Abstract
The containment of the invasive species is a widespread problem in the environmental management, with a significant economic impact. We analyze an optimal control model which aims to find the best temporal resource allocation strategy for the removal of an invasive species. We derive the optimality system in the state and control variables and we use the phase-space analysis to provide qualitative insights about the behavior of the optimal solution. Finally, for the state-costate variables which satisfy a boundary-valued nearly-Hamiltonian system, we propose exponential Lawson symplectic approximations applied in the forward-backward form. The numerical results related to an example of invasive plant considered in Baker, et al. (Nat Resour Model 31(4):e12190, 2018), confirm the qualitative findings provided by the state-control analysis.
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References
Baker, C.M.: Target the source: optimal spatiotemporal resource allocation for invasive species control. Conserv. Lett. 10, 41–48 (2016)
Baker, C.M., Bode, M.: Placing invasive species management in a spatiotemporal context. Ecol. Appl. 26(3), 712–725 (2016)
Baker, C. M., Diele, F., Lacitignola, D., Marangi, C., Martiradonna, A.: Optimal control of invasive species through a dynamical systems approach. Nonlinear Anal. Real World Appl. 49, 45–70 (2018)
Baker, C.M., Diele, F., Marangi, C., Martiradonna, A., Ragni, S.: Optimal spatiotemporal effort allocation for invasive species removal incorporating a removal handling time and budget. Nat. Resour. Model. 31(4), e12190 (2018)
Diele, F., Marangi, C., Ragni, S.: SB3A splitting for approximation of invariants in time-dependent Hamiltonian systems. Appl. Math. Comput. 217, 2798–2807 (2010)
Diele, F., Marangi, C., Ragni, S.: Exponential Lawson integration for nearly Hamiltonian systems arising in optimal control. J. Math. Comput. Simul. 81(5), 1057–1067 (2011)
European Commission Staff, WORKING DOCUMENT IMPACT ASSESSMENT Accompanying the document Proposal for a Council and European Parliament Regulation on the prevention and management of the introduction and spread of invasive alien species, 2013, http://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:52013SC0321
Hairer, E., Lubich, C., Wanner G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006)
Lawson, J.D.: Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372–380 (1967)
Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Models, 1st edn. Chapman & Hall/CRC, Boca Raton (2007)
Leonard, D., Van Long, N.: Optimal Control Theory and Static Optimization in Economics. Cambridge University Press, Cambridge (1992)
Magnea, U., Sciascia, R., Paparella, F., Tiberti, R., Provenzale, A.: A model for high-altitude alpine lake ecosystems and the effect of introduced fish. Ecol. Model. 251, 211–220 (2013)
Marangi, C., Casella, F., Diele, F., Lacitignola, D., Martiradonna, A., Provenzale, A., Ragni, S.: Mathematical tools for controlling invasive species in Protected Areas. In Mathematical Approach to Climate Change and its Impacts. Springer INdAM Series. Springer Cham. 38, 211–237 (2020)
Martiradonna, A., Diele, F., Marangi, C.: Analysis of state-control optimality system for invasive species management. In Anal. Prob. Appl. and Comput. Trends in Mathematics. Birkhäuser, Cham. 3–13 (2019)
Martiradonna, A., Diele, F., Marangi, C.: COINS (COntrol of INvasive Species); R routine for the optimal control of invasive species, ECOPOTENTIAL Project (2018). https://github.com/CnrIacBaGit/COINSvlabrepo
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelize, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)
Acknowledgements
This work has been carried out within the H2020 project ECOPOTENTIAL: Improving Future Ecosystem Benefits Through Earth Observations’. The project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 641762.
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Martiradonna, A., Diele, F., Marangi, C. (2020). Optimal Control of Invasive Species with Budget Constraint: Qualitative Analysis and Numerical Approximation. In: Aguiar, M., Braumann, C., Kooi, B., Pugliese, A., Stollenwerk, N., Venturino, E. (eds) Current Trends in Dynamical Systems in Biology and Natural Sciences. SEMA SIMAI Springer Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-41120-6_8
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DOI: https://doi.org/10.1007/978-3-030-41120-6_8
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