (§2). In the PCSS (see (a) at $t = 20$), both, control $a$ (feeding) and stock $v$, are large at the right boundary, and the canonical path goes from a FCSS to a PCSS. (b) FISH (§3). The PCSS can be seen as a type of marine reserve, with little fishing effort $E$ and high stock in the middle of the domain; however, the harvest $h = Ev$ is almost constant in the domain. (c) POLL (§4). High pollution at the left, hence low consumption. In all three cases, the control of the system from the initial states maximizes the profit, and all three PCSSs can be considered as patterned optimal steady states (POSSs)
(a) One patch phase portrait for FEED, parameters (26), ${\delta}{ = }0.3$. (b) Top: 2P continuation in ${\delta}$ with $D{ = }0.25$, with the black branch correponding to the FCSS, and the blue branch a PCSS. Bottom: ODI indicator function $h_2$ along the black branch. (c) canonical path from (the states of) the FCSS to p1/pt36. (d) value diagram of p1/pt36. (e, f) ${\Omega} = (-1.5,1.5)$, bifurcation diagram and sample solutions, $D{ = }0.25$
FISH. (a) Plots of $J_c(h)$ and of $f(v,E)$ for illustration. (b) Phase portrait for the 1P problem near the nontrivial CSS, $\rho = 0.03$. (c) The larger scale phase portrait; two saddle points (red dots), and an unstable node at $(v,{\lambda}) = (150000,0)$; the black arrows indicate the flow on the invariant ${\lambda} = 0$ axis. (d) The ODI indicator function $h_2$ for continuation of the CSS from (b) in $\rho$
FISH 2P. (a) bifurcation diagrams. FCSS branch (black), and PCSS branch (blue, respectively blue (patch 1) and red (patch2)). (b, c) canonical paths to ${{\hat{u}}^{{\rm{{ {{\rm{FCSS}}}}}}}}$ and ${{\hat{u}}^{{\rm{{ {{\rm{PCSS}}}}}}}}$ at $\rho = 0.04$, both starting from ${\hat{v}}_0 = 10^5(0.5,1.5)$, and both giving almost equal values $J \approx 6.39*10^9$, while $J({{\hat{u}}^{{\rm{{ {{\rm{FCSS}}}}}}}})<J({{\hat{u}}^{{\rm{{ {{\rm{PCSS}}}}}}}})$. (d) value diagrams for the FCSS (d1) and the PCSS (d2) at $\rho = 0.04$, and both together in (d3)
Dynamics (a) and values (b) for the greedy overfishing choice $E(t) = h_*/v(t)$ for (22), leading to extinction of $v$ in finite time, and yielding suboptimal values $J(v_0)$, see Remark 3.2
FISH 1D, ${\Omega} = (-2.5,2.5)$, $D = 0.01$, other parameters as before. (a) bifurcation diagrams. (b) PCSS sample solution. (c) $h$ on the canonical path to the PCSS (see Fig. \ref{f0}(b) for $E$ and $v$), starting near (but not in) the ${\hat{v}}$ from the FCSS
FISH on ${\Omega} = (-l_x,l_x)^2$, $l_x = 1.25$, with $D = 0.01$. (a) bifurcation diagram, $J_c$ and $\max v$ over $\rho$, FCCS branch (black), and two primary PCSS branches, spots ($\mathtt{b1}$, blue), and stripes ($\mathtt{b2}$, red). (b, c) A sample stripe, with $E$, and sample spots, unphysical at pt20 ($\mathtt{u1}$ means $v$). (d) canonical path to b2/9, top row $E$, bottom row $v$, with $t$ as indicated. Right: $J_{ca}$ along the canonical path, with value $J = 8.235*10^9$, and value $J_1 = 8.0002*10^9$ of the target
POLL 1P problem. (a) phase portrait, $\rho = 0.015$. (b) Continuation of the CSS in $\rho$. (c) canonical path to the CSS at $\rho = 0.02$ for $s(0) = 0.5$; high initial consumption, decreasing $c$ and $J_c$, increasing $s$. (d) $s(0) = 2.25$, increasing $c$ and $J$, decreasing $s$
POLL, 2P, $D = 0.008$, (a) bifurcation diagram. (b) Value diagram for the PCSS ${{\hat{u}}^{{\rm{{ {{\rm{PCSS}}}}}}}}$ at $\rho = 0.027$ (marked by the circle). The domain of attraction of ${{\hat{u}}^{{\rm{{ {{\rm{PCSS}}}}}}}}$ extends almost towards the diagonal. For $v_1(0)>v_2(0)$ we get the mirror image by controlling to $M{{\hat{u}}^{{\rm{{ {{\rm{PCSS}}}}}}}}$. (c, d) two canonical paths to ${{\hat{u}}^{{\rm{{ {{\rm{PCSS}}}}}}}}$ at $\rho = 0.027$
POLL, ${\Omega}{ = }(-1.5,1.5)$, $D{ = }0.01$. (a, b) bifurcation diagrams and a sample solution. (c) Initial states $v_0(x){ = }{{\hat{v}}^{{\rm{{ {{\rm{FCSS}}}}}}}}{-}x/5$ and $v_1(x){ = }{{\hat{v}}^{{\rm{{ {{\rm{FCSS}}}}}}}}{-}x/4$, and ${v_{\max}}$. For $v_0$, there exists a canonical path to ${{\hat{u}}^{{\rm{{ {{\rm{PCSS}}}}}}}}$ at b1/17 in (d), but not for $v_1$
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