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The Baldwin effect under multi-peaked fitness landscapes: Phenotypic fluctuation accelerates evolutionary rate

Phenotypic fluctuations and plasticity can generally affect the course of evolution, a process known as the Baldwin effect. Several studies have recast this effect and claimed that phenotypic plasticity acceler- ates evolutionary rate (the Baldwin expediting effect); however, the validity of this claim is still controversial. In this study, we investi- gate the evolutionary population dynamics of a quantitative genetic model under a multi-peaked fitness landscape, in order to evaluate the validity of the effect. We provide analytical expressions for the evolutionary rate and average population fitness. Our results indicate that under a multi-peaked fitness landscape, phenotypic fluctuation always accelerates evolutionary rate, but it decreases the average fit- ness. As an extreme case of the trade-off between the rate of evolution and average fitness, phenotypic fluctuation is shown to accelerate the error catastrophe, in which a population fails to sustain a high-fitness peak. In the context of our findings, we discuss the role of phenotypic plasticity in adaptive evolution.


I. INTRODUCTION
Phenotypic plasticity is a common process, whereby phenotypic changes occur during an organism's life cycle. In some cases, phenotypic plasticity can emerge as a response to environmental variation. For example, the desert locust transitions from a solitary to a gregarious phase, depending on population density [1,2]. In addition, a phenotype can fluctuate randomly, for example, as a result of the stochasticity of gene expression, exemplified by recent studies in microorganisms [3][4][5][6] and animal development [7][8][9]. Such processes are also interpreted as phenotypic plasticity, and are ubiquitous in nature. Given this, the role of random phenotypic plasticity in adaptive evolution warrants further consideration. Do phenotypic changes in individuals have any impacts across generations, and thus, influence their evolution? At first glance, phenotypic plasticity does not seem to affect evolutionary processes because only genotypes, rather than phenotypes, are heritable. Non-heritable phenotypes acquired during the life cycle, however, can affect evolution through natural selection because the natural selection acts on acquired phenotypes. This does not mean Lamarckism -heritability of acquired phenotypes, but indicates that genetically determined plasticity, i.e., an ability to change phenotypes, can affect the evolutionary process. This process is referred to as the Baldwin effect, which was first introduced by Baldwin [10], and later recast by Simpson [11]. Several studies [9,[12][13][14][15][16][17] have revealed that this effect is advantageous under a fluctuating environment, as phenotypic plasticity can allow organisms to survive sudden environmental changes, and therefore, facilitate adaptation to new environments. However, whether phenotypic plas- * saito@complex.c.u-tokyo.ac.jp, Tel.: +81-3-5454-6732 ticity is advantageous throughout evolution, even under fixed environments, is still elusive.
Can phenotypic plasticity accelerate evolutionary rate? This question stems from a recent incarnation of the Baldwin effect known as the "Baldwin expediting effect" [18]. Indeed, the seminal work by Hinton and Nowlan [19] showed that phenotypic plasticity accelerates evolutionary rate even in fixed environments, although, the validity of these findings is controversial. For example, several subsequent studies have shown that phenotypic plasticity can decelerate evolutionary rate [18,20,21], whereas others have claimed that it accelerates evolutionary rate [19,[22][23][24][25][26][27][28]. However, it is important to note that the studies [18,20] in which it was concluded that phenotypic plasticity leads to a deceleration of evolutionary rate dealt with uni-modal Gaussian fitness landscapes [29]. Recently, some studies pointed out that, under a multi-peaked fitness, phenotypic plasticity can smoothen fitness valleys, and thus, the escape from a local maximum can be enhanced, leading to the acceleration of evolution [26,[30][31][32]. In fact, the acceleration of evolution under a multi-peaked fitness landscape has been confirmed numerically [22,24,27].
The first analytical result addressing evolution under a multi-peaked fitness landscape in a fixed environment was provided by Borenstein et al. [26], in which they showed that phenotypic plasticity accelerates evolution. Instead of a mutation-selection process, they considered a single random walker on a multi-peaked fitness landscape, to mimic evolutionary population dynamics in genotypic space; therefore, their study did not distinguish between the behaviors of a population and an individual. Thus, in their random walker model the following three points were left unaddressed; first, average population fitness has not been obtained analytically; second, behaviors of the distribution of a population cannot be discussed in their model; and third, the relationship between the acceleration of evolution and average popula-tion fitness is unclear. These points should be clarified in order to fully comprehend the role of phenotypic fluctuation (plasticity) in evolutionary population dynamics.
In this study, we investigate evolutionary population dynamics, rather than the single random walker model, under multi-peaked landscapes in a fixed environment, and study the role of phenotypic fluctuation on the evolutionary rate. We provide analytical results for the relationship between phenotypic fluctuation, evolutionary rate, and average population fitness by considering the distribution of genotypes in a population. First, we assess whether a population located at a fitness peak can remain around the peak over generations to maintain high fitness. We show that for both large phenotypic fluctuations and high mutation rates, populations fail to keep descendants near the peak over multiple generations, and therefore, they are unable to maintain a high average fitness. Second, we evaluate the rate at which a population moves between fitness peaks and derive an analytical expression representing the dependency of evolutionary rate on phenotypic fluctuation. The obtained result indicates that phenotypic fluctuation always enhances diffusion in genotypic space, and thus, it accelerates evolution under multi-peaked fitness landscapes.
One consequence of our results is that for a large phenotypic fluctuation, the failure of a population to concentrate near a fitness peak results in a decrease in average fitness. This can be regarded as "error catastrophe." The concept of error catastrophe refers to the collapse of a population that maintains high fitness because of a high mutation rate [33,34]. Our results demonstrate that error catastrophe occurs not only because of a high mutation rate, but also because of large phenotypic fluctuations. Furthermore, our results illustrate an apparent relationship between error catastrophe and the Baldwin effect, in that error catastrophe is explained as an extreme case of a trade-off between an acceleration of evolutionary rate and a decrease in fitness. These findings reveal novel aspects of the role of phenotypic plasticity in evolution.
The organization of this paper is as follows. In Sec. II A, we explain how phenotypic fluctuation modifies fitness landscapes. In Sec. II B, we introduce our model of evolutionary population dynamics with phenotypic fluctuation and derive an equation for the time evolution of a population distribution in genotypic space. In Sec III A and Sec III B, we introduce a simple periodic fitness landscape as a multi-peaked fitness landscape. In Sec. III A, we investigate whether a population can keep its descendants around a peak of fitness, and can sustain high average fitness. In Sec. III B, we investigate how phenotype fluctuation affects evolutionary rate by using a transformation of the model equation into the Schrödinger equation and adopting WKB approximation. In Sec. III C, we investigate a quasi-periodic fitness landscape. In Sec. IV, we include concluding remarks.

