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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
##### References:
 [1] W. Brock and A. Xepapadeas, Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control, Journal of Economic Dynamics and Control, 32 (2008), 2745-2787.  doi: 10.1016/j.jedc.2007.08.005.  Google Scholar [2] W. Brock and A. Xepapadeas, Pattern formation, spatial externalities and regulation in coupled economic–ecological systems, Journal of Environmental Economics and Management, 59 (2010), 149-164.  doi: 10.1016/j.jeem.2009.07.003.  Google Scholar [3] C. W. Clark, Mathematical Bioeconomics, John Wiley & Sons, New York, second edition, 1990.  Google Scholar [4] G. Feichtinger and R. Hartl, Optimale Kontrolle Ökonomischer Prozesse, Walter der Gruyter, 1986. doi: 10.1515/9783110856149.  Google Scholar [5] B. A. Forster, Optimal pollution control with a nonconstant exponential rate of decay, Environment Economics and Management, 2 (1975), 1-6.  doi: 10.1016/0095-0696(75)90016-9.  Google Scholar [6] M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.  Google Scholar [7] D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, Optimal Control of Nonlinear Processes: With Applications in Drugs, Corruption, and Terror, Springer, 2008. doi: 10.1007/978-3-540-77647-5.  Google Scholar [8] D. Grass and H. Uecker, Optimal management and spatial patterns in a distributed shallow lake model, Electr. J. Differential Equations, 2017, Paper No. 1, 21 pp.  Google Scholar [9] D. Grass, H. Uecker and T. Upmann, Optimal fishery with coastal catch, Natural Resource Modelling, 32 (2019), e12235, 32 pp. doi: 10.1111/nrm.12235.  Google Scholar [10] W. Hediger, Sustainable development with stock pollution, Environment and Development Economics, 14 (2009), 759-780.  doi: 10.1017/S1355770X09005282.  Google Scholar [11] R. B. Hoyle, Pattern Formation, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511616051.  Google Scholar [12] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman Hall, 2007.  Google Scholar [13] J. D. Murray, Mathematical Biology, Biomathematics, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar [14] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, 1962.  Google Scholar [15] A. K. Skiba, Optimal growth with a convex-concave production function, Econometrica, 46 (1978), 527-539.  doi: 10.2307/1914229.  Google Scholar [16] O. Tahvonen and C. Withagen, Optimality of irreversible pollution accumulation, Journal of Economic Dynamics and Control, 20 (1996), 1775-1795.  doi: 10.1016/0165-1889(95)00909-4.  Google Scholar [17] N. Tauchnitz, The Pontryagin maximum principle for nonlinear optimal control problems with infinite horizon, J. Optim. Theory Appl., 167 (2015), 27-48.  doi: 10.1007/s10957-015-0723-y.  Google Scholar [18] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar [19] H. Uecker, Optimal harvesting and spatial patterns in a semi arid vegetation system, Natural Resource Modelling, 29 (2016), 229-258.  doi: 10.1111/nrm.12089.  Google Scholar [20] H. Uecker, Hopf bifurcation and time periodic orbits with pde2path – algorithms and applications, Comm. in Comp. Phys, 25 (2019), 812-852.  doi: 10.4208/cicp.oa-2017-0181.  Google Scholar [21] H. Uecker, Numerical Continuation and Bifurcation in Nonlinear PDEs, SIAM, 2021. Google Scholar [22] H. Uecker and H. de Wit, Infinite time–horizon spatially distributed optimal control problems with pde2path – algorithms and tutorial examples, arXiv: 1912.11135, (2019). Google Scholar [23] H. Uecker, D. Wetzel and J. D. M. Rademacher, pde2path – a Matlab package for continuation and bifurcation in 2D elliptic systems, NMTMA, 7 (2014), 58-106.  doi: 10.4208/nmtma.2014.1231nm.  Google Scholar [24] T. Upmann, H. Uecker, L. Hammann and B. Blasius, Optimal stock enhancement activities for a spatially distributed renewable resource, Journal of Economic Dynamics & Control, 123 (2021), 104060 17 pp. doi: 10.1016/j.jedc.2020.104060.  Google Scholar [25] F. Wirl, Optimal accumulation of pollution: Existence of limit cycles for the social optimum and the competitive equilibrium, Journal of Economic Dynamics and Control, 24 (2000), 297-306.  doi: 10.1016/S0165-1889(98)00070-0.  Google Scholar

