State monitoring of a population system in changing environment

Abstract

For Lotka-Volterra population systems, a general model of state monitoring is presented. The model includes time-dependent environmental effects or direct human intervention (“treatment”) as control functions and, instead of the whole state vector, the densities of certain indicator species (distinguished or lumped together) are observed. Mathematical systems theory offers a sufficient condition for local observability in such systems. The latter means that, based on the above (dynamic) partial observation, the state of the population can be recovered, at least near equilibrium. The application of this sufficient condition is illustrated by three-species examples such as a one-predator two-prey system and a simple food chain.

References

  1. Bernard, O., G. Sallet and A. Sciandra. 1998. Nonlinear observers for a class of biological systems: Application to validation of a phytoplanktonic growth model. IEEE Trans. on Automatic Control, 43, No. 8, August, 1056–1065.

    Google Scholar 

  2. Farkas, G. 1998a. Local controllability of reactions. J. Math. Chemistry 24: 1–14.

    CAS  Article  Google Scholar 

  3. Farkas, G. 1998b. On local observability of chemical systems. J. Math. Chemistry 24: 15–22.

    CAS  Article  Google Scholar 

  4. Freedman, H. 1980. Deterministic Mathematical Models in Population Ecology. Marcel Dekker, New York.

    Google Scholar 

  5. Gámez, M.; R. Carreño, A. Kósa and Z. Varga. 2002. Observability in strategic models of selection. Biosystems (submitted)

    Google Scholar 

  6. Gragnani, A. 2002. The role of toxicants on predator-prey dynamics. 5th ESMTB Conference on Mathematical Modelling and Computing in Biology and Medicin. 2–6 July, 2002, Milano, Italy. Abstracts, p. 179. (Proceedings with full papers in preparation)

    Google Scholar 

  7. Lee, E. B. and L. Markus. 1971. Foundations of Optimal Control Theory. Wiley, New York.

    Google Scholar 

  8. Lotka, A. J. 1925. Elements of Physical Biology. Dover, New York.

    Google Scholar 

  9. Metz, J. A. J. 1977. State space model for animal behaviour. Ann. Syst. Res. 6:65–109.

    Article  Google Scholar 

  10. Metz, J. A. J. and O. Diekmann (eds.) 1986. The Dynamics of Physiologically Structured Populations, Springer Lecture Notes in Biomathematics 68.

    Google Scholar 

  11. Nagy, G., S. Romagnoli, Z. Varga and L. Venzi. 2002. Le condizioni di ottimalità per la determinazione delle catture di pesce. XXXIX Convegno annuale delia SIDEA, Società Italiana di Economia Agraria, Firenze 12–14 settembre.

    Google Scholar 

  12. Petrosjan, L. A. and V. V. Zakharov. 1997. Mathematical Models in Environmental Policy Analysis. Nova Science Publishers, Inc. N.Y USA.

    Google Scholar 

  13. Scarelli, A. andZ. Varga. 2002. Controllability of selection-mutation systems. Bio Systems 65: 113–121.

    Article  Google Scholar 

  14. Scudo, F. and Y. R. Ziegler (eds.) 1978. The Golden Age of Mathematical Ecology. Springer Verlag, Berlin.

    Google Scholar 

  15. Szigeti, F., C. Vera and Z. Varga. 2002. Nonlinear system inversion applied to ecological monitoring. 15-th IFAC World Congress on Automatic Control, Barcelona 2002 (full paper, accepted).

    Book  Google Scholar 

  16. Varga, Z. 1989. On controllability of Fisher’s model of selection. In: C. M. Dafermos, G. Ladas and G. Papanicolaou (eds.). Differential Equations. Marcel Dekker Publ., New York, pp. 717–723.

    Google Scholar 

  17. Varga, Z. 1992. On observability of Fisher’s model of selection. Pure Mathematics and Applications. Ser. B., 3: 15–25.

    Google Scholar 

  18. Volterra, V. 1931. Leçons sur la théorie mathématique de la lutte pour la vie. Gauthier-Villars, Paris.

    Google Scholar 

  19. Yodzis, P. 1989. Introduction to Theoretical Ecology. Harper and Row, New York.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Z. Varga.

Rights and permissions

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Cite this article

Varga, Z., Scarelli, A. & Shamandy, A. State monitoring of a population system in changing environment. COMMUNITY ECOLOGY 4, 73–78 (2003). https://doi.org/10.1556/ComEc.4.2003.1.11

Download citation

Keywords

  • Lotka-Volterra model
  • Non-linear system
  • Observability
  • Predator/prey systems