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Semigroups of Operators -Theory and Applications pp 197–211Cite as

Dynamics of Spatial Logistic Model: Finite Systems

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 113))

Abstract

The spatial logistic model is a system of point entities (particles) in \(\mathbb {R}^d\) which reproduce themselves at distant points (dispersal) and die, also due to competition. The states of such systems are probability measures on the space of all locally finite particle configurations. In this paper, we obtain the evolution of states of ‘finite systems’, that is, in the case where the initial state is supported on the subset of the configuration space consisting of finite configurations. The evolution is obtained as the global solution of the corresponding Fokker-Planck equation in the space of measures supported on the set of finite configurations. We also prove that this evolution preserves the existence of exponential moments and the absolute continuity with respect to the Lebesgue-Poisson measure.

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References

  1. Albeverio, S., Kondratiev, Yu. G., and Röckner, M.: Analysis and geometry on configuration spaces. J. Func. Anal. 154, 444–500 (1998)

    Google Scholar 

  2. Dobrushin, R. L., Sinai, Y. G., and Sukhov, Y. M.: Dynamical systems of statistical mechanics. In: Itogi Nauki, pp. 235–284. VINITI (1985); eng. transl. in: Sinai Ya. G. (ed.) Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics, II, Encyclopaedia Math. Sci., Springer, Berlin Heidelberg 1989

    Google Scholar 

  3. Finkelshtein, D., Kondratiev, Y., Kozitsky, Y., and Kutoviy, O.: Stochastic evolution of a continuum particle system with dispersal and competition. Eur. Phys. J. Special Topics 216, 107–116 (2103)

    Google Scholar 

  4. Finkelshtein, D., Kondratiev, Y., Kozitsky, Y., and Kutoviy, O.: The statistical dynamics of a spatial logistic model and the related kinetic equation. arXiv:1401.0557v1, 2014 (to appear in Math. Models Methods Appl. Sci.)

  5. Finkelshtein, D., Kondratiev, Y., and Kutoviy, O.: An operator approach to Vlasov scaling for some models of spatial ecology. Methods Funct. Anal. Topology 19 108–126 (2013)

    Google Scholar 

  6. Finkelshtein, D., Kondratiev, Y., and Kutoviy, O.: Semigroup approach to birth-and-death stochastic dynamics in continuum. J. Funct. Anal. 262, 1274–1308 (2012)

    Google Scholar 

  7. Finkelshtein, D. L., Kondratiev, Yu. G., and Kutoviy, O.: Individual based model with competition in spatial ecology. SIAM J. Math. Anal. 41, 297–317 (2009)

    Google Scholar 

  8. Finkelshtein, D. L., Kondratiev, Yu. G., and Oliveira, M. J.: Markov evolution and hierarchical equations in the continuum. I: One-component systems. J. Evol. Equ. 9 197–233 (2009)

    Google Scholar 

  9. Fournier, N. and Méléard, S.: A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 1880–1919 (2004)

    Google Scholar 

  10. Kondratiev, Yu. and Kuna, T.: Harmonic analysis on configuration space. I. General theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5, 201–233 (2002)

    Google Scholar 

  11. Kondratiev, Yu, Kutoviy, O, and Pirogov, S.: Correlation functions and invariant measures in continuous contact model. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11, 231–258 (2008)

    Google Scholar 

  12. Kondratiev, Yu. G. and Skorokhod, A. V. On contact models in continuum. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9, 187–198 (2006)

    Google Scholar 

  13. Neuhauser, C.: Mathematical challenges in spatial ecology. Notices of AMS. 48 (11), 1304–1314 (2001)

    Google Scholar 

  14. Ovaskainen, O., Finkelshtein, D., Kutoviy, O, Cornell, S, Bolker, B, and Kondratiev, Yu.: A general mathematical framework for the analysis of spatiotemporal point processes. Theor. Ecol. 7, 101–113 (2014)

    Google Scholar 

  15. Parthasarathy, K.P. (1967) Probability Measures on Metric Spaces. New York – London: Probability and Mathematical Statistics 3, Academic Press, Inc

    Google Scholar 

  16. Thieme, H.R. and Voigt, J.: Stochastic semigroups: their construction by perturbation and approximation. In Positivity IVtheory and applications, 135–146, Tech. Univ. Dresden, Dresden, 2006

    Google Scholar 

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Acknowledgments

The author thanks Yuri Kondratiev for fruitful discussions on the subject of this work. He is also grateful for the support provided by the DFG through SFB 701: “Spektrale Strukturen und Topologische Methoden in der Mathematik”, by the ZiF Research Group “Stochastic Dynamics: Mathematical Theory and Applications” (Universität Bielefeld), and by the European Commission under the project STREVCOMS PIRSES-2013–612669.

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Authors and Affiliations

  1. Institute of Mathematics, Maria Curie-Skłodowska University, 20–031, Lublin, Poland

    Yuri Kozitsky

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  1. Yuri Kozitsky
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Correspondence to Yuri Kozitsky .

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Editors and Affiliations

  1. School of Mathematics,Statistics and Computer Science, Institute of Mathematics,Technical University of Łódź, University of KwaZulu-Natal, Durban, Poland

    Jacek Banasiak

  2. Lublin University of Technology, Lublin, Poland

    Adam Bobrowski

  3. University of Warsaw, Institute of Applied Mathematics and Mechanics, Warsaw, Poland

    Mirosław Lachowicz

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Kozitsky, Y. (2015). Dynamics of Spatial Logistic Model: Finite Systems. In: Banasiak, J., Bobrowski, A., Lachowicz, M. (eds) Semigroups of Operators -Theory and Applications. Springer Proceedings in Mathematics & Statistics, vol 113. Springer, Cham. https://doi.org/10.1007/978-3-319-12145-1_12

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