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Probabilistic analysis of replicator–mutator equations

Part of: Mathematical biology in general Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  23 March 2022

Lijun Bo  and
Huafu Liao
Lijun Bo*
Affiliation:
Xidian University and University of Science and Technology of China
Huafu Liao*
Affiliation:
National University of Singapore
*
*Postal address: School of Mathematics and Statistics, Xidian University, Xi’an 710071, China; School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China. Email address: lijunbo@ustc.edu.cn
**Postal address: Department of Mathematics, National University of Singapore, Singapore 119076, Singapore. Email address: mathuaf@nus.edu.sg
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Abstract

This paper discusses a general class of replicator–mutator equations on a multidimensional fitness space. We establish a novel probabilistic representation of weak solutions of the equation by using the theory of Fokker–Planck–Kolmogorov (FPK) equations and a martingale extraction approach. We provide examples with closed-form probabilistic solutions for different fitness functions considered in the existing literature. We also construct a particle system and prove a general convergence result to the unique solution of the FPK equation associated with the extended replicator–mutator equation with respect to a Wasserstein-like metric adapted to our probabilistic framework.

Keywords

Replicator–mutator equationsprobabilistic representationFPK equationsmartingale extractionparticle system

MSC classification

Primary: 92B05: General biology and biomathematics
Secondary: 60H10: Stochastic ordinary differential equations 60G46: Martingales and classical analysis
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 1 , March 2022 , pp. 167 - 201
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Alfaro, M. and Carles, R. (2014). Explicit solutions for replicator–mutator equations: extinction versus acceleration. SIAM J. Appl. Math. 74, 1919–1934.10.1137/140979411CrossRefGoogle Scholar
Alfaro, M. and Carles, R. (2017). Replicator–mutator equations with quadratic fitness. Proc. Amer. Math. Soc. 145, 5315–5327.10.1090/proc/13669CrossRefGoogle Scholar
Alfaro, M. and Veruete, M. (2019). Evolutionary branching via replicator–mutator equations. J. Dynam. Differential Equat. 31, 2029–2052.10.1007/s10884-018-9692-9CrossRefGoogle Scholar
Bürger, R. (1988). Perturbations of positive semigroups and applications to population genetics. Math. Z. 197, 259–272.10.1007/BF01215194CrossRefGoogle Scholar
Bürger, R. (1998). Mathematical properties of mutation-selection models. Genetica 102, article no. 279.10.1007/978-94-011-5210-5_23CrossRefGoogle Scholar
Del Moral, P. and Miclo, L. (2000). A Moran particle system approximation of Feynman–Kac formulae. Stoch. Process. Appl. 86, 193–216.10.1016/S0304-4149(99)00094-0CrossRefGoogle Scholar
Duffie, D., FilipoviĆ, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.10.1214/aoap/1060202833CrossRefGoogle Scholar
Fleming, W. H. (1979). Equilibrium distributions of continuous polygenic traits. SIAM J. Appl. Math. 36, 148–168.10.1137/0136014CrossRefGoogle Scholar
Fournier, N. and Guillin, A. (2015). On the rate of convergence in Wasserstein distance of the empirical measure. Prob. Theory Relat. Fields 162, 707–738.10.1007/s00440-014-0583-7CrossRefGoogle Scholar
Health, D. and Schweizer, M. (2000). Martingale versus PDEs in finance: an equivalence result with examples. J. Appl. Prob. 37, 947–957.10.1239/jap/1014843075CrossRefGoogle Scholar
Kac, M. (1956). Foundations of kinetic theory. In Proc. 3rd Berkeley Symp. Math. Statist. Prob., Vol. III, University of California Press, Berkeley, pp. 171–197.10.1525/9780520350694-012CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Kimura, M. (1965). A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Nat. Acad. Sci. USA 54, 731–736.10.1073/pnas.54.3.731CrossRefGoogle Scholar
LuÇon, E. and Stannat, W. (2014). Mean field limit for disordered diffusions with singular interactions. Ann. Appl. Prob. 24, 1946–1993.10.1214/13-AAP968CrossRefGoogle Scholar
Mandelkern, M. (1989). Metrization of the one-point compactification. Proc. Amer. Math. Soc. 107, 1111–1115.10.1090/S0002-9939-1989-0991703-4CrossRefGoogle Scholar
Manita, O. A., Romanov, M. S. and Shaposhnikov, S. V. (2015). On uniqueness of solutions to nonlinear Fokker–Planck–Kolmogorov equations. Nonlinear Anal. 128, 199–226.10.1016/j.na.2015.08.008CrossRefGoogle Scholar
Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge University Press.10.1017/CBO9780511526244CrossRefGoogle Scholar
Qin, L. and Linetsky, V. (2016). Positive eigenfunctions of Markovian pricing operators: Hansen–Scheinkman factorization, Ross recovery, and long-term pricing. Operat. Res. 64, 99–117.10.1287/opre.2015.1449CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York.10.1007/978-3-662-06400-9CrossRefGoogle Scholar
Rouzine, I., Brunet, E. and Wilke, C. (2008). The traveling-wave approach to asexual evolution: Muller’s ratchet and speed of adaptation. Theoret. Pop. Biol. 73, 24–46.10.1016/j.tpb.2007.10.004CrossRefGoogle ScholarPubMed
Rouzine, I., Wakeley, J. and Coffin, J. (2003). The solitary wave of asexual evolution. Proc. Nat. Acad. Sci. USA 100, 587–592.10.1073/pnas.242719299CrossRefGoogle Scholar
Sniegowski, P. and Gerrish, P. (2010). Beneficial mutations and the dynamics of adaptation in asexual populations. Phil. Trans. R. Soc. B 365, 1255–1263.10.1098/rstb.2009.0290CrossRefGoogle ScholarPubMed
Strook, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, New York.Google Scholar
Sznitman, A. S. (1991). Topics in propagation of chaos. In École d’ÉtÉ de ProbabilitÉs de Saint-Flour XIX—1989, Springer, Berlin, Heidelberg, pp. 165–251.Google Scholar
Tsimring, L., Levine, H. and Kessler, D. (1996). RNA virus evolution via a fitness-space model. Phys. Rev. Lett. 76, 4440–4443.10.1103/PhysRevLett.76.4440CrossRefGoogle Scholar
Villani, C. (2003). Topics in Optimal Transportation. American Mathematical Society, Providence, RI.10.1090/gsm/058CrossRefGoogle Scholar
Xu, L. P. (2018). Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials. Ann. Appl. Prob. 28, 1136–1189.10.1214/17-AAP1327CrossRefGoogle Scholar