A. Modulated fitness by phenotypic fluctuation
Let us begin with a discussion of how fitness landscapes are modified by phenotypic fluctuation (also see [18,20,30]). Note that the phenotypic fluctuation here is sometimes referred to the environmental variance in the context of population genetics and it may also be regarded as an example of phenotypic plasticity. Here, we analyze asexual population dynamics of a single-locus quantitative genetics model. Suppose that an individual organism has two types of a one-dimensional quantitative trait: a heritable trait of genotype g and a non-heritable trait of phenotype x. Genotype g is inherited from its parent, whereas phenotype x is given through a genotypephenotype mapping function p(x|g; Σ) with fluctuation magnitude Σ. Fitness is assigned to each phenotype x and thus is expressed as F (x), given that natural selection acts on phenotypes x rather than genotypes g. By calculating the average with respect to x, the effective fitness assigned to the genotype g is given as the following equation: where Ω is the domain of phenotype value x. For simplicity, p(x|g; Σ) is assumed to be a Gaussian distribution with mean g and variance Σ, and Ω is assumed to be Ω = (−∞, ∞); . ( From this simplification, Eq. 1 leads to dx. The above equation reveals that phenotypic fluctuations make a fitness landscape flatter, and its peaks, lower. Examples of such modulation effects are shown in Fig. 1(a) and Fig. 1(b): By flattening the fitness landscape, fitness peak values are decreased, and thus it acts as a cost of phenotypic fluctuation.

