×

Rare mutations in evolutionary dynamics. (English) Zbl 1350.92033

Summary: In this paper we study the effect of rare mutations, driven by a marked point process, on the evolutionary behavior of a population. We derive a Kolmogorov equation describing the expected values of the different frequencies and prove some rigorous analytical results about their behavior. Finally, in a simple case of two different quasispecies, we are able to prove that the rarity of mutations increases the survival opportunity of the low fitness species.

MSC:

92D15 Problems related to evolution
92D25 Population dynamics (general)
35R09 Integro-partial differential equations
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Amadori, A. L., Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach, Differential Integral Equations, 16, 7, 787-811 (2003) · Zbl 1052.35083
[2] Amadori, A. L.; Briani, M.; Natalini, R., A non-local rare mutations model for quasispecies and Prisoner’s dilemma: numerical assessment of qualitative behaviour, European J. Appl. Math., 1-24 (2015), published online: 20 July 2015
[3] Athreya, K. B.; Kliemann, W.; Koch, G., On sequential construction of solutions of stochastic differential equations with jump terms, Systems Control Lett., 10, 2, 141-146 (1988) · Zbl 0636.60062
[4] Calzolari, A.; Nappo, G., Sulla costruzione di un processo di puro salto (1996), University of Roma “La Sapienza”, Technical Report
[5] Calzolari, A.; Nappo, G., Counting observations: a note on state estimation sensitivity with an \(L^1\)-bound, Appl. Math. Optim., 44, 2, 177-201 (2001) · Zbl 0983.60029
[6] Champagnat, N.; Ferrière, R.; Méléard, S., From individual stochastic processes to macroscopic models in adaptive evolution, Stoch. Models, 24, sup. 1, 2-44 (2008) · Zbl 1157.60339
[7] Dieckmann, U.; Law, R., The dynamical theory of coevolution: a derivation from stochastic ecological processes, J. Math. Biol., 34, 5-6, 579-612 (1996) · Zbl 0845.92013
[8] Eigen, M.; Schuster, P., The Hypercycle: A Principle of Natural Self-organization (1979), Springer-Verlag
[9] Hofbauer, J.; Sigmund, K., Evolutionary Games and Population Dynamics (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0914.90287
[10] Jourdain, B.; Méléard, S.; Woyczynski, W., Lévy flights in evolutionary ecology, J. Math. Biol., 65, 4, 677-707 (2012) · Zbl 1311.92136
[11] Lamperti, J., Stochastic Processes: A Survey of the Mathematical Theory, Appl. Math. Sci., vol. 23 (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0365.60001
[12] Maynard Smith, J.; Price, G. R., The logic of animal conflict, Nature, 246, 15-18 (1973) · Zbl 1369.92134
[13] Nowak, M. A., Evolutionary Dynamics. Exploring the Equations of Life (2006), The Belknap Press of Harvard University Press: The Belknap Press of Harvard University Press Cambridge, MA · Zbl 1115.92047
[14] Stadler, P.; Schuster, P., Mutation in autocatalytic reaction networks, J. Math. Biol., 30, 6, 597-632 (1992) · Zbl 0776.92007
[15] Taylor, P. D.; Jonker, L. B., Evolutionary stable strategies and game dynamics, Math. Biosci., 40, 1-2, 145-156 (1978) · Zbl 0395.90118
[16] Traulsen, A.; Claussen, J. C.; Hauert, C., Coevolutionary dynamics in large, but finite populations, Phys. Rev. E, 74 (2006), Article number 011901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.