Abstract
Natural selection is an elegantly simple concept but one that can manifest in complex ways. I review how the basic model of single-trait viability selection has been extended to more complex forms of selection on multiple traits and on reaction norms. Fitness is defined as the expected lifetime reproductive success for individuals with a given genotype or phenotype over a given range of environments. Since the reproductive success realized by any individual will include a stochastic departure from this expectation, selection is therefore a consistent difference in fitness between organisms with different characteristics. A clear distinction is drawn between selection, which can act on any phenotypic difference, and the response to selection, which can occur only if phenotypic differences are heritable. This distinction separates the action of natural selection in filtering variation from the origin of the novel variants on which selection acts. Since selection frequently acts on standing genetic variation or on conditionally neutral variation, both of which accumulate in populations before the imposition of selection, such variation accumulates independently of its fitness effects under the subsequent selection regime. Recent discussions of “Lamarckian” inheritance must be carefully circumscribed to avoid the implication of directed mutation, for which there is no evidence.
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The concept of fitness as a measure for a species’ success in natural selection is central to the theory of evolution. We here investigate how reproduction rates which are not constant but vary in response to environmental fluctuations, influence a species’ prosperity and thereby its fitness. Interestingly, we find that not only larger growth rates but also reduced sensitivities to environmental changes substantially increase the fitness. Thereby,
depending on the noise level of the environment, it might be an evolutionary successful strategy to minimize this sensitivity rather than to optimize the reproduction speed. Also for neutral evolution, where species with exactly the same properties compete, variability in the growth rates plays a crucial role. The time for one species to fixate is strongly reduced in the presence of environmental noise. Hence, environmental fluctuations constitute a possible explanation for effective population
sizes inferred from genetic data that often are much smaller than the census population size. Since its formulation by Darwin and Wallace, the theory of evolution and its explanation for the ongoing development of different species became a paradigm of modern
biology1,2. Herbert Spencer’s famous expression “survival of the fittest”3 provides an appealing and concise summary of the concept of natural selection. However, it leaves aside
one of the most complex yet important aspects of evolutionary theory, namely identifying the factors determining the fitness of a species4,5: fittest individuals are by definition the ones which prevail but the reasons facilitating their survival are not obvious. Even leaving aside the
difficulties arising due to the genotype-phenotype mapping6, it is far from trivial to identify a species’ fitness function and its dependence on measurable ecological quantities. Examples of determinants for evolutionary fitness are reproduction-related quantities like birth rate, viability, number of offspring and span of fertility. All of those directly influence the
amount of genes that an individual will transmit to the future population [either carried by the individual itself or its offspring]. Importantly, all those factors depend on the specific environment in a species’ habitat. This fact is strongly related to the concept of niches: in each niche a different species has potentially the largest fitness and outcompetes less adapted ones. Other ecological factors like population structure and composition also have bearing on this issue. Therefore,
traditional fitness concepts solely based on growth rate and viability were extended by frequency-dependent7,8 or inclusive fitness approaches9. Environmental
conditions are not constant but vary on almost every time scale and pattern. Whole niches change with time and space10 but also within a well-defined niche constant environmental conditions seem to be the exception rather than the norm. For instance, the availability of different nutrients, the presence of detrimental substances or other external factors like temperature, all strongly
influence reproduction/survival and occur on a broad range of time scales11. The relevance of environmental fluctuations for evolutionary dynamics was demonstrated in many different contexts, e.g. general consequences of environmental noise on growth and
extinction12,13,14,15,16,17,18,19,20,21,22,23,24, more specific scenarios like the influence of environmental noise on evolutionary game theory or predator-prey
models25,26, the invasion dynamics of new species27, its interplay with phenotypic
variations28, phenotypic plascitity29 or role environmental tolerance30. Evolutionary strategies to actively cope with variable environmental conditions like phenotypic
heterogeneity or bet-hedging have been extensively studied as
well31,32,33,34,35,36. The scope of this paper is to quantitatively understand the impact of fluctuating reproduction rates on evolutionary dynamics. Specifically, the interplay of such dynamics with demographic fluctuations was not fully elucidated yet. The latter becomes especially
important as the crossover between selection driven and fluctuation driven evolution is a major focus of modern research on evolutionary
dynamics37,38,39,40. Environmental fluctuations potentially