B. The Discrete Time Model and The Approximated Continuous Time Equation
Once F (x) is given, the effective fitness assigned to individual organisms with genotype g is obtained in Eq. (3). We here consider the evolutionary population dynamics of N individual organisms with nonoverlapping generations. Suppose that N t (g; Σ) is the frequency distribution of the population at t-th generation, and each individual can produce descendants that survive in the next generation at a rate f (g)/ f . Here, · represents an average over the entire population, (e.g., f ≡ Ω N t (g; Σ)f (g; Σ)dg/ N t (g; Σ)dg). Parental organisms are removed from the population for the next generation. Due to mutations, genotypes of t + 1-th generation g ′ deviate from the mother's genotype g as g ′ = g + ξ, where ξ is a random number drawn from a Gaussian distribution with a mean zero and variance D g . This D g corresponds to mutation rate. This effect of mutations on the distribution N (g) can be written using the mutation operator M g, By combining both effects of natural selection and mutations, whole evolutionary dynamics is expressed as With an assumption of sufficiently small D g , this discrete time equation (4) is approximated by a continuous time equation governed by non-linear Fokker-Planck-like equation: [35] ∂N (t,g;Σ) ∂t which is a function of continuous time. From this approximation, we can obtain time evolution of a set of moment of N (t, g; Σ); for instance III. RESULTS

A. Localized-extended transition in a periodic fitness landscape
Here we investigate the model under a simple multipeaked landscape, namely, a periodic one given as F (x) = 1 + cos αx. Hence, the effective fitness landscape is expressed as f (g) = 1 + exp(−α 2 Σ/2) cos αg from Eq. (3). First, we investigate whether the population initially located at a peak of fitness can stay around the peak over generations to maintain high fitness. To examine this, we assume that at the initial state t = 0 all individuals in a population are located at g = 0, and that the genotype distribution can be approximated by a Gaussian distribution with a mean value G(t) (= g ) and variance V g (t) (= g 2 − g 2 ) at any generation. This simplification allows us to calculate the time evolution of N (t, g; Σ) from Eq. (6) and Eq. (7), resulting in The average population fitness is thus obtained as At the initial state t = 0, we assume G(0) = 0 and V g (0) = 0. Thus, Eq. (8) leads to dG/dt = 0 (i.e., G(t) = 0) and The above equation has a fixed point solution when where V * g gives the minimum of dV g /dt as given by Here, W (·) is the Lambert W-function defined by the inverse function of f (W ) = W e W . A finite V g at equilibrium indicates that the population is localized around one of the peaks of fitness. When the inequality (11) is not satisfied, V g in Eq. (10) always increases in time, and thus, the variance of N (t, g; Σ) goes to infinity, leading to a broad distribution of N (t, g; Σ) in genotype space. This qualitative change in the variance V g can be referred to as the localized-extended transition. Simpler representation of the condition Eq. (11) is obtained by using the approximation 1 + exp[− ] ≃ 1, which is reasonable around a transition point because variance V g is sufficiently large near the transition point. It yields the condition for a finite variance at equilibrium, namely, the condition for the localized state as follows: To examine the validity of these arguments, we compare Eq. (11) and Eq. (13) with an individual-based simulation (i.e., direct numerical simulation of Eq. (4)). Figure 2 shows that both the estimates agree rather well with the results from the individual-based simulation.
It should be noted that the Gaussian approximation fails when the population is separated into two subgroups and localized at different peaks of fitness. Because of this failure, the variance of population estimated in the individual-based simulation does not coincide with the approximated value V eq g calculated as a solution of Eq. (10). To remedy this failure, we take account of only one peak with the largest subpopulation, as shown in Appendix B, in which the procedure to determine the boundary between the localized and extended phases from the individual-based simulation is also described.
The average population fitness is theoretically estimated as follows: by solving dV g /dt = 0 in Eq. (10) by using a quasi-Newton method, we compute V eq g , and then the average population fitness is obtained by Eq. (9). The estimated average fitness is shown in Fig. 3. We also estimate the average population fitness by using an alternative approximation method, a harmonic potential approximation, which is given below (see Eq. (20) and also Eq. (D4) in Appendix D). These results are shown in Fig. 3, and agree rather well with the results obtained from the individual-based simulation.
Both approximated estimations and results from the individual-based simulation show that in the extended phase, the average population fitness takes a value that is as low as that without selection pressure, namely, f = 1. Such a drop in a high fitness value can be interpreted as "error catastrophe", i.e., the population fails to sustain high fitness due to high mutation rate or large phenotypic fluctuations. In contrast, the population in the localized phase is concentrated around the peak position of the fitness, and thus can sustain a high average fitness.
With the present analysis based on the Gaussian approximation, the population dynamics in relatively short time scale are evaluated to examine whether the population is concentrated around the peak. For a longer time scale, population can jump through the fitness valleys, even in the localized phase as we show in the next section.

