zbMATH — the first resource for mathematics

Allee optimal control of a system in ecology. (English) Zbl 1411.93198
Summary: The Allee threshold of an ecological system distinguishes the sign of population growth either towards extinction or to carrying capacity. In practice, human interventions can tune the Allee threshold for instance thanks to the sterile male technique and the mating disruption. In this paper, we address various control problems for a system described by a diffusion-reaction equation regulating the Allee threshold, viewed as a real parameter determining the unstable equilibrium of the bistable nonlinear reaction term. We prove that this system is the mean field limit of an interacting system of particles in which the individual behaviour is driven by stochastic laws. Numerical simulations of the stochastic process show that the propagation of population is governed by travelling wave solutions of the macroscopic reaction-diffusion system, which model the fact that solutions, in bounded space domains, reach asymptotically an equilibrium configuration.
An optimal control problem for the macroscopic model is then introduced with the objective of steering the system to a target travelling wave. Using well-known analytical results and stability properties of travelling waves, we show that well-chosen piecewise constant controls allow to reach the target approximately in sufficiently long time. We then develop a direct computational method and show its efficiency for computing such controls in various numerical simulations. Finally, we show the effectiveness of the obtained macroscopic optimal controls in the microscopic system of interacting particles and we discuss their advantage when addressing situations that are out of reach for the analytical methods. We conclude the paper with some open problems and directions for future research.
Reviewer: Reviewer (Berlin)

93E20 Optimal stochastic control
93E03 Stochastic systems in control theory (general)
92D40 Ecology
35K57 Reaction-diffusion equations
90C15 Stochastic programming
Full Text: DOI
[1] Albi, G.; Pareschi, L.; Toscani, G.; Zanella, M., Active Particles, 1, Recent advances in opinion modeling: control and social influence, 49-98, (2017), Birkhäuser
[2] Annunziato, M.; Borzì, A., A Fokker-Planck control framework for multidimensional stochastic processes, J. Comput. Appl. Math., 237, 487-507, (2013) · Zbl 1251.35196
[3] Aronson, D. G.; Weinberger, H. F., Partial Differential Equations & Related Topics, 446, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, 5-49, (1975), Springer
[4] Aronson, D. G.; Weinberger, H. F., Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30, 33-76, (1978) · Zbl 0407.92014
[5] Arrieta, J. M.; López-Fernández, M.; Zuazua, E., Approximating travelling waves by equilibria of non-local equations, Asymptot. Anal., 3, 145-186, (2012) · Zbl 1248.35038
[6] Aydogdu, A.; Caponigro, M.; McQuade, S.; Piccoli, B.; Duteil, N. P.; Rossi, F.; Trélat, E., Active Particles, 1, Interaction network, state space, and control in social dynamics, 99-140, (2017), Birkhäuser
[7] Barthel, W.; John, C.; Tröltzsch, F., Optimal boundary control of a system of reaction diffusion equations, J. Appl. Math. Mech., 12, 966-982, (2010) · Zbl 1375.49030
[8] Barton, N.; Turelli, M., Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of allee effects, Am. Nat., 178, E48-E75, (2011)
[9] K. Beauchard and M. Morancey, Local controllability of 1D Schrödinger equations with bilinear control and minimal time, arXiv:1208.5393. · Zbl 1281.93016
[10] Bellomo, N.; Bellouquid, A.; Knopoff, D., From the microscale to collective crowd dynamics, Multiscale Model. Simul., 11, 943-963, (2013) · Zbl 1280.90019
[11] Bellomo, N.; Preziosi, L., Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. Comput. Modelling, 32, 413-452, (2000) · Zbl 0997.92020
[12] Bellomo, N.; Winkler, M., A degenerate chemotaxis system with flux limitation: maximally extended solutions and absence of gradient blow-up, Comm. Partial Differential Equations, 42, 436-473, (2017) · Zbl 1430.35166
[13] Bellouquid, A.; Delitala, M., Mathematical Modeling of Complex Biological Systems, (2006), Birkhũser · Zbl 1178.92002
[14] Bertsekas, D. P., Nonlinear Programming, 1-60, (1999), Athena Scientific
[15] Bisi, M.; Desvillettes, L., From reactive Boltzmann equations to reaction-diffusion systems, J. Statist. Phys., 124, 881-912, (2006) · Zbl 1134.82323
[16] Bliman, P. A.; Vauchelet, N., Establishing traveling wave in bistable reaction-diffusion system by feedback, IEEE Control Syst. Lett., 1, 62-67, (2017)
[17] Bonnans, J. F.; Gilbert, J. C.; Lemaréchal, C.; Sagastizábal, C. A., Numerical Optimization: Theoretical and Practical Aspects, (2006), Springer Science \(\&\) Business Media · Zbl 1108.65060
[18] Burini, D.; Gibelli, L.; Outada, N., Active Particles, 1, A kinetic theory approach to the modeling of complex living systems, 229-258, (2017), Birkhäuser
[19] Cannarsa, P.; Floridia, G.; Khapalov, A. Y., Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign, J. Math. Pures Appl., 108, 425-458, (2017) · Zbl 1370.93140
[20] Carrillo, J. A.; Choi, Y. P.; Perez, S. P., Active Particles, 1, A review on attractive-repulsive mean fields for consensus in collective behavior, 259-298, (2017), Birkhäuser
[21] Casas, E.; Ryll, C.; Tröltzsch, F., Sparse optimal control of the Schlögl and Fitzhugh-Nagumo systems, Comput. Methods Appl. Math., 1, 1-29, (2013) · Zbl 1393.49019
[22] Chalub, F. A.; Markowich, P. A.; Perthame, B.; Schmeiser, C., Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142, 123-141, (2004) · Zbl 1052.92005
[23] Coron, J. M., Control and Nonlinearity, (2007), Amer. Math. Soc.
