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Ecolog y Letters. 2023;26:540–5 48.wileyonlinel ibrary.com/journal/ele
INTRODUCTION
Individuals in any population vary in their life courses, ex-
emplified by differences in lifespan, reproduction and phe-
notypic characteristics (Endler,1986; Hartl & Clark,2007;
Steiner & Tuljapurkar, 2012; Tuljapurkar et al., 2009).
Classical evolutionary theories, founded in seminal work
by Fisher (1930), Wright (1931) and Haldane (1927, 1932),
explain such variation by genotypic variation, environ-
mental variation or their interaction. According to these
theories, if environments are constant over many genera-
tions, selection should erode genotypic variation by select-
ing for very few adaptive phenotypes and their associated
genotypes; in population genetic terms, additive genetic
variation should erode. However, neutral molecular vari-
ation maintains some genetic diversity without substan-
tial phenotypic variation, if the phenotypes are selected
(Crow & Kimura, 1970; Kimura, 1968). In consequence,
in a constant environment, individual variation in phe-
notypic characteristics and life courses should decline
if phenotypes are linked to fitness and trade- offs among
life- history traits do not balance each other and thereby
maintain phenotypic variation. These predictions are chal-
lenged by the observation that even isogenic individuals,
originating from parental populations that have lived for
many generations in highly controlled lab conditions, ex-
hibit high levels of variation among individual life courses
and phenotypes, even for phenotypes that directly link to
fitness and that are under selection (Flatt, 2020; Jouvet
et al.,2018; Steiner et al.,2019). Similarly, in less controlled
genetic and environmental conditions, environmental
variation, genotypic variation and their interaction only
account for a small fraction of the total observed pheno-
typic variation in fitness components (Snyder et al.,2 021;
Snyder & Ellner,2 018; Steiner et al.,2021; van Daalen &
Caswell, 2020). For systems where such a decomposition
of genotypic, environmental and other stochastic variation
is challenging because of a lack of accurate data, similar
amounts of total phenotypic variation are observed as in
more controlled systems (Finch & Kirkwood,2000; Snyder
& Ellner,2016; van Daalen & Caswell,2020). The question
arises, how can such high levels of phenotypic variation
be maintained, knowing that basic evolutionary theories
do not readily predict the persistence of such high levels
LETTER
Adaption, neutrality and life- course diversity
Ulrich KarlSteiner1 | ShripadTuljapurkar2
Received: 19 Septembe r 2022
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Revis ed: 16 January 2023
|
Acce pted: 18 Janu ary 2023
DOI: 10.1111/ele.14174
1Institute of Biology, Freie Un iversität
Berlin, Ber lin, G ermany
2Depa rtme nt of Biolog y, Stanford
University, Stan ford, Ca lifor nia, USA
Correspondence
Ulr ich Karl Stein er, Institute of Biology,
Freie Universit ät Berl in, Königin- Luise Str.
1– 3. 14195 Berli n, Germany.
Emai l: ulrich.steiner@fu-berlin.de
Funding information
Deutsche Forschungsgemeinschaft, Grant/
Award Numb er: 430170797
Editor: Robin Snyder
Abstract
Heterogeneity among individuals in fitness components is what selection acts
upon. Evolutionary theories predict that selection in constant environments acts
against such heterogeneity. But observations reveal substantial non- genetic and
also non- environmental variability in phenotypes. Here, we examine whether there
is a relationship between selection pressure and phenotypic variability by analysing
structured population models based on data from a large and diverse set of species.
Our findings suggest that non- genetic, non- environmental variation is in general
neither truly neutral, selected for, nor selected against. We find much variations
among species and populations within species, with mean patterns suggesting
nearly neutral evolution of life- course variability. Populations that show greater
diversity of life courses do not show, in general, increased or decreased population
growth rates. Our analysis suggests we are only at the beginning of understanding
the evolution and maintenance of non- genetic non- environmental variation.
KEYWOR DS
COMADRE, COMPADRE, demographic stochasticity, indiv idual heterogeneity, li fe- history
evolution, matrix population models, phenotypic variance, sensitivity
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STEINER and TULJAPURKAR
of variability (Barton et al., 2017; Bell, 2010; Flatt, 2020;
Melbourne & Hastings,2008). From an empirical point of
view, estimates of heritability of functional traits and re-
sulting expectations of trait shifts frequently do not match
observed fluctuations in phenotypic traits of natural pop-
ulations (Coulson et al., 2008, 2010; Flatt, 2020). These
challenges in explaining observed variability only by geno-
types, environments and their interaction lead us to the
view that non- genetic and non- environmental processes
generate and contribute to the high levels of variation in
phenotypes and life courses among individuals (Jouvet
et al., 2 018; Snyder et al., 2021; Snyder & Ellner, 2018;
Steiner et al.,2019, 2021; van Daalen & Caswell,2020).