B. Escape rate analysis in periodic fitness landscape
Next we focus on the population dynamics of the localized phase. The population shows diffusion in genetic space, and maintains a high fitness value for longer time scale. Figure 4 Fig. 4(b). In the figure, [G 2 ] increases in proportion to t 1 , indicating that even in the localized phase the genotype in population performs random walks by jumping across a fitness valley. Through such random walks, a population can search for novel genotypes with a higher fitness value in genotype space without suffering from the error catastrophe. Thus, the diffusion coefficient of this random walk gives a measure for the rate of evolutionary innovations, in other words, evolvability. It should be noted that the deterministic diffusion in genotype space is expected to be observed for infinite population size, although the results from the individual based simulation show stochastic behavior due to finite population size effect.
The diffusion process in genotype space constitutes a sequence of jumps across fitness valleys. As Fig. 1(b) shows, phenotypic fluctuation tends to make fitness valleys shallower, and thus enhances the jump probability from a peak to a neighboring peak. Here, we analytically estimate the jump probability per unit time Γ defined as π/α −π/α N (t, g; Σ)dg ∼ e −2Γt , i.e., with initial condition N (t = 0, g = 0; Σ) = 1. Note that the jump probability from a peak of fitness to a neighboring peak per unit time is one half of the escape rate in which the jumps to both neighboring peaks have to be considered. With a separation ansatz N (t, g; Σ) = e λt φ(g; Σ), Eq. (5) becomes the following eigenvalue problem: where φ(g; Σ) is rewritten as φ(g) and D is defined as D = D g /2. By interpreting f as an external control parameter R = f , we obtain where Λ is given as After appropriate transformations of variables [36], Eq (16) is transformed into where If we use t sc = −i t and m sc = 2 /2d, this agrees with the familiar form of Schrödinger equation. The Schrödinger form Eq. (18) with WKB approximation [37,38] allows us to estimate the jump probability between a peak of fitness and another peak as an escape rate from a double well potential, assuming that the probability distribution is located only in the left well at t = 0, as is illustrated in Fig. 5. Details of the derivation are given in Appendix D. Now, we consider the same fitness landscape in Sec. III A: f (g) = 1 + χ cos αg and χ = exp(−α 2 Σ/2), where χ decreases as phenotypic fluctuation Σ increases. Using quasi equilibrium approximation (see details in Appendix D), the average population fitness R for this landscape can be derived as follows: This evaluation of average fitness R at the quasi equilibrium agrees rather well with the results from the individual-based simulation, as is shown in Fig. 3 (the thick line represents the evaluation in Eq. (20) ), indicating that the quasi equilibrium approximation is valid. Using WKB approximation and R estimated above, we can evaluate the jump probability per unit time Γ as Note that g = b is a point where the integrand becomes zero i.e., b satisfies The effective diffusion coefficient D e is given by The factor 2 in the above equation appears since the escape rate from a peak of fitness to the left neighbor peak has to be incorporated, as well as that to the right neighbor peak. Equations (21) and Eq. (24) describe the relation between phenotypic fluctuation Σ and evolvability D e . To examine the validity of the estimation of the escape rate 2Γ in Eq (21), we compare Eq. (21) with that obtained numerically by the individual-based simulation of Eq. (4) (details of the estimation of the escape rate from the individual-based simulation are explained in Appendix C). As shown in Fig. 6, the two agree rather well. These results shown in Fig. 6 indicate that phenotypic fluctuation always accelerates the evolutionary rate, while large phenotype fluctuation decreases the average population fitness, as is shown in Fig. 3. Therefore, the evolvability and fitness values are in a trade-off relationship.

C. Mixed cosine potential
To demonstrate that our analysis is not limited to the simple periodic potential, we consider the quasi-periodic fitness landscape; F (x) = A 1 (1 + cos α 1 x) + A 2 (1 + cos α 2 x). From Eq. (1), the effective fitness landscape can be calculated as  +D g (26) Nullcline analysis of Eq. (26) is given in Fig. 8(a) -(c), against the three values of Σ: for small Σ (Fig. 8(a)), a population can be localized at around each local maximum of fitness; for middle value of Σ ( Fig. 8(b)), a population can be localized only at around a larger local maximum of fitness; for large Σ (Fig. 8(c)), all fixed points disappear, and V g diverges, which indicates the