influence both neutral and selection driven evolution rendering a proper understanding essential to grasp the dynamics. Here, we investigate this issue by combining analytical calculations and stochastic simulations. Thereby, we show that an individual’s sensitivity to environmental changes contributes substantially to its fitness: a reduced sensitivity increases the fitness and may compensate for large disadvantages in the average reproduction rate. We also find that fluctuating environments
influence neutral evolution where they can cause much quicker fixation times than naïvely expected. As we explain in detail in the following, this finding has interesting consequences for the interpretation of the effective population size which is typical characteristic to quantify randomness in an evolutionary process. Finally, we show that our results hold not only for very quickly fluctuating environment but also for switching rates up to the time scale of reproduction. To understand the impact of variable environmental conditions, we first consider an extension of a model introduced by May18, which is an evolutionary process based on fluctuating birth rates. Different species are defined by their specific traits which influence both their average reproduction rate as well as their sensitivity on
environmental changes.The model assumes that populations grow logistically, i.e. the population grows exponentially if the total population size is small but reaches a finite maximal size after a while which is set by limited resources. In contrast to standard logistic or Verhulst dynamics we here consider growth rates which are not necessarily constant but may fluctuate due to environmental changes. Mathematically such a scenario can be modeled by decoupled birth-death dynamics for each trait,
S, with noise stemming from environmental changes and demographic fluctuations [the latter were not considered in ref. 18]. The dynamics is described by the following stochastic differential equations for the total number of individuals, NS, of type S:Abstract
Results
The first term is purely deterministic and accounts for reproduction and death events according to standard logistic
growth41,42. In more detail, an individual of type S reproduces at a growth rate, νS while the death rates are assumed to be identical for all traits. Population growth is bounded and deaths rates increase with the total population size
Let us now consider the role of environmental noise. We assume that the environment directly acts on the reproduction rates as illustrated in Fig. 1A. Thereby, variable environmental conditions can be modeled by fluctuating birth rates:
where
Noise Correlation and Fokker-Planck Description
As it will turn out, the correlation level of the noise is crucial for some important features of the model.
Environmental noise acting on the growth rate can influence several species at the same time or act independently on each species. To capture such different noise correlation levels we introduce a correlation parameter,
To analyze the evolutionary dynamics and its dependence on both environmental and demographic noise, it is useful to study
the Fokker-Planck equation [FPE] associated to Eq. [1]. This equation cannot only be used to derive crucial quantities for the evolutionary process like fixation probabilities and times but also offers the possibility to distinguish the contributions of Darwinian fitness and neutral evolution as we are going to explain the the following section. In the remain of this section, the FPE for the
relative abundance,
where
where
Sensitivity to Environmental Changes as an Evolutionary Disadvantage
With the one-dimensional FPE at hand, the evolutionary dynamics and
especially the impact of environmental noise can be investigated. The equation has two terms: The first one proportional to
To grasp the consequences of different sensitivities to environmental changes, we first discuss the case of distinct environmental sensitivities, defined by
where
Even though environmental variability causes a drift term favoring the trait which is less sensitive to environmental changes48, the interplay between drift and diffusion term has to be understood to predict the evolutionary outcome. Indeed, the
environmental contribution to the drift caused by
with
In
Fig. 2 we show the fixation probability for different values of s and
[A] Fixation
probability, Pfix, depending on selection strength, s, and variability
The general case of both species having variable birth rates yields analogous results: a selection advantage for the species with less sensitivity on the environment. Importantly, the
selection disadvantage due to environmental noise does not scale with the carrying capacity as most demographic fluctuation effects do, i.e. the mechanism is effective irrespectively of the population size. When investigating only one species, the condition
An alternative interpretation of our results on the fixation probability is as follows. As mentioned above, the parameters σ or Δ depend on two factors: the sensitivity of a trait’s growth rate on environmental conditions and the strength of the environmental fluctuations themselves. For a given sensitivity of an individual on the environment, the ordinate Δ in Fig. 2 then corresponds to the strength of environmental fluctuations. While for weakly fluctuating environments a growth advantage is more beneficial, the situation is different for strong environmental variations. Then it is more advantageous to minimize the sensitivity to those variations rather than to optimize the growth rate. Interestingly one can construe this result in the context of game theory: decreasing the sensitivity to environmental changes also means to optimize the worst-case-scenario outcome because the average birth rate is the least reduced when the variability is small. In game theory, this corresponds to the MaxiMin strategy which was shown to be very successful in many fields as finance, economy or behavioral psychology49,50. In the field of evolutionary dynamics another example of a MaxiMin strategy is bacterial chemotaxis, where it was proposed that bacteria track chemoattractants trying to maximize their minimal uptake51.
