Electronic Journal of Probability

The polymorphic evolution sequence for populations with phenotypic plasticity

Martina Baar and Anton Bovier

Full-text: Open access


In this paper we study a class of stochastic individual-based models that describe the evolution of haploid populations where each individual is characterised by a phenotype and a genotype. The phenotype of an individual determines its natural birth- and death rates as well as the competition kernel, $c(x,y)$ which describes the induced death rate that an individual of type $x$ experiences due to the presence of an individual or type $y$. When a new individual is born, with a small probability a mutation occurs, i.e. the offspring has different genotype as the parent. The novel aspect of the models we study is that an individual with a given genotype may expresses a certain set of different phenotypes, and during its lifetime it may switch between different phenotypes, with rates that are much larger then the mutation rates and that, moreover, may depend on the state of the entire population. The evolution of the population is described by a continuous-time, measure-valued Markov process. In [4], such a model was proposed to describe tumor evolution under immunotherapy. In the present paper we consider a large class of models which comprises the example studied in [4] and analyse their scaling limits as the population size tends to infinity and the mutation rate tends to zero. Under suitable assumptions, we prove convergence to a Markov jump process that is a generalisation of the polymorphic evolution sequence (PES) as analysed in [9, 11].

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 72, 27 pp.

Received: 5 August 2017
Accepted: 3 July 2018
First available in Project Euclid: 27 July 2018

Permanent link to this document

Digital Object Identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J85: Applications of branching processes [See also 92Dxx] 92D25: Population dynamics (general)

adaptive dynamics canonical equation large population limit mutation-selection individual-based model

Creative Commons Attribution 4.0 International License.


Baar, Martina; Bovier, Anton. The polymorphic evolution sequence for populations with phenotypic plasticity. Electron. J. Probab. 23 (2018), paper no. 72, 27 pp. doi:10.1214/18-EJP194. https://projecteuclid.org/euclid.ejp/1532678635

Export citation


  • [1] K. B. Athreya. Some results on multitype continuous time Markov branching processes. Ann. Math. Stat., 39:347–357, 1968.
  • [2] K. B. Athreya and P. E. Ney. Branching processes. Die Grundlehren der mathematischen Wissenschaften, Vol. 196. Springer-Verlag Berlin Heidelberg, 1972.
  • [3] M. Baar, A. Bovier, and N. Champagnat. From stochastic, individual-based models to the canonical equation of adaptive dynamics - in one step. Ann. Appl. Probab., 27:1093–1170, 2017.
  • [4] M. Baar, L. Coquille, H. Mayer, M. Hölzel, M. Rogava, T. Tüting, and A. Bovier. A stochastic model for immunotherapy of cancer. Scientific Reports, 6:24169, 2016.
  • [5] V. Bansaye and S. Méléard. Stochastic models for structured populations. Scaling limits and long time behavior, volume 1 of Mathematical Biosciences Institute Lecture Series. Stochastics in Biological Systems. Springer, Cham; MBI Mathematical Biosciences Institute, Ohio State University, Columbus, OH, 2015.
  • [6] H. J. E. Beaumont, J. Gallie, C. Kost, G. C. Ferguson, and P. B. Rainey. Experimental evolution of bet hedging. Nature, 462:90–93, 2009.
  • [7] B. Bolker and S. W. Pacala. Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol., 52(3):179 – 197, 1997.
  • [8] B. M. Bolker and S. W. Pacala. Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Am. Nat., 153(6):575–602, 1999.
  • [9] N. Champagnat. A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch. Proc. Appl., 116(8):1127–1160, 2006.
  • [10] N. Champagnat, P.-E. Jabin, and S. Méléard. Adaptation in a stochastic multi-resources chemostat model. J. Math. Pures Appl., 101(6):755–788, 2014.
  • [11] N. Champagnat and S. Méléard. Polymorphic evolution sequence and evolutionary branching. Prob. Theory Rel., 151(1-2):45–94, 2011.
  • [12] P. Collet, S. Méléard, and J. A. J. Metz. A rigorous model study of the adaptive dynamics of Mendelian diploids. J. Math. Biol., 67(3):569–607, 2013.
  • [13] D. L. DeAngelis and V. Grimm. Deangelis dl, grimm v. individual-based models in ecology after four decades. F1000Prime Reports, 6(39), 2014.
  • [14] U. Dieckmann and R. Law. Moment approximations of individual-based models. In U. Dieckmann, R. Law, and J. A. J. Metz, editors, The geometry of ecological interactions: simplifying spatial complexity, pages 252–270. Cambridge University Press, 2000.
  • [15] P. Dupuis and R. S. Ellis. A weak convergence approach to the theory of large deviations. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., New York, 1997.
  • [16] S. N. Ethier and T. G. Kurtz. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986.
  • [17] N. Fournier and S. Méléard. A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab., 14(4):1880–1919, 2004.
  • [18] M. I. Freidlin and A. D. Wentzell. Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg, 3rd edition, 2012.
  • [19] G. Fusco and A. Minelli. Phenotypic plasticity in development and evolution: facts and concepts. Phil. Trans. Royal Soc. London B: Biol. Sci., 365:547–556, 2010.
  • [20] V. Grimm and S. F. Railsback. Individual-based modeling and ecology. Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton, NJ, 2005.
  • [21] M. Hölzel, A. Bovier, and T. Tüting. Plasticity of tumour and immune cells: a source of heterogeneity and a cause for therapy resistance? Nat. Rev. Cancer, 13(5):365–376, 2013.
  • [22] H. Kesten and B. P. Stigum. Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math. Stat., 37:1463–1481, 1966.
  • [23] H. Kesten and B. P. Stigum. A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Stat., 37:1211–1223, 1966.
  • [24] H. Kesten and B. P. Stigum. Limit theorems for decomposable multi-dimensional Galton-Watson processes. J. Math. Anal. Appl., 17:309–338, 1967.
  • [25] J. Landsberg, J. Kohlmeyer, M. Renn, T. Bald, M. Rogava, M. Cron, M. Fatho, V. Lennerz, T. Wölfel, M. Hölzel, and T. Tüting. Melanomas resist T-cell therapy through inflammation-induced reversible dedifferentiation. Nature, 490(7420):412–416, 10 2012.
  • [26] S. Pénisson. Conditional limit theorems for multitype branching processes and illustration in epidemiological risk analysis. PhD thesis, Universität Potsdam, Potsdam (Germany), 2010.
  • [27] B. A. Sewastjanow. Verzweigungsprozesse. R. Oldenbourg Verlag, Munich-Vienna, 1975.