(i) Localized-localized transition
For the simplicity of discussion, we assume α 1 > α 2 in the following argument. Here, we define the localizedlocalized transition as the transition where a population cannot be localized at around the lowest local maximum of V (x). Thus, we investigate whether a population can be localized around G lm , which satisfies In other words, we determine the conditions on which V g has finite value for t → ∞ for a population at G = G lm from the following equation: Solving the above equation with an initial condition V g (0) = 0 by numerical integration, we determine the transition point where V g diverges. In Fig. 9, the blue line shows the localized-localized transition point obtained by the numerical integration. Intuitive estimation is provided by assuming that the critical value of V g isṼ g ∼ π 2α1 2 , which corresponds to the square of the width of a smaller peak. From this assumption, the transition point can be evaluated by .
(30) The above estimate agrees well with the results of the numerical integration of Eq. (29) (see Fig. 9).

(ii) Localized-extended transition
To investigate the localized-extended transition, we consider the behaviors of the population distribution located at the global maximum G = 0. We assume that, near the transition point, V g is large enough to consider e −α 2 1 (Σ+Vg )/2 as negligible, but small enough to interpret e −α 2 2 (Σ+Vg )/2 as a finite value. From this assumption, Eq. (26) can be rewritten as Here, the same procedures to derive Eq. (11) and Eq. (13) are applied. In Fig. 9, we show the estimated localizedextended transition point.

D. General Rugged Landscape
Next, we confirm our scenario that phenotypic fluctuation accelerates evolution for a general rugged fitness landscape numerically. Here, we consider a landscape F (x) = 1 + M k α k M j αj cos 2πk L x + 2πL k φ k , where α k and φ k are uniform random number ranging over [0,1]. Although this landscape is periodic with period L, it can be regarded as a random landscape within length L. From Eq. (1), the effective fitness landscape can be calculated as (32) Figure 10-(a) shows f (g, Σ) for M = 10, L = 20. Using this effective fitness landscape, we perform the individual-based simulation. As shown in Fig. 10-(b), the time course of mean square displacement, [G 2 ], of the mean genotypic value G of each generation indicates that diffusion constant in genotype space increases monotonically with Σ. This demonstrates that phenotypic fluctuation enhances the evolutionary rate (For some case of F (x), the diffusion shows non-monotonic change against Σ for small values of Σ (data not shown), but for large Σ, it shows monotonic increase with Σ).