Neutral Evolution
Beside contributing to the fitness, environmental variability also influences fixation probability and time in the case of neutral evolution, i.e.
While the correlation parameter
does not qualitatively influence results discussed so far, it plays an important role for neutral evolution. Interestingly, for fully correlated noise
Let us first consider the fixation probability [cf. Eq. [6]], see
Fig. 3A. While for the standard situation of no environmental noise [or fully correlated noise] a linear dependence of the fixation probability on the initial fraction of a trait, x0 is observed,
Fixation probability [panel A] and time [panel B] in the neutral case.
Solid lines indicate analytical results for the two typical cases of perfectly correlated and uncorrelated noises ε = 1 and ε = 0. Parameters are:
While the behavior of the fixation probability is mainly due to the stable fixed point, the situation is more intricate for the extinction or fixation time which is another important quantity to describe evolutionary processes. As mentioned above, a
coexistence fixed point is expected to increase the extinction time while a larger random drift decreases it. To ultimately understand the influence of environmental fluctuations, we therefore calculated the extinction time,
where
The result for not fully correlated noise [ε < 1] differs again from the
non-fluctuating/fully correlated scenario,
Individual Based Model
To further investigate the impact of variable environmental conditions, we introduce an Individual Based Model [IBM]. Such individual or agent based models serve as powerful tools to study evolutionary processes. They intrinsically include demographic noise as reproduction and death events are explicitly modeled. Additionally the IBM here serves as a proof of principle that linear multiplicative noise can be realistically expected when considering birth rates which depend on fluctuating environments. Since we model the environment and its fluctuations now explicitly, we can vary the environmental switching rate and thereby study the so far discussed phenomena beyond the white-noise limit, i.e. for environments which change slower. Even though, a particular choice for the IBM is made, we want to stress that the results presented above hold for any microscopic model whose macroscopic representation is given by Eq. [1], i.e. where the birth rate is subject to fluctuations thereby leading to linearly multiplicative noise.
In the IBM each individual reproduces according to its
experienced environments. The simplest version of the model is that only the current environment influences the birth rate but to show that our results are more rigorous we also include more realistic scenarios where an individual’s environment history matters. To be more specific, the reproduction rate of an individual, i, at time t, depends a priori on the history of environmental conditions experienced during its lifetime
The growth rate
of a species depends on the previously experienced environments. Before considering that, let us first discuss how the growth rate depends on a particular constant environment E. This quantity, the instantaneous growth rate
[A] Illustration of the instantaneous growth rate depending on the environment. Both species have the same average reproduction rate ϕ1 = ϕ2 = ϕ but species 1 is more sensitive to environmental changes [ω1 > ω2] [B] Comparison of the IBM and the Langevin model. We show the fixation time for neutral evolution for
x0 = 0.5 vs the environmental switching rate 1/τ. Dots correspond to the IBM [m = 0] for different values of
with ϕS the ordinate of the inflection point,
Let us now consider changing environments and individuals whose current growth rate depends also on previously experienced
environments. The reproduction rate
where the memory parameter
Mapping
To compare the results of the microscopic individual based model to the effective stochastic model, Eq. [1], the parameters of both models have to be mapped. In this section, we briefly explain how such a mapping can be obtained but results in the following sections can be understood without those details. For simplicity let us consider
the case
Note that the variability in the growth rate not only results in
Results Beyond the White-Noise Limit
For a detailed comparison of the IBM with the analytics derived in the first part of this paper, we simulate the IBM with a modified Gillespie algorithm updating reproduction rates after every environmental change43. As shown in Fig. 3A,B, results for fixation probability and time, are in excellent agreement with analytic solutions [Eqs. [6] and [7]]. In particular, the sigmoidal shape of the fixation probability is well reproduced by the IBM, supporting the existence and importance of linear multiplicative noise.
