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Hopf bifurcations in the full SKT model and where to find them
Optimal spatial patterns in feeding, fishing, and pollution
Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, D-26129 Oldenburg, Germany |
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.
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. |
[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. |
[3] |
C. W. Clark, Mathematical Bioeconomics, John Wiley & Sons, New York, second edition, 1990. |
[4] |
G. Feichtinger and R. Hartl, Optimale Kontrolle Ökonomischer Prozesse, Walter der Gruyter, 1986.
doi: 10.1515/9783110856149. |
[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. |
[6] |
M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8167-8. |
[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. |
[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. |
[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. |
[10] |
W. Hediger,
Sustainable development with stock pollution, Environment and Development Economics, 14 (2009), 759-780.
doi: 10.1017/S1355770X09005282. |
[11] |
R. B. Hoyle, Pattern Formation, Cambridge University Press, 2006.
doi: 10.1017/CBO9780511616051.![]() ![]() ![]() |
[12] |
S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman Hall, 2007. |
[13] |
J. D. Murray, Mathematical Biology, Biomathematics, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[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. |
[15] |
A. K. Skiba,
Optimal growth with a convex-concave production function, Econometrica, 46 (1978), 527-539.
doi: 10.2307/1914229. |
[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. |
[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. |
[18] |
A. M. Turing,
The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[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. |
[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. |
[21] |
H. Uecker, Numerical Continuation and Bifurcation in Nonlinear PDEs, SIAM, 2021. |
[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). |
[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. |
[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. |
[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. |
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. |
[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. |
[3] |
C. W. Clark, Mathematical Bioeconomics, John Wiley & Sons, New York, second edition, 1990. |
[4] |
G. Feichtinger and R. Hartl, Optimale Kontrolle Ökonomischer Prozesse, Walter der Gruyter, 1986.
doi: 10.1515/9783110856149. |
[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. |
[6] |
M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8167-8. |
[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. |
[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. |
[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. |
[10] |
W. Hediger,
Sustainable development with stock pollution, Environment and Development Economics, 14 (2009), 759-780.
doi: 10.1017/S1355770X09005282. |
[11] |
R. B. Hoyle, Pattern Formation, Cambridge University Press, 2006.
doi: 10.1017/CBO9780511616051.![]() ![]() ![]() |
[12] |
S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman Hall, 2007. |
[13] |
J. D. Murray, Mathematical Biology, Biomathematics, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[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. |
[15] |
A. K. Skiba,
Optimal growth with a convex-concave production function, Econometrica, 46 (1978), 527-539.
doi: 10.2307/1914229. |
[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. |
[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. |
[18] |
A. M. Turing,
The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[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. |
[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. |
[21] |
H. Uecker, Numerical Continuation and Bifurcation in Nonlinear PDEs, SIAM, 2021. |
[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). |
[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. |
[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. |
[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. |










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