J Stat Phys (2018) 172:126–142
Environmental Stochasticity and the Speed of Evolution
· David A. Kessler
Nadav M. Shnerb
Received: 17 July 2017 / Accepted: 21 February 2018 / Published online: 2 March 2018
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Biological populations are subject to two types of noise: demographic stochastic-
ity due to ﬂuctuations in the reproductive success of individuals, and environmental variations
that affect coherently the relative ﬁtness of entire populations. The rate in which the aver-
age ﬁtness of a community increases has been considered so far using models with pure
demographic stochasticity; here we present some theoretical considerations and numerical
results for the general case where environmental variations are taken into account. When the
competition is pairwise, ﬁtness ﬂuctuations are shown to reduce the speed of evolution, while
under global competition the speed increases due to environmental stochasticity.
Keywords Speed of evolution · Population genetics · Fixation probability · Demographic
stochasticity · Environmental stochasticity
Life forms evolve via a continuous process of competition and selection. Malthusian dynamics
drives any population to its abundance limit, set by the carrying capacity of its environment
and interaction with other populations. In this state the process becomes a zero-sum game
and the relative ﬁtnesses of individuals govern their chances to stay alive and to reproduce.
When the ﬁtness is an inherited feature, as in the case where it reﬂects the characteristics of a
genotype, a new strain (species, haplotype) appears each time a signiﬁcant mutation happens.
The abundances of strains with beneﬁcial mutations and higher ﬁtness grow, on average, at
the expense of inferior strains. If the supply of beneﬁcial mutations is unlimited, this process
leads to a continuous increase in the average ﬁtness of the whole community.
The rate in which the average ﬁtness increases has been considered by many authors [1–
4]. It was shown that the average ﬁtness grows linearly in time, and the dependence of its
Nadav M. Shnerb
Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel