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One can always find a trade-off shape that keeps the [MATH] constant among strategies; for the case of reproductive strategies, this condition is met when litter size decreases linearly as adult survival increases – the linear trade-off described above. This is an ideal setup to identify other players in the evolutiona... |
, for instance, used it to demonstrate that environmental variability may select for iteroparity in a density-dependent environment. |
We first use this setup and study the impact of neutral processes on the evolutionary dynamics of two classes of life-history strategies associated with generation time: reproductive strategies and development duration. We confirm that the strategies considered, although they yield different generation times, are neutr... |
. This defines the conditions where turnover bias should be an important determinant of the evolution of life-history strategies. |
Results Strategies with different generation times and equal [MATH] are neutral We first consider a monomorphic population living in a stable environment with limited resources. The population is constituted by [MATH] juveniles and [MATH] adults, whose dynamics can be described by the discrete system ( ): |
[EQUATION] where [MATH] is the survival of juveniles becoming adults, [MATH] the lifetime fecundity, and [MATH] a density-dependent egg (or newborn) survival. There are two ways of changing the generation time in this model: by increasing adult longevity – by increasing the probability that an adult survives until the ... |
Model 1. The reproduction strategy is described by the continuous variable [MATH] , the survival of adults from one reproductive event to the next. Adults with [MATH] are semelparous: they reproduce once and die. Iteroparity arises as soon as [MATH] is above [MATH] , and the mean number of reproductive events increases... |
Each egg or newborn produced survives with the density-dependent probability [MATH] , which decreases as the overall number produced, [MATH] , increases. We do not assign a specific density-dependence function to [MATH] at this point; Instead, we assume that this function eventually yields a non-zero equilibrium where ... |
[EQUATION] for model 1 – the system ( ) corresponds to ( ) with [MATH] . By solving the system, we show that at equilibrium the product of the lifetime potential fecundity [MATH] and of the density-dependent egg survival [MATH] equals [MATH] . This property also emerges when [MATH] is formulated explicitly (SI text 1 a... |
We now model the evolutionary dynamics of the reproduction strategy by considering the fate of a single mutant with strategy [MATH] appearing in a monomorphic population (with strategy [MATH] ) whose dynamics are described by the system ( ). We use here the adaptive dynamics framework, which commonly assumes that mutat... |
. As we have seen, at this equilibrium the resident’s lifetime fecundity [MATH] equals [MATH] . Another classical assumption of adaptive dynamics is that the resident population is large enough for the mutant to be negligibly rare at the beginning of the invasion. The density-dependent parameter [MATH] thus only depend... |
[EQUATION] where [MATH] is the number of surviving eggs laid by the mutant at time [MATH] . In the appendix, we show that all mutants have equal growth rates – regardless of the resident they compete with – such that no reproduction strategy should increase (or decrease) in frequency in response to selection. Reproduct... |
Model 2. The generation time can also be impacted by the duration of development, which increases as [MATH] increases in equation ( ). With [MATH] , the dynamics of the mutant – with strategy [MATH] – is described by the following system: |
[EQUATION] Here again, we find that the mutant’s growth rate is insensitive to the mutant and the resident strategies (see appendix). This confirms that strategies with equal [MATH] s but different generation times are neutral. In the following section, we study their neutral evolutionary dynamics. |
The neutral evolution of slow/fast strategies When changes in genotype frequencies are not – or little – governed by selection, neutral processes like genetic drift may affect evolution in finite populations. Here we study the neutral evolutionary dynamics of strategies with different generation times using a Markov ch... |
Reproduction strategies. As before, the number of offspring produced by an adult with reproduction strategy [MATH] at any time step equals [MATH] , and each survives with probability [MATH] . With probability [MATH] , an offspring carries a mutation that changes its reproduction strategy. Here again, we assume that [MA... |