[24] Coron, J.-M.; Trélat, E., Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim., 43, 549-569, (2004) · Zbl 1101.93011
[25] Coron, J.-M.; Trélat, E., Global steady-state stabilization and controllability of 1D semilinear wave equations, Commun. Contemp. Math., 8, 535-567, (2006) · Zbl 1101.93039
[26] Cristiani, E.; Piccoli, B.; Tosin, A., Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9, 155-182, (2011) · Zbl 1221.35232
[27] Durrett, R., Crabgrass, measles, and gypsy moths: an introduction to modern probability, Bull. Amer. Math. Soc., 18, 117-143, (1988) · Zbl 0653.60095
[28] Durrett, R., Lectures on Probability Theory, Ten lectures on particle systems, 97-201, (1995), Springer · Zbl 0840.60088
[29] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of Inverse Problems, (1996), Springer Science & Business Media · Zbl 0859.65054
[30] Erneux, T.; Nicolis, G., Propagating waves in discrete bistable reaction-diffusion systems, Phys. D, 67, 237-244, (1993) · Zbl 0787.92010
[31] Fernández, L. A.; Khapalov, A. Y., Controllability properties for the one-dimensional heat equation under multiplicative or non-negative additive controls with local mobile support, ESAIM Control Optim. Calc. Var., 4, 1207-1224, (2012) · Zbl 1262.35119
[32] Fife, P. C.; McLeod, J. B., The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65, 335-361, (1977) · Zbl 0361.35035
[33] Filbet, F.; Laurenot, P.; Perthame, B., Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50, 189-207, (2005) · Zbl 1080.92014
[34] Fisher, R. A., The wave of advance of advantageous genes, Ann. Eugen., 7, 355-369, (1937) · JFM 63.1111.04
[35] Fleming, W. H.; Rishel, R. W., Deterministic and Stochastic Optimal Control, (1975), Springer-Verlag · Zbl 0323.49001
[36] Forsgren, A.; Gill, P. E.; Wright, M. H., Interior methods for nonlinear optimization, SIAM Rev., 4, 525-597, (2002) · Zbl 1028.90060
[37] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A mathematical programming language, Technical Report, AT&T Bell Laboratories, Murray Hill (1987). · Zbl 0701.90062
[38] Francesco, M. D.; Fagioli, S.; Rosini, M. D.; Russo, G., Active Particles, 1, Follow-the-leader approximations of macroscopic models for vehicular and Pedestrian flows, 333-378, (2017), Birkhäuser
[39] F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuka, FreeFem\(+ +\) Manual (2005), http://www.freefem.org/.