The fundamental question we address here is whether
such a non- genetic, non- environmental- driven variation
is truly neutral, selected for or against. In the case of
neutral variation, the follow- up question would be, how
is such neutral variation maintained (Demetrius, 1974)?
Here we do not decompose variance in genetic, environ-
mental, phenotypic plastic (gene- by- environment) and
neutral contributions to life- course variability, as pre-
viously done for datasets that have the needed depth of
information or by making assumptions about partition-
ing (Snyder et al., 2021; Snyder & Ellner, 2 018; Steiner
et al.,2021). Instead, we aim at quantifying the selective
forces on the processes that generate variation among
life courses by relying on the analysis of structured
population models (Steiner & Tuljapurkar, 2020). We
describe this approach in the following section starting
with structured populations and associated life courses.
In any structured population, a life course of an indi-
vidual can be described by a sequence of stages that ends
with death (Caswell,2001). These stage sequences, or life-
course trajectories, differ among individuals in length,
i.e. age at death, and in the sequence and frequency of
stages experienced. Stages can comprise many traits,
e.g. size in development (say of newborns, juveniles, sub-
adult and adults), reproductive state (say immatures,
non- breeders and levels of reproductive output), other
traits (such as behaviour, morphology, physiology and
gene expression) and even non- biological features (say
location or physical environment). Obviously, models
simplify phenotypes to one or a few traits, but even so
trait values will change during ontogeny within an indi-
vidual and among individuals may not follow the same
time sequence. Thus, stage sequences, life courses and
phenotypes are linked and so is their diversity. After
birth, there is a growth of diversity in stage sequences
and a corresponding growth of diversity in phenotypic
characteristics. In this sense, the rate at which sequences
of stages diversify with increasing length, quantif ied by
population entropy (Hernández- Pacheco & Steiner,2017;
Steiner & Tuljapurkar,2020; Tuljapurkar et al.,2009), is
also useful as a measure of phenotypic diversity.
When describing such stage sequences or life courses
in population models, all individuals often start in the
same newborn stage, thereby discarding differences in
(often unknown) birth characteristics. But individuals
can also be born into one of a few stages, as frequently
modelled for plants (e.g. sexual reproductive: seed or
seedling; clonal reproductive: offshoot). After being
born life diversifies in stage sequences followed, and
hence, phenotypic characteristics with increasing age
and the rate at which these sequences of stages diversify
with increasing length can be quantified by population
entropy (Hernández- Pacheco & Steiner,2 017; Steiner &
Tu l j ap u r k a r, 2020; Tuljapurkar et al.,2009). In age- only
structured models, the length of life is the only aspect
that varies among individuals, but the stage sequence is
the same among individuals; if an individual survives, it
simply enters the next age class as any other surviving
individual does without differentiating characteristics.
Demetrius' entropy (1974) quantifies the variability in
reproductive output of such age- only structured popu-
lations with increasing age, and Demetrius' entropy con-
trasts with Keyfitz's entropy (actually, the latter is not
mathematically an entropy) that also applies to age- only
structured populations but quantifies changes in life ex-
pectancy caused by changes in age- specific mortality.
Here, we use a measure of entropy that is a generalization
of Demetrius' entropy, in that the population entropy we
use emphasizes differences in reproduction/survival/
growth generated by stage transition dynamics. In stage-
structured population models, high population entropy
leads to highly diverse life courses in short times and low
entropy leads to few distinct life courses that groups of
individuals follow (Hernández- Pacheco & Steiner,2017).
To be precise, entropy measures the rate of diversifica-
tion in stage sequences of a cohort. As this rate relates to
the diversification of life courses of such a cohort, it also
relates to the diversity of life courses at different ages of a
cohort. Not only the life courses, i.e. the stage sequences,
but also their rate of diversification are determined by
the stage transition rates (Caswell,2001). To quantify the
contributions of each stage transition to the rate of diver-
sification of life courses (Steiner & Tuljapurkar, 2020),
we can perturb each stage transition rate, i.e. elements of
the population matrix model, and then compute the con-
tributions of these perturbations to population entropy.
Of course, such estimation of the sensitivity of each
transition rate to the population entropy does not reveal
anything about f itness— λ, the rate at which a population
grows (Caswell,2001; Steiner & Tuljapurkar,2020).
However, the desired linkage to fitness is revealed by
the sensitivities of the population growth rate, λ, to the
same perturbations of the transition rates of the model. If
one then examines the correlation between the sensitivities
of entropy and fitness— both are estimated for each transi-
tion rate of a given model— , we can link the rate (process)
of life- course diversification and selective forces (Figure1)
(Steiner & Tuljapurkar, 2020). To expand on this argu-
ment, if a perturbation of a stage transition parameter in a
model leads to both an increase in entropy and population
growth rate (fitness λ), selection for greater diversification
542
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LIFE - COU RSE DIV ERSITY
of life courses is favoured, whereas, if a negative correlation
between these sensitivities occurs, selection against diver-
sification is suggested, and if there is no correlation be-
tween the two sensitivities, the observed variability among
life courses may be neutral. We base our interpretation on
the idea that selection should act more strongly on stage
transitions that have higher sensitivities with respect to
population growth, λ, and hence, fitness (Pfister,1998). To
illustrate the concept, imagine a mutation that changes a
stage transition rate (fertility rate or other stage transition
rate), if this change in transition probabilities inf luences
fitness, λ, more than changes in other transition rates, it
should be under stronger selection than those transition
rates that only have little inf luence on fitness.
To evaluate how the diversity in life courses is selected
upon— positively, negatively or neutral— , we explore
the correlation of the sensitivity with respect to entropy
and the sensitivity with respect to population growth
for a large variety of species and taxa for which pop-
ulation projection models have been collected within
the COMADRE and COMPADRE databases (Jones
et al.,2022; Salguero- Gómez et al.,2016) (COMPADRE
& COMADRE Plant Matrix Databases, 2022). Available
from: https://www.compa dre- db.org; accessed 7.3.2022.
We estimate for each transition rate of each population
projection model the sensitivity with respect to entropy
and population growth, then correlate these two sensi-
tivities for each projection model and compare these
correlations across species, taxa, phyla, ontology, age
(models containing at least one class that is based on age;
in our case, it needs to be in addition to the stage struc-
ture), organism type and matrix dimension for plants
and animals. We find that both in plants and animals,
substantial variation in the correlation between the two
sensitivities among species exists and we find a very weak
or no overall correlation between sensitivities, suggest-
ing close to neutral evolution of life- course variability.
We also address a different question, whether popula-
tions or species with high rates of life- course diversifica-
tion exhibit high f itness compare d to those that diversi fy at
a lower rate in their life courses. Such investigation might
be understood in terms of adaptive niche differentiation
or specialization (Hernández- Pacheco & Steiner, 2 017).
Here, our findings suggest that matrices with high rates
of diversification (higher entropy) do not show increased
or reduced fitness. Note, only a single entropy and a single
population growth rate are calculated per matrix, while
for each of the many transition rates (non- zero matrix el-
ements), sensitivities can be calculated. Overall, we find
that populations that diversify at higher rates in their life
courses do not show increased or decreased population
growth rates and selective forces seem not to increase or
decrease life- course diversification.
MATERIALS AND METHODS
Of the 3317 population matrices in the COMADRE animal
database and the 8708 matrices in the COMPADRE plant
database, we selected matrix models that were ergodic and
irreducible (1350 and 5823, respectively). Of these, we se-
lected only matrices that had for each stage (each matrix
column, Figure1) at least two non- zero elements (one of
them could be a reproductive stage). This resulted in 37
matrices on 11 animal species, and 2144 matrices on 262
plant species. The extreme reduction in the animal matrix
number ref lects that many of these animal matrices are
sparse matrices, for instance, age- structured- only (Leslie)
population matrices. Note, most of the animal matrix
models are coming from marine organisms that show slow
growth, such as corals, sponges and tunicates, hence, the
animal data are highly biased and not representative of all
animals. This bias is not generated by theoretical limita-
tions but rather by a lack of data on animal populations
FIGUR E 1 Sketch: for each population matri x, we est imated for each element (here exe mplified by element k3,2) an integrated sensitivity
with re spect to entropy (∆Hk3,2) and with resp ect to f itne ss (∆λk3,2) by increasing (perturbing) element k3,2 by amount b and simultaneously
reducing elements k1, 2; k2,2; k4,2 by b/n with n=(number of non- zero colum n elements) − 1. Such integrated s ensitivities were then computed for
each matrix element ki,j and for both types of sensitivities. For each population m atri x we fitted a linear model th rough data points based on
these two typ es of sensitivities from each of the ki, j elements. Each l ine in Figure2 corresponds to such a correlation model.
-b/n
-b/n
+b
-b/n
Populaon matrix A
Ferlity of each individual in stage j
Transion rates of stage j (column) individuals to stage i(row)
H = entropy of Populaon matrix
quanfies rate of diversificaon of life courses
λ= Populaon growth rate of Populaon matrix
quanfies fitness of populaon
Example of perturbaon of matrix element k3,2 by amount b
Compensaon of perturbaon of element k3,2 in element k1,2; k2,2; k4,2
Integrated sensivity entropy ∆Hk3,2= ∆H (Matrix with posive perturbaon in k3,2 and compensaon in k1,2 ;k2,2 ;k4,2)
Integrated sensivity fitness ∆ λ k3,2= ∆λ(Matrix with posive perturbaon in k3,2 and compensaon in k1,2 ;k2,2 ;k4,2)
∆H
∆λ
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STEINER and TULJAPURKAR
and formulation of non- sparse stage- structured animal
matrix models. Nevertheless, we end up with a biased and
relatively small sample of animals.