We further assume that mutations produce a small change in the reproduction strategy, such that a population with strategy [MATH] can only mutate to the immediately lower [MATH] or higher [MATH] . The probability of transition to either of these states equals: |
[EQUATION] and the probability of remaining in state [MATH] equals: [EQUATION] For the first and last of the [MATH] states in the Markov chain, only the probabilities of transition [MATH] and [MATH] can be calculated with equation ( ), and the probabilities of remaining in those states are [MATH] and [MATH] , respectiv... |
The evolutionary dynamics of the reproduction strategy can thus be described by the Markov chain represented in figure . Using equations ( ) and ( ), one can create a Markov chain of any length by dividing the range of possible reproduction strategies into a given number of adjacent values of [MATH] . We used the set [... |
At equilibrium, the most likely reproduction strategy has the highest level of iteroparity ( [MATH] ; Fig. ). Semelparity ( i.e. |
[MATH] ) is, among all possible reproductive strategies, the least likely to be observed: for example it would be observed less than [MATH] of the time on a very long time series where many neutral mutations would have fixed, or in less than [MATH] populations in [MATH] at equilibrium. It should be noted that interpret... |
[MATH] , or more than [MATH] reproductive events on average – occur about [MATH] of the time. Turnover bias. Slow strategies – e.g. high degrees of iteroparity – are more likely to evolve neutrally as a result of a bias that we call ‘turnover bias’. For instance, a population with reproduction strategy [MATH] produces ... |
. Therefore, a population that switches from state [MATH] to state [MATH] is less likely to switch back to [MATH] than a population switching from [MATH] to [MATH] (see fig. ). This yields a small bias towards increasing the degree of iteroparity at each step, which does not stop until the highest degree of iteroparity... |
Duration of development. The neutral evolutionary dynamics of reproduction strategies is easily understood when considering how the Adults class is filled at each timestep: a proportion [MATH] remain through adult survival, while a proportion [MATH] are newly made through reproduction. Only the latter mutate (but see t... |
Turnover bias in polymorphic populations The models described in the two sections above rely on the assumption that mutations are rare, so that the population is only transiently polymorphic. Moreover, in the neutral Markov chain, mutants can only reach neighboring reproduction strategies, even though the degree of ite... |
At each timestep, a juvenile with genotype [MATH] can either remain a juvenile (with probability [MATH] ) or attempt to become an adult (probability [MATH] ); then it survives with probability [MATH] . An adult with genotype [MATH] survives with probability [MATH] at each time step and produces [MATH] eggs on average. ... |
We ran [MATH] replicate simulations of the evolution of [MATH] and [MATH] with [MATH] [MATH] and [MATH] . The simulations were initiated with a single genotype with [MATH] or [MATH] (fig. [MATH] ). Both traits have very similar evolutionary dynamics: initially, the replicate populations diverge and exhibit a large rang... |
Genotypes with a slow turnover evolve under a wide range of parameter values, including even higher mutation rates, lower fecundity and different initial values of [MATH] or [MATH] (see SI text 2 and fig. S5). We also studied another form of density-dependency and obtained very similar results to those presented in fig... |
It is worth noting that the population size is not strictly constant in this model: the density dependent process can yield population sizes above or below its expected equilibrium. This may impact the resulting evolutionary dynamics, because fast strategies perform better when the whole population growth rate is above... |
Age-dependent mutation rate Above we considered a constant mutation rate among the various litters produced by an individual. By doing so, we neglected the well-known fact that, in many species, the mutation rate can increase with one or both parents’ age |
. This increase is likely due to a large extent to an increase in the number of germline cell divisions . This raises two distinct issues: that iteroparous individuals may be biased towards producing more mutants later in life, and that maybe this yields an increase in the overall mutation rate as the degree of iteropa... |
The second issue is more problematic and needs to be resolved here: an increase in the mutation rate with the degree of iteroparity could, in theory, revert the neutral dynamics and make semelparity the most likely outcome under the linear trade-off. It is trivial, from equations ( ) and ( ), that a mutation rate [MATH... |
We assume that a batch of gametes is produced each time an individual reproduces, of a size proportional to the number of offspring it is expected to yield at this reproductive season (SI text 3). The overall number of gametes produced is thus constant among strategies, yet our model shows that producing them at once –... |
How may this phenomenon affect the neutral evolution of reproduction strategies? In the previous sections, we showed that highly iteroparous strategies were scarcely replaced due to their long generation times. A higher mean mutation rate of iteroparous individuals might (over)compensate the bias and favor the evolutio... |
Turnover bias vs. selection It is legitimate to ask whether turnover bias can have an impact on the evolutionary dynamics of generation time in the presence of selection. Consider first a two-allele haploid model where the two alleles yield the two reproduction strategies [MATH] and [MATH] [MATH] ). Turnover bias favor... |
[EQUATION] such that it is equally probable to evolve to [MATH] from a population where [MATH] is fixed than the reverse. Assuming that the fitness advantage of [MATH] over [MATH] equals [MATH] (and the fitness disadvantage of [MATH] |
[MATH] ), we can calculate [EQUATION] The equilibrium in equation is thus obtained for: [EQUATION] [MATH] increases as the difference between [MATH] and [MATH] increases – i.e. when turnover bias is more acute – and when the population size decreases and selection consequently becomes less efficient. For instance, in a... |
Now consider, as we did in previous sections, that reproduction strategies can vary continuously , distributed along a trade-off between litter size and longevity. The shape of the trade-off determines the selective value of any strategy competing with others. A convex trade-off appropriately sets a selection gradient ... |
The evolutionarily expected reproduction strategies are represented in fig. . In large populations ( [MATH] ), semelparity evolves at very low values of [MATH] – as soon as the trade-off becomes non-linear in fig. . In smaller populations, the level of iteroparity [MATH] decreases with [MATH] , at a slower rate when th... |
vs [MATH] ). This result is typical of evolutionary dynamics where neutral and selective processes act in opposite directions: selection is most efficient in large populations – favoring the evolution of semelparity in the present case – such that turnover bias has a small effect on the evolutionary dynamics in this si... |
Discussion Life-history strategies that influence the generation time but not the lifetime reproductive success are known to be neutral in a density-dependent environment yielding stable population dynamics |
, suggesting a minor role of the generation time in their evolution. Here we show that a bias favors the evolution of the slowest possible life-history strategy – that with the longest generation time. Indeed, individuals adopting (fast) strategies with short generation times produce many offspring per time unit (and t... |
Turnover bias – as we called it – is an unescapable part of the evolutionary dynamics of strategies with different generation times. This bias impacts the occurrence of mutations but not their fixation; we actually detected it by building an “origin-fixation model” (OFM) that separates the two processes. OFMs are rarel... |
. To our knowledge, only Proulx and Day have built an OFM to study the evolution of a trait associated with the generation time. They have considered differences in fixation probabilities among strategies, but not in their rates of mutation appearance per time unit, rendering it impossible to detect the bias. Interesti... |
An expected feature of biases of the mutational process is that their contribution to evolutionary dynamics is maximum when alternative alleles have similar fixation probabilities, that is when allele fixation is mainly due to neutral processes like genetic drift |
. Our results indeed indicate that turnover bias impacts the evolutionary outcome more strongly when the selective values of the strategies in competition are close and/or when the population is small. When selection strongly favors the fixation of one specific strategy (or of a range thereof), turnover bias is still a... |
), while its role at the population level may be contingent on specific population parameters. The importance of turnover bias in nature thus relies on how often selection is inefficient among strategies with different turnovers. It might be a very special and rare situation, but the factual truth is that we lack empir... |
– evolutionary epidemiologists have studied the virulence-persistence trade-off using mechanistic models , but we are not aware of their application to other trade-offs. |
We have ignored other, potentially important selection pressures that may contribute to the evolutionary dynamics of strategies associated with the generation time. Bet-hedging strategies, for instance, evolve as a response to environmental fluctuations and are often characterized by longer generation times than those ... |
. This prediction contrast with a recent study by Mitteldorf and Martins showing that a fluctuating environment can select for short generation times, by creating repeated episodes of directional selection. This gives a selective advantage to genotypes adapting quickly, which can be achieved by fast strategies as they ... |
. Purging selection in presence of these mutations would likely favor genotypes producing fewer mutants. This might create a selection pressure for slow strategies, even in stable environments. |
We have not considered the genetic architecture of the evolving traits, as it cannot change turnover bias directly. Turnover bias results from traits impacting the generation time, regardless of their genetic, physiological or developmental components. Nonetheless, the genetic architecture may play a (distinct) major r... |
but they are rarely considered for phenotypic traits (but see ). Mutation biases are similar to turnover bias in the sense that they affect the origin of mutations but not their fixation. |
Overall, evolutionary ecology typicallly focused on the complex interactions of organisms with their biotic and abiotic environment and the selection pressure that results, leaving the study of neutral processes and biases in the production of mutants to theoretical population genetics. Our hybrid model makes novel pre... |
Material and methods Simulation procedure We simulated a population of a variable number [MATH] of genotypes with different reproduction strategies [MATH] . Each genotype [MATH] is represented by [MATH] juveniles and [MATH] adults. At a given timestep, adults with genotype [MATH] produce a number of eggs [MATH] sampled... |
Each offspring of genotype [MATH] can mutate with probability [MATH] . When this event occurs, the number of genotypes [MATH] is incremented by [MATH] and the new genotype [MATH] has [MATH] [MATH] , and [MATH] [MATH] is sampled from a normal distribution with mean [MATH] and standard deviation [MATH] ; mutants with [MA... |
After reproduction, the number of surviving adults of genotype [MATH] is sampled from a binomial with [MATH] trials and probability [MATH] . These will constitute the adults with genotype [MATH] at the next timestep, together with the offspring produced at the previous timestep that survive, whose number is sampled fro... |
In each replicate simulation, we simulated the process described above during [MATH] timesteps, and recorded [MATH] [MATH] and [MATH] for each of the [MATH] genotypes present in the population. We ran [MATH] replicate simulations for each parameter set explored; the distribution of the mean values of [MATH] in the [MAT... |
Appendix: mutant’s asymptotic growth rate Initially, the mutant is a juvenile so [MATH] and [MATH] . The mutant population may grow from this point, and its ability to do so is given by the Lyapunov exponent, or invasion fitness, denoted [MATH] |
[EQUATION] where [MATH] Because [MATH] is equal for all residents, the invasion dynamics of the mutant are independent of the resident’s strategy – either [MATH] for the model 1 or [MATH] for model 2 (see system ). For model 1, the asymptotic growth rate of the mutant can be calculated as the first eigenvalue of the ma... |
[EQUATION] The mutant’s asymptotic growth rate is calculated as the largest eigenvalue of [MATH] (defined in eq. ), which can be obtained by solving [MATH] . This yields the characteristic polynomial: |
[EQUATION] which simplifies to: [EQUATION] Hence the polynomial has roots [MATH] and [MATH] [MATH] for [MATH] so the asymptotic growth rate of the mutant equals [MATH] |
The transition matrix for model 2 can be written: [EQUATION] Solving [MATH] , we obtain the characteristic polynomial: [EQUATION] |
which mirrors equation ( 14 ) above: here, too, the mutant’s asymptotic growth rate equals [MATH] , regardless of its strategy. Figures |