Finally, the IBM
enables us to study the environmental switching rate. This is of main interest as previous results were obtained using a white-noise approximation and strictly hold only for very rapidly fluctuating environments. In Fig. 4B, the dependency on τ of the extinction time in the neutral case for
All in all, our analysis of the IBM beyond the white noise limit confirms that there is a broad parameter regime where environmental fluctuations play a crucial role for both neutral evolution and the fitness functions. For fluctuations up to the timescale of reproduction events [marked by the vertical gray line], the description introduced above is valid. Nutrients and other metabolically important substances can vary on time scales quicker than reproduction. Therefore, we expect effects as discussed above to play a crucial role for evolutionary dynamics.
Conclusion
We quantitatively demonstrated that environmental variability has crucial impact on evolutionary fitness. Our results do not rely on details of microscopic models but are rather derived from a macroscopic model whose only key assumption is that the birth rate of individuals is not constant but fluctuates. This assumption automatically leads to linearly multiplicative noise which gives rise to all discussed effects.
First, we quantified the role of reduced sensitivity to environmental changes and determined how it increases the fitness. Even though the increase stems from noise, its amplitude does not drop as the total population size increases. Therefore, such a mechanism is effective also for very large populations, contrary to most other fluctuation-based effects. By studying the interplay of the resulting evolutionary dynamics with random drift, we confirmed the importance of that fitness contribution and showed that those contributions are visible in a broad parameter regime. Importantly, even though fluctuation driven the fitness contribution due to environmental noise is of the same order of magnitude than the contribution due to different growth rates and present for all population sizes. This finding is of great interest when thinking about whether a generalist or a specialist is evolutionary favored57,58. We can quantify that depending on the level of environmental noise two regimes are present: For strongly fluctuating environments it strongly pays off to be less sensitive to such changes [to be generalist] while for little environmental fluctuations is more beneficial to reproduce as quick as possible [be a specialist].
In addition, we showed that the timescale of extinction in the neutral case is strongly affected by environmental noise. That provides a possible contribution to the reduction of effective population sizes, which are often found experimentally to be much smaller than the census population size. The reason is that environmental fluctuations increase the random drift that automatically results in smaller effective populations size, even if the source of the larger fluctuations is not demographic noise. Finally, we investigated individual based models that generate the linear multiplicative noise considered here. We thereby demonstrate that our description holds for fluctuation time scales up to the time scale of reproduction events.
As a future perspective, it will be of interest to study other forms of multiplicative noise in more detail, e.g. a noise in the death rate γ that would lead to a nonlinear dependency of the noise on the number of individuals. Also the interplay between the noise-induced frequency dependence discussed here and the one resulting from payoff matrixes in standard evolutionary game theory is worth further investigation. Finally, the question as to how fluctuation effects are influenced by reproduction rates that depend on time - a realistic model extension - remains open. With no environmental fluctuations, such rates would result in a smaller standard deviation of the expected time of reproduction and could potentially further increase the strength of the effects on fitness that we presented here.
Additional Information
How to cite this article: Melbinger, A. and Vergassola, M. The Impact of Environmental Fluctuations on Evolutionary Fitness Functions. Sci. Rep. 5, 15211; doi: 10.1038/srep15211 [2015].
Supplementary Material
Supplementary Information:
Acknowledgments
We thank Jonas Cremer for valuable discussions and comments on the manuscript. A.M. acknowledges the German Academic Exchange Service [DAAD] for financial support.
Footnotes
The authors declare no competing financial interests.
Author Contributions A.M. and M.V. designed and performed the research and wrote the manuscript.
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