IV. DISCUSSION
In the present study, the evolutionary population dynamics of a simple quantitative genetics model under a multi-peaked fitness landscape is investigated by focusing on the role of phenotypic fluctuation on the evolutionary process. The main results of this study are summarized in the following points (i)-(iv): (i) The relationship between average fitness and phenotypic fluctuation. Analytical expression is obtained either by Gaussian approximation of a population distribution or by a harmonic potential approximation. By the Gaussian approximation, average fitness is given by Eq. (9) with a solution of ODE Eq. (10) that requires numerical computation, while by the harmonic potential approximation, the average fitness is analytically estimated as Eq. (20) (or Eq. (D4)). As is shown in Fig. 3, both approximations agree rather well with the results from the individual-based simulation of evolutionary population dynamics described by Eq. (4) and indicate that average population fitness decreases as the magnitude of phenotypic fluctuation Σ increases.
(ii) Error catastrophe due to large phenotypic fluctuation. Because of either a high mutation rate or large phenotypic fluctuation, a population fails to keep its descendants around a peak of fitness, and thus is unable to sustain a high-fitness phenotype. Such a decrease in fitness is known as "error catastrophe." Although the error catastrophe usually refers to the failure in main-taining the population due to a high mutation rate, our result demonstrates that the error catastrophe can also occur as a result of phenotypic fluctuation. In our model, phenotypic fluctuation always enhances the risk of error catastrophe, whereas an interesting exception to this has been reported in which phenotypic fluctuation can suppress the error catastrophe under a step-like fitness landscape in a high-dimensional genotype space [39].
(iii) Dependence of evolutionary rate on the magnitude of phenotypic fluctuation. We provide an analytical expression of the rate at which a population jumps from one peak to another peak in the multi-peaked fitness landscape, which is expressed as the evolutionary rate. The obtained expression Eq. (21) shows that phenotypic plasticity generally enhances evolutionary rate. Such an acceleration has been noted previously [22,24,26,27,30]; however, our results provide the first analytical expression including both the effects of natural selection and mutation. We also confirm numerically that the phenotypic fluctuation accelerates evolution for a general rugged fitness landscape, and thus the present conclusion is not limited to a periodic fitness landscape but is expected to be valid for general landscapes, such as Fig. 10-(a).
Our claim of the acceleration of evolution by phenotypic fluctuation is different from Fisher's fundamental theorem [40], which claims that genotypic variance is proportional to evolutionary rate. In Fisher's theorem, the phenotypic plasticity or fluctuations are not considered, and the effect of mutation is also ignored, whereas in our results, genetic variance, phenotypic effect, and the effect of mutation are incorporated. A possible relationship between the phenotypic fluctuation and genotypic variance was also discussed in [5].
(iv) Phenotypic fluctuation exhibits trade-off relationships with the acceleration of evolution and degradation of fitness. This conclusion is reached by combining the results (i), (ii), and (iii). This trade-off relationship is illustrated by both Fig. 3 and Fig. 6, in which greater phenotypic fluctuation results in a smaller fitness value and a larger evolutionary rate. In addition, the transition to the extended phase (the error catastrophe) for larger phenotypic fluctuation can be interpreted as an extreme situation of this trade-off. It is well established that an increase in mutation rate leads to the same tradeoff relationship between the evolutionary rate and degradation of fitness, and that error catastrophe occurs as an extreme case of the trade-off. Our results indicate that phenotypic fluctuation introduces an additional mutation rate, although the origin of the phenotypic fluctuation is essentially different from that of mutation. Note that this "additional mutation rate" effect of phenotypic fluctuation works only in case of multi-peaked fitness landscape, but not in case of a unimodal fitness landscape [18,20,21].
Throughout the present study, we interpret random phenotype fluctuation as a kind of phenotypic plasticity (i.e., the phenotypic distribution of a single genotype is independent of environment and fitness (see Eq. (2))). Several previous studies addressing the Baldwin effect also adopted random fluctuation as phenotypic plasticity [18,20,26,30]. Our results indicate that even random phenotypic fluctuation accelerates the evolutionary rate under a multi-peaked fitness landscape. Another interpretation of the phenotypic plasticity is that phenotype distribution of a given genotype depends on the fitness landscape, referred to as responsive plasticity [19,22,23,[25][26][27]. The difference between phenotypic fluctuation and responsive plasticity is only in the modulation of a fitness landscape by phenotypic changes, and therefore, the techniques that we adopted in the present study (e.g., the transformation of Fokker-Plancklike equation into the Schrödinger equation, WKB approximation) are applicable after a modulated fitness is obtained; these techniques are generally useful for investigations of evolutionary dynamics [39,[41][42][43].
In our model, we assume that both genotype and phenotype are represented as a one-dimensional continuous value and the mapping between them is straightforward (assumptions often employed in quantitative genetics [18,20,44]). Because such simplicity could miss potential importance of high dimensionality and complexity in real genotype-phenotype mapping, we give brief remarks here. A high dimensional binary genotype model with a simple fitness landscape was adopted in some old studies [19,23], especially the work by Hinton and Nowlan [19] demonstrated the validity of Baldwin effect in the model. Recently, relevance of phenotypic fluctuation (noise in developmental process) to mutational robustness has been studied by models with high-dimensional genotype space and with non-trivial genotype-phenotype mapping [45][46][47]. Although these studies did not intend to validate the Baldwin effect, some of their results share a mechanism of the acceleration effect of evolution; their conclusion suggested that smoothening fitness landscape by phenotypic fluctuation prevents from trapping in local maximum of fitness, which is also important for the Baldwin effect. For this reason, it is reasonable to expect that the Baldwin effect is valid even for the case with high dimensional genotype space under a rugged fitness landscape. To confirm this statement, however, further studies are required.
In this paper we study an infinite population model in which population is always genetically polymorphic and there is no fixation event of a single genotype. It will be important for future research to extend our study to include cases with finite populations, in which both the genetic drift and phenotypic fluctuations have to be taken into account seriously.
So far, mounting experimental evidence supports that fitness landscapes in real organisms are multi-peaked [48][49][50]. In addition, it has also been shown that stochasticity and random fluctuations in phenotypes are common in living systems [3][4][5][6]. In the presence of such fluctuation and multi-peaked fitness landscape, our analytical study has demonstrated the relationship between phenotypic fluctuation and evolutionary rate, as well as the relationship between the Baldwin effect and error catastrophe. Therefore, our findings have broad implications for the study of evolutionary dynamics.