Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1735–1774 | Cite as

On the stochastic evolution of finite populations



This work is a systematic study of discrete Markov chains that are used to describe the evolution of a two-types population. Motivated by results valid for the well-known Moran (M) and Wright–Fisher (WF) processes, we define a general class of Markov chains models which we term the Kimura class. It comprises the majority of the models used in population genetics, and we show that many well-known results valid for M and WF processes are still valid in this class. In all Kimura processes, a mutant gene will either fixate or become extinct, and we present a necessary and sufficient condition for such processes to have the probability of fixation strictly increasing in the initial frequency of mutants. This condition implies that there are WF processes with decreasing fixation probability—in contradistinction to M processes which always have strictly increasing fixation probability. As a by-product, we show that an increasing fixation probability defines uniquely an M or WF process which realises it, and that any fixation probability with no state having trivial fixation can be realised by at least some WF process. These results are extended to a subclass of processes that are suitable for describing time-inhomogeneous dynamics. We also discuss the traditional identification of frequency dependent fitnesses and pay-offs, extensively used in evolutionary game theory, the role of weak selection when the population is finite, and the relations between jumps in evolutionary processes and frequency dependent fitnesses.


Stochastic processes Population genetics Fixation probabilities Perron–Frobenius property Time-inhomogeneous Markov chains Stochastically ordered processes 

Mathematics Subject Classification

92D15 92D25 15B51 60J10 



FACCC was partially supported by FCT/Portugal Strategic Project UID/MAT/00297/2013 (Centro de Matemática e Aplicações, Universidade Nova de Lisboa) and by a “Investigador FCT” grant. FACCC is also indebted to Alexandre Baraviera (Universidade Federal do Rio Grande do Sul, Brazil) and Charles Johnson (College of William and Mary, USA) for useful discussions in preliminary ideas of this work. MOS was partially supported by CNPq under grants # 308113/2012-8, # 486395/2013-8 and # 309079/2015-2. MOS also thanks the hospitality of the Universidade Nova de Lisboa and the partial support under grant UID/MAT/00297/2013. MOS further thanks preliminary discussions of some the ideas in this work with the working group in evolutionary game theory at Universidade Federal Fluminense. Both authors also thank useful comments from Henry Laurie (Cape Town University), Alan Hastings (University of California at Davis), the handling editor, and an anonymous referee which helped to improve the original manuscript.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Departamento de Matemática and Centro de Matemática e Aplicações, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaCaparicaPortugal
  2. 2.Departamento de Matemática AplicadaUniversidade Federal FluminenseNiteróiBrasil

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