Authors

Stefano Massaroli, Federico Califano, Angela Faragasso, Atsushi Yamashita, Hajime Asama

Abstract

Scientists have long been attracted to mechanisms surrounding the predator–prey system. The Lotka–Volterra (LV) model is the most popular formalism used to investigate the dynamics of this system. LV equations present non-linear dynamics that exhibit periodic oscillations in both prey and predator populations. In practical situations, it is useful to stabilise the system asymptotically to a desired set point (population) wherein the two species coexist by fashioning specific control actions. This control strategy can be beneficial for problems that can arise when there is a risk of extinction of one of the species and human intervention must be planned. One natural and well-established theory for describing systems obeying energy balance laws is the port-Hamiltonian modeling, an extension of classical Hamiltonian mechanics to systems endowed with control and observation. The LV model can be formally represented as a non-linear mechanical oscillator employing the canonical equations of Hamilton. This special mathematical structure aids planning and designing efficient control actions. The proposed strategy employs a systematic procedure to efficiently plan biological control actions and bypass species extinction through asymptotic stabilisation of populations.

Citation

  • Journal: IEEE Control Systems Letters
  • Year: 2020
  • Volume: 4
  • Issue: 4
  • Pages: 964–969
  • Publisher: Institute of Electrical and Electronics Engineers (IEEE)
  • DOI: 10.1109/lcsys.2020.2997741

BibTeX

@article{Massaroli_2020,
  title={{On the Extinction–Free Stabilization of Predator-Prey Dynamics}},
  volume={4},
  ISSN={2475-1456},
  DOI={10.1109/lcsys.2020.2997741},
  number={4},
  journal={IEEE Control Systems Letters},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Massaroli, Stefano and Califano, Federico and Faragasso, Angela and Yamashita, Atsushi and Asama, Hajime},
  year={2020},
  pages={964--969}
}

Download the bib file

References

  • LaSalle, J. Some Extensions of Liapunov’s Second Method. IRE Transactions on Circuit Theory vol. 7 520–527 (1960) – 10.1109/tct.1960.1086720
  • Tahara, T. et al. Asymptotic stability of a modified Lotka-Volterra model with small immigrations. Scientific Reports vol. 8 (2018) – 10.1038/s41598-018-25436-2
  • Gavina, M. K. A. et al. Multi-species coexistence in Lotka-Volterra competitive systems with crowding effects. Scientific Reports vol. 8 (2018) – 10.1038/s41598-017-19044-9
  • Solomon, S. & Richmond, P. Stable power laws in variable economies; Lotka-Volterra implies Pareto-Zipf. The European Physical Journal B - Condensed Matter vol. 27 257–261 (2002) – 10.1140/epjb/e20020152
  • Cornu, M., Billoir, E., Bergis, H., Beaufort, A. & Zuliani, V. Modeling microbial competition in food: Application to the behavior of Listeria monocytogenes and lactic acid flora in pork meat products. Food Microbiology vol. 28 639–647 (2011) – 10.1016/j.fm.2010.08.007
  • Brew, J. S. An alternative to Lotka-Volterra competition in coarse-grained environments. Theoretical Population Biology vol. 25 265–288 (1984) – 10.1016/0040-5809(84)90010-8
  • Neuhauser, C. & Fargione, J. E. A mutualism–parasitism continuum model and its application to plant–mycorrhizae interactions. Ecological Modelling vol. 177 337–352 (2004) – 10.1016/j.ecolmodel.2004.02.010
  • CRUYWAGEN, G. C. et al. THE MODELLING OF DIFFUSIVE TUMOURS. Journal of Biological Systems vol. 03 937–945 (1995) – 10.1142/s0218339095000836
  • Holt, R. D. & Pickering, J. Infectious Disease and Species Coexistence: A Model of Lotka-Volterra Form. The American Naturalist vol. 126 196–211 (1985) – 10.1086/284409
  • Nutku, Y. Hamiltonian structure of the Lotka-Volterra equations. Physics Letters A vol. 145 27–28 (1990) – 10.1016/0375-9601(90)90270-x
  • Plank, M. Hamiltonian structures for the n-dimensional Lotka–Volterra equations. Journal of Mathematical Physics vol. 36 3520–3534 (1995) – 10.1063/1.530978
  • Nicholls, F. M. & Barbosa, K. A. State feedback regulation on port-Hamiltonian systems: a convex based approach. 2018 IEEE International Conference on Automation/XXIII Congress of the Chilean Association of Automatic Control (ICA-ACCA) 1–6 (2018) doi:10.1109/ica-acca.2018.8609834 – 10.1109/ica-acca.2018.8609834
  • Gause, G. F. Experimental Demonstration of Volterra’s Periodic Oscillations in the Numbers of Animals. Journal of Experimental Biology vol. 12 44–48 (1935) – 10.1242/jeb.12.1.44
  • Nunna, K., Sassano, M. & Astolfi, A. Constructive Interconnection and Damping Assignment for Port-Controlled Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 60 2350–2361 (2015) – 10.1109/tac.2015.2400663
  • Lotka, A. J. Contribution to the Theory of Periodic Reactions. The Journal of Physical Chemistry vol. 14 271–274 (1910) – 10.1021/j150111a004
  • Huffaker, C. B. Experimental studies on predation: Dispersion factors and predator-prey oscillations. Hilgardia vol. 27 343–383 (1958) – 10.3733/hilg.v27n14p343
  • Alvarez, J., Alvarez-Ramirez, J., Espinosa-Perez, G. & Schaum, A. Energy shaping plus damping injection control for a class of chemical reactors. Chemical Engineering Science vol. 66 6280–6286 (2011) – 10.1016/j.ces.2011.09.002
  • Stenseth, N. Chr., Falck, W., Bjørnstad, O. N. & Krebs, C. J. Population regulation in snowshoe hare and Canadian lynx: Asymmetric food web configurations between hare and lynx. Proceedings of the National Academy of Sciences vol. 94 5147–5152 (1997) – 10.1073/pnas.94.10.5147
  • Holling, C. S. Some Characteristics of Simple Types of Predation and Parasitism. The Canadian Entomologist vol. 91 385–398 (1959) – 10.4039/ent91385-7
  • CHIVERS, W. J. & HERBERT, R. D. THE EFFECTS OF VARYING PARAMETER VALUES AND HETEROGENEITY IN AN INDIVIDUAL-BASED MODEL OF PREDATOR-PREY INTERACTION. Advances in Complex Systems vol. 06 441–456 (2003) – 10.1142/s0219525903000980
  • Bacaër, N. A Short History of Mathematical Population Dynamics. (Springer London, 2011). doi:10.1007/978-0-85729-115-8 – 10.1007/978-0-85729-115-8
  • Li, J. Control Schemes to Reduce Risk of Extinction in the Lotka-Volterra Predator-Prey Model. Journal of Applied Mathematics and Physics vol. 02 644–652 (2014) – 10.4236/jamp.2014.27071
  • volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memoria della Reale Accademia Nazionale dei Lincei (1926)
  • Plank, M. Bi-Hamiltonian systems and Lotka - Volterra equations: a three-dimensional classification. Nonlinearity vol. 9 887–896 (1996) – 10.1088/0951-7715/9/4/004
  • Ortega, R., Jiang, Z. P. & Hill, D. J. Passivity-based control of nonlinear systems: a tutorial. Proceedings of the 1997 American Control Conference (Cat. No.97CH36041) 2633–2637 vol.5 (1997) doi:10.1109/acc.1997.611933 – 10.1109/acc.1997.611933
  • vulpiani, Chaos From Simple Models to Complex Systems (2010)
  • Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
  • Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
  • Ortega, R., Astolfi, A., Bastin, G. & Rodriguez, H. Stabilization of food-chain systems using a port-controlled Hamiltonian description. Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334) (2000) doi:10.1109/acc.2000.878579 – 10.1109/acc.2000.878579
  • van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002