show all references

##### References:
 [1] W. Brock and A. Xepapadeas, Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control, Journal of Economic Dynamics and Control, 32 (2008), 2745-2787.  doi: 10.1016/j.jedc.2007.08.005.  Google Scholar [2] W. Brock and A. Xepapadeas, Pattern formation, spatial externalities and regulation in coupled economic–ecological systems, Journal of Environmental Economics and Management, 59 (2010), 149-164.  doi: 10.1016/j.jeem.2009.07.003.  Google Scholar [3] C. W. Clark, Mathematical Bioeconomics, John Wiley & Sons, New York, second edition, 1990.  Google Scholar [4] G. Feichtinger and R. Hartl, Optimale Kontrolle Ökonomischer Prozesse, Walter der Gruyter, 1986. doi: 10.1515/9783110856149.  Google Scholar [5] B. A. Forster, Optimal pollution control with a nonconstant exponential rate of decay, Environment Economics and Management, 2 (1975), 1-6.  doi: 10.1016/0095-0696(75)90016-9.  Google Scholar [6] M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.  Google Scholar [7] D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, Optimal Control of Nonlinear Processes: With Applications in Drugs, Corruption, and Terror, Springer, 2008. doi: 10.1007/978-3-540-77647-5.  Google Scholar [8] D. Grass and H. Uecker, Optimal management and spatial patterns in a distributed shallow lake model, Electr. J. Differential Equations, 2017, Paper No. 1, 21 pp.  Google Scholar [9] D. Grass, H. Uecker and T. Upmann, Optimal fishery with coastal catch, Natural Resource Modelling, 32 (2019), e12235, 32 pp. doi: 10.1111/nrm.12235.  Google Scholar [10] W. Hediger, Sustainable development with stock pollution, Environment and Development Economics, 14 (2009), 759-780.  doi: 10.1017/S1355770X09005282.  Google Scholar [11] R. B. Hoyle, Pattern Formation, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511616051.  Google Scholar [12] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman Hall, 2007.  Google Scholar [13] J. D. Murray, Mathematical Biology, Biomathematics, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar [14] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, 1962.  Google Scholar [15] A. K. Skiba, Optimal growth with a convex-concave production function, Econometrica, 46 (1978), 527-539.  doi: 10.2307/1914229.  Google Scholar [16] O. Tahvonen and C. Withagen, Optimality of irreversible pollution accumulation, Journal of Economic Dynamics and Control, 20 (1996), 1775-1795.  doi: 10.1016/0165-1889(95)00909-4.  Google Scholar [17] N. Tauchnitz, The Pontryagin maximum principle for nonlinear optimal control problems with infinite horizon, J. Optim. Theory Appl., 167 (2015), 27-48.  doi: 10.1007/s10957-015-0723-y.  Google Scholar [18] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar [19] H. Uecker, Optimal harvesting and spatial patterns in a semi arid vegetation system, Natural Resource Modelling, 29 (2016), 229-258.  doi: 10.1111/nrm.12089.  Google Scholar [20] H. Uecker, Hopf bifurcation and time periodic orbits with pde2path – algorithms and applications, Comm. in Comp. Phys, 25 (2019), 812-852.  doi: 10.4208/cicp.oa-2017-0181.  Google Scholar [21] H. Uecker, Numerical Continuation and Bifurcation in Nonlinear PDEs, SIAM, 2021. Google Scholar [22] H. Uecker and H. de Wit, Infinite time–horizon spatially distributed optimal control problems with pde2path – algorithms and tutorial examples, arXiv: 1912.11135, (2019). Google Scholar [23] H. Uecker, D. Wetzel and J. D. M. Rademacher, pde2path – a Matlab package for continuation and bifurcation in 2D elliptic systems, NMTMA, 7 (2014), 58-106.  doi: 10.4208/nmtma.2014.1231nm.  Google Scholar [24] T. Upmann, H. Uecker, L. Hammann and B. Blasius, Optimal stock enhancement activities for a spatially distributed renewable resource, Journal of Economic Dynamics & Control, 123 (2021), 104060 17 pp. doi: 10.1016/j.jedc.2020.104060.  Google Scholar [25] F. Wirl, Optimal accumulation of pollution: Existence of limit cycles for the social optimum and the competitive equilibrium, Journal of Economic Dynamics and Control, 24 (2000), 297-306.  doi: 10.1016/S0165-1889(98)00070-0.  Google Scholar
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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