Supplementary information SI text 1: Density-dependent population dynamics Here we study the population dynamics described by the system in the main paper, replacing [MATH] by two density-dependency functions. |
Density-dependency function 1: [MATH] Here the population is monomorphic, so the number of eggs produced at a given timestep, [MATH] equals [MATH] . We simulated the system with [MATH] and [MATH] , starting with [MATH] and [MATH] . As shown in fig. S1 , this function yields a stable dynamic equilibrium when [MATH] , wi... |
We ran simulations in the stable regime ( [MATH] [MATH] ) for [MATH] generations with [MATH] [MATH] and [MATH] (fig. S2 ). The population equilibrium is indeed stable and quickly reached. At the equilibrium, the density-dependent lifetime fecundity [MATH] equals [MATH] , as expected from the more general theoretical ap... |
Density-dependency function 2: [MATH] This function yields a sigmoidal relationship between [MATH] and the number of eggs produced, [MATH] – as before, the population is monomorphic at this point so [MATH] . Two parameters control the shape of the function: [MATH] is the number of eggs at which [MATH] , and [MATH] cont... |
We also ran simulations in the stable regime with this function ( [MATH] [MATH] [MATH] ) for [MATH] generations with [MATH] [MATH] and [MATH] (fig. S4 ). At the equilibrium, the density-dependent lifetime fecundity [MATH] equals [MATH] , as with the first function above and as we show should generally be expected in re... |
SI text 2: Evolutionary dynamics Density-dependency function 1: [MATH] We simulated the evolution of [MATH] in [MATH] replicate populations, as described in the Material and methods section of the main paper. We used different parameter sets, showing that neither the mutations rate [MATH] , the potential fecundity [MAT... |
Density-dependency function 2: [MATH] In this section, the function for the density-dependent is replaced by [MATH] . Otherwise the simulation procedure is exactly the same as that described in the Material and methods section in the main text and in the previous section. Changing [MATH] does not impact the results: in... |
Constant population size Our aim here is to maintain the size of the population strictly constant to a value [MATH] . We sample the number of surviving adults of each genotype (subscript [MATH] ) from a binomial with probability [MATH] , and the number of juveniles with genotype [MATH] that become adults from a binomia... |
[EQUATION] The evolutionary dynamics obtained with a constant population size are show in fig. S7 SI text 3: Age-dependent mutation rate |
In this section, we detail the calculations of the age-specific mutation rate, and of the resulting average mutation rate per gamete, when gametes need to be produced on several occasions in a lifetime. Since gametes are typically short-lived, we will consider that a new batch of gametes needs to be produced at each re... |
A critical number of cells in the germline needs to be reached before the first gametes can be produced through meiosis. Here we assume that half the germline cells are used to produce a batch of gametes, the others being kept for future production – this is true in man, for which each primordial (Ad) spermatogonia div... |
. Thus the number of germline cells needs to reach a critical threshold equal to twice the number of cells required for a batch (see fig. S9 ). Then each germline cell destined to immediate production yields [MATH] gametes through meiosis. Thus, from the number of gametes in a batch ( [MATH] ), one can obtain the numbe... |
[EQUATION] where [MATH] is the number of germline cells used to produce the first batch of gametes. Substituting for [MATH] , we obtain [MATH] |
Age-distribution of the mutation rate. The first batch is produced from cells mutated [MATH] times. They will experience another division with DNA replication – and mutation – during meiosis I, and another division without DNA replication (meiosis II), so the mutation rate of batch [MATH] [MATH] , equals [MATH] times t... |
Then one division needs to occur each time another batch is produced, such that the mutation rate of batch [MATH] [MATH] ) equals |
[EQUATION] The last batch has the same mutation rate as the previous one because the remaining germline cells are used, with no further DNA replication required, so that [MATH] |
Mean mutation rate. Equation ( S19 ) gives the mutation rate in each batch, so we can now calculate the mean probability of mutation across all gametes produced continuously: |
