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This study introduces a new framework to approximate complex Markov processes in finite populations. It offers an efficient method for analyzing stochastic dynamics in social and biological systems, overcoming computational limitations.

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Area of Science:

  • Mathematical Biology
  • Computational Biology
  • Theoretical Ecology

Background:

  • Dynamical phenomena in finite populations often rely on Markov processes.
  • These processes can become computationally complex and prohibitive with increasing population size or states.

Purpose of the Study:

  • To develop a general framework for approximating stationary distributions of discrete Markov processes.
  • To enable efficient analysis of stochastic dynamics in large or complex systems.

Main Methods:

  • Developed a hierarchy of approximations for stationary distributions.
  • Applied to discrete Markov processes with time-invariant transition probabilities.
  • Framework accommodates systems with a large number of states.

Main Results:

  • The framework provides an efficient method for studying stochastic effects in social and biological communities.
  • It overcomes limitations of existing methods, particularly for systems with stable polymorphic configurations.
  • Demonstrated general applicability across interdisciplinary problems.

Conclusions:

  • The developed formalism offers a versatile and efficient approach to analyzing complex dynamical systems.
  • It is relevant for understanding evolutionary dynamics, social systems, and other interdisciplinary challenges.
  • The method effectively handles stochastic effects and large state spaces.