We limited the analysis to matrices with at least two
non- zero elements per stage to evaluate perturbations
(sensitivities) that do not trade- off against survival, but
against other stage transitions or reproductive rates
(Figure1). We call these sensitivities integrated sensitiv-
ities (Steiner & Tuljapurkar,2020). Each integrated sen-
sitivity evaluates by how much a perturbation of amount
b, in one focal matrix element k, influences population
entropy, H, and population growth, λ, when simultane-
ously all other non- zero elements in the given stage (col-
umn) are reduced by b/n, with n equals the number of
non- zero elements in a column minus the focal element.
Note, integrated sensitivities can have positive or nega-
tive signs, i.e. they can increase or decrease entropy or λ.
For more details on entropy and integrated sensitivities,
also see the Supplemental Information where we give an
illustrative example of our estimation (SI 1).
Before we estimated the integrated sensitivities, we
transformed the absorbing population projection matri-
ces into Markov chains (Tuljapurkar,1982) (SI 1). We then
computed for each of the 41,812 non- zero matrix elements
their integrated sensitivities with respect to population
entropy and population growth rate λ on the plant matri-
ces and 602 non- zero elements of the animal matrices. As
the integrated sensitivities had very heavy tail distribu-
tions on both tails, we excluded extreme values that more
likely arose from biologically unrealistic matrix param-
eter entries. Note, transition rates that were close to 1
or 0 did not result in extremely integrated sensitivities
(FigureS3). We excluded extreme values of integrated
sensitivities that exceeded three times the standard de-
viation for integrated sensitivities of entropy (13 animal
matrix elements; 811 plant matrix elements) and values
on integrated sensitivities of lambda that exceeded three
times the standard deviation for the animal data (16), or
0.02 (a less conservative value than 3 × SD) for the plant
data (559), leaving 577 integrated sensitivities and 40,640
integrated sensitivities, respectively, for the animal and
plant data analysis (4 and 198 were outliers for both in-
tegrated sensitivities, of respectively, animal or plant
data). Resulting distributions, after the outlier removal,
remained heavy tailed.
For statistical testing, we fitted linear models (despite
symmetrically long tails on both sides of the residual dis-
tribution) and used model comparison based on Akaike's
information criterion (AIC). We defined a difference
in AIC >2 as substantial better support (Burnham &
Anderson,2004). We evaluated the model fit and the as-
sumptions using diagnostic plots.
For each matri x, we also computed popu lation matrix-
level entropy and population growth rate, λ; note there
is one value of entropy and population growth for each
matrix. We also correlated sensitivities with respect to
entropy and those with respect to lambda for the 37 ani-
mal matrices and 2144 plant matrices against each other.
Model comparisons were done using AIC (Burnham &
Anderson,2004).
RESULTS
We show the integrated sensitivities of entropy and
those of lambda across all animal species in our data
in Figure2a. The figure shows no evident correlation
between these sensitivities (supported by statistical
analysis, Table1: Model 1 [null model with only an inter-
cept], vs. Model 2 [simple regression of the two sensitivi-
ties, slope −0.084], both models receive equal support).
Hence, neither selection for nor against higher or lower
rates of diversification of life courses is observed in ani-
mals. For plants, we find a weak positive correlation
FIGUR E 2 Correlating integ rated sensitivities with respect to entropy and that with respe ct to lambda for animal populations (a) and
plant populations (b). Each line fits the correlat ion for one population (one matrix model). Line colors ref lect the dif ferent species as more
than one matri x model can be f itted per sp ecies (e.g. different years, or populations). For the plant data (b) the numb er of species is too large to
differentiate among the species. For better visibility CI (confidence inter vals) are not plotted.
−1e−03
−5e−04
0e+00
5e−04
1e−03
−0.002−0.0010.0000.0010.002
Integrated Sensitivity Entropy
Integrated sensitvity Lambda
Species
Acropora cervicornis
Acropora downingi
Acropora hyacinthus
Amphimedon compressa
Botrylloides violaceus
Gorgonia ventalina
Haliotis rufescens
Pocillopora damicornis
Sigmodon hispidus
Spongia graminea
Xestospongia muta
(a)
−0.02
−0.01
0.00
0.01
0.02
−0.002 −0.001 0.0000.001 0.002
Integrated Sensitivity Entropy
Integrated Sensitivity Lambda
(b)
544
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LIFE - COU RSE DIV ERSITY
(Tab l e1: Model 