[EQUATION] Semelparous populations produce a single batch, such that [MATH] . Interestingly, producing two batches instead of a single one gives the exact same result, because the two batches are produced simultaneously, even though one is going to be used before the other. Once [MATH] , the mean mutation rate for game... |
# Source: arxiv 1504.08117 # Title: Average Convergence Rate of Evolutionary Algorithms # Sections: all # Downloaded: 2026-03-03T01:57:25.097218+00:00 |
Average Convergence Rate of Evolutionary Algorithms Abstract In evolutionary optimization, it is important to understand how fast evolutionary algorithms converge to the optimum per generation, or their convergence rates. This paper proposes a new measure of the convergence rate, called the average convergence rate. It... |
Index Terms: evolutionary algorithms, evolutionary optimization, convergence rate, Markov chain, matrix analysis Introduction Evolutionary algorithms (EAs) belong to iterative methods. As iterative methods, a fundamental question is their convergence rates: how fast does an EA converge to the optimum per generation? Ac... |
schmitt2001importance and it is found that the convergence rate is determined by the second largest eigenvalue of the transition matrix. The other category is based on Doeblin’s condition. The upper bound on the convergence rate is derived using Deoblin’s condition in he1999convergence As to continuous optimization, th... |
The convergence rate in previous studies ming2006convergence suzuki1995markov he1999convergence schmitt2001importance is based on Markov chain theory. Suppose that an EA is modeled by a finite Markov chain with a transition matrix [MATH] , in which a state is a population he2003towards . Let [MATH] be the probability d... |
he1999convergence . The goal is to obtain a bound [MATH] such that [MATH] . But to obtain a closed form of [MATH] often is difficult in both theory and practice. |
The current paper aims to seek a convergence rate satisfying two requirements: it is easy to calculate the convergence rate in practice while it is possible to make a rigorous analysis in theory. Inspired from conventional iterative methods varga2009matrix , a new measure of the convergence rate, called the average con... |
II Definition and Calculation Consider the problem of minimizing (or maximizing) a function [MATH] . An EA for solving the problem is regarded as an iterative procedure (Algorithm ): initially construct a population of solutions [MATH] ; then generate a sequence of populations [MATH] [MATH] [MATH] and so on. This proce... |
Algorithm 1 An EA with an archive 1: initialize a population of solutions [MATH] and set [MATH] 2: an archive records the best solution in [MATH] |
3: while the archive doesn’t include an optimal solution do 4: generate a new population of solutions [MATH] 5: update the archive if a better solution is generated; |
6: [MATH] 7: end while The fitness of population [MATH] is defined by the best fitness value among its individuals, denoted by [MATH] . Since [MATH] is a random variable, we consider its mean value [MATH] Let [MATH] denote the optimal fitness. The fitness difference between [MATH] and [MATH] is [MATH] The convergence r... |
[EQUATION] Since [MATH] , calculating the above ratio is unstable in practice. Thus a new average convergence rate for EAs is proposed in the current paper. |
Definition 1 Given an initial population [MATH] , the average (geometric) convergence rate of an EA for [MATH] generations is [EQUATION] |
If [MATH] , let [MATH] . For the sake of simplicity, [MATH] is short for [MATH] The rate represents a normalized geometric mean of the reduction ratio of the fitness difference per generation. The larger the convergence rate, the faster the convergence. The rate takes the maximal value of 1 at [MATH] |
Inspired from conventional iterative methods varga2009matrix , Definition 3.1] , the average (logarithmic) convergence rate is defined as follows: |
[EQUATION] Formula ( ) is not adopted since its value is [MATH] at [MATH] . But in most cases, average geometric and logarithmic convergence rates are almost the same. Let [MATH] . Usually [MATH] and [MATH] , then [MATH] |
In practice, the average convergence rate is calculated as follows: given [MATH] with [MATH] known in advance, 1. Run an EA for [MATH] times ( [MATH] ). |
2. Then calculate the mean fitness value [MATH] as follows, [EQUATION] where [MATH] denotes the fitness [MATH] at the [MATH] th run. The law of large numbers guarantees ( ) approximating to the mean fitness value [MATH] when [MATH] tends towards [MATH] |
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