The European Physical Journal Special Topics

, Volume 216, Issue 1, pp 107–116 | Cite as

Stochastic evolution of a continuum particle system with dispersal and competition

Micro- and mesoscopic description
  • Dmitri FinkelshteinEmail author
  • Yuri KondratievEmail author
  • Yuri KozitskyEmail author
  • Oleksandr KutoviyEmail author
Regular Article


A Markov evolution of a system of point particles in ℝ d is described at micro- and mesoscopic levels. The particles reproduce themselves at distant points (dispersal) and die, independently and under the effect of each other (competition). The microscopic description is based on an infinite chain of equations for correlation functions, similar to the BBGKY hierarchy used in the Hamiltonian dynamics of continuum particle systems. The mesoscopic description is based on a Vlasov-type kinetic equation for the particle’s density obtained from the mentioned chain via a scaling procedure. The main conclusion of the microscopic theory is that the competition can prevent the system from clustering, which makes its description in terms of densities reasonable. A possible homogenization of the solutions to the kinetic equation in the long-time limit is also discussed.


Correlation Function European Physical Journal Special Topic Microscopic Theory BBGKY Hierarchy Poisson Measure 
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© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Institute of Mathematics, National Academy of Sciences of UkraineKiev-4Ukraine
  2. 2.Fakultät für Mathematik, Universität BielefeldBielefeldGermany
  3. 3.Instytut Matematyki, Uniwersytet Marii Curie-SklodowskiejLublinPoland

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