The polymorphic evolution sequence for populations with phenotypic plasticity. (English) Zbl 1415.92120

Summary: 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 [the first author et al., A stochastic model for immunotherapy of cancer. Sci. Rep. 6, 24169 (2016)], 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 [the first author et al., loc. cit.] 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 [N. Champagnat, Stochastic Processes Appl. 116, No. 8, 1127–1160 (2006; Zbl 1100.60055); with S. Méléard, Probab. Theory Relat. Fields 151, No. 1–2, 45–94 (2011; Zbl 1225.92040)].


92D10 Genetics and epigenetics
92D15 Problems related to evolution
60J85 Applications of branching processes
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[1] K. B. Athreya. Some results on multitype continuous time Markov branching processes. Ann. Math. Stat., 39:347–357, 1968. · Zbl 0169.49202
[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. · Zbl 1371.92094
[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. · Zbl 0890.92020
[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. · Zbl 1100.60055
[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. · Zbl 1322.92052
[11] N. Champagnat and S. Méléard. Polymorphic evolution sequence and evolutionary branching. Prob. Theory Rel., 151(1-2):45–94, 2011. · Zbl 1225.92040
[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. · Zbl 1300.92075
[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. · Zbl 0904.60001
[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. · Zbl 0592.60049
[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. · Zbl 1060.92055
[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. · Zbl 1085.92043
[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. · Zbl 0203.17402
[23] H. Kesten and B. P. Stigum. A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Stat., 37:1211–1223, 1966. · Zbl 0203.17401
[24] H. Kesten and B. P. Stigum. Limit theorems for decomposable multi-dimensional Galton-Watson processes. J. Math. Anal. Appl., 17:309–338, 1967. · Zbl 0203.17501
[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.
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