[40] Hillen, T.; Othmer, H. G., The diffusion limit of transport equations derived from velocity jump processes, SIAM J. Appl Math., 61, 751-775, (2000) · Zbl 1002.35120
[41] Hodgkin, A. L.; Huxley, A. F., A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117, 500-544, (1952)
[42] Iriberri, N.; Uriarte, J.-R., Minority language and the stability of bilingual equilibria, Ration. Soc., 24, 442-462, (2012)
[43] Itô, Y.; Kawamoto, H., Number of generations necessary to attain eradication of an insect pest with sterile insect release method: A model study, Res. Popul. Ecol. (Kyoto), 20, 216-226, (1979)
[44] Kanarek, A. R.; Webb, C. T., Allee effects, adaptive evolution, and invasion success, Evol. Appl., 2, 122-135, (2010)
[45] Kàrn’y, M., Towards fully probabilistic control design, Automatica, 32, 1719-1722, (1996) · Zbl 0868.93022
[46] Keener, J. P.; Sneyd, J., Mathematical Physiology, (1998), Springer · Zbl 0913.92009
[47] Klassen, W.; Curtis, C. F., Sterile Insect Technique, History of the sterile insect technique, 3-36, (2005), Springer
[48] Knipling, E. F., Sterile-male method of population control, Science, 130, 902-904, (1959)
[49] Laplante, J. P.; Erneux, T., Propagation failure in arrays of coupled bistable chemical reactors, J. Phys. Chem., 96, 4931-4934, (1992)
[50] Lewis, M. A.; Kareiva, P., Allee dynamics and the spread of invading organisms, Theor. Popul. Biol., 2, 141-158, (1993) · Zbl 0769.92025
[51] Liebhold, A. M.; Tobin, P. C., Population ecology of insect invasions and their management, Annu. Rev. Entomol., 53, 387-408, (2008)
[52] Lions, J. L., Contrôle Optimal de Systemes Gouvernés par des Équations aux Dérivées Partielles, (1968), Gauthier-Villars · Zbl 0179.41801
[53] Masi, A. D.; Ferrari, P. A.; Lebowitz, J., Reaction-diffusion equations for interacting particle systems, J. Statist Phys., 44, 589-644, (1986) · Zbl 0629.60107
[54] Masi, A. D.; Presutti, E.; Vares, M. E., Escape from the unstable equilibrium in a random process with infinitely many interacting particles, J. Statist Phys., 44, 645-696, (1986) · Zbl 0652.60110
[55] Murray, J. D., Mathematical Biology. II Spatial Models and Biomedical Applications, 18, 1-43, (2001), Springer-Verlag
[56] Nagumo, J.; Yoshizawa, S.; Arimoto, S., Bistable transmission lines, IEEE Trans. Circuit Theory, 3, 400-412, (1965)
[57] Neuhauser, C., Mathematical challenges in spatial ecology, Notices Amer. Math. Soc. 48, 11, 1304-1314, (2002) · Zbl 1128.92328
[58] Okubo, A.; Levin, S. A., Diffusion and Ecological Problems, Modern Perspectives, 10-30, (2001), Springer
[59] Perthame, B., Mathematical tools for kinetic equations, Bull. Amer. Math. Soc., 41, 2, 205-244, (2004) · Zbl 1151.82351
[60] Perthame, B., Parabolic Equations in Biology, Parabolic equations in biology, 1-21, (2015), Springer International Publishing
[61] Phillips, D. L., A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach., 9, 84-97, (1962) · Zbl 0108.29902
[62] C. Pouchol, E. Trélat and E. Zuazua, Phase portrait control 1D monostable and bistable reaction-diffusion equations, preprint, arXiv:1805.10786v1 (2018), 30 pp.
[63] Raymond, J.-P.; Zidani, H., Hamiltonian pontryagin’s principles for control problems governed by semilinear parabolic equations, Appl. Math. Optim., 2, 143-177, (1999) · Zbl 0922.49013
[64] Ruszczyński, A. P., Nonlinear Optimization, (2006), Princeton Univ. Press
[65] Ryll, C.; Löber, J.; Martens, S.; Engel, H.; Tröltzsch, F., Control of Self-Organizing Nonlinear Systems, Analytical, optimal, and sparse optimal control of traveling wave solutions to reaction-diffusion systems, 189-210, (2016), Springer International Publishing
[66] Stevens, A., The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61, 183-212, (2000) · Zbl 0963.60093
[67] Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equation, (2004), Society for Industrial and Applied Mathematics · Zbl 1071.65118
[68] Strugarek, M.; Vauchelet, N., Reduction to a single closed equation for 2-by-2 reaction-diffusion systems of Lotka-Volterra type, SIAM J. Appl. Math., 76, 2060-2080, (2016) · Zbl 1355.35108
[69] Taylor, C. R., Determining optimal sterile male release strategies, Environ. Entomol., 5, 87-95, (1976)
[70] Tits, A. L.; Wächter, A.; Bakhtiari, S.; Urban, T. J.; Lawrence, C. T., A primal-dual interior-point method for nonlinear programming with strong global and local convergence properties, SIAM J. Optim., 1, 173-199, (2003) · Zbl 1075.90078
[71] Tröltzsch, F., Optimal Control of Partial Differential Equations, 265-312, (2010), Amer. Math. Soc.
[72] A. Wächter, An interior point algorithm for large-scale nonlinear optimization with applications in process engineering, PhD thesis, Carnegie Mellon University (2002).
[73] Winkler, M.; Bellomo, N., Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Amer. Math. Soc. Ser. B, 4, 31-67, (2017) · Zbl 1367.35044
[74] Ycart, B., Modèles et Algorithmes Markoviens, (2002), Springer Science & Business Media
[75] Zuazua, E., Handbook of Differential Equations: Evolutionary Equations, Controllability and observability of partial differential equations: some results and open problems, 527-621, (2007), North-Holland · Zbl 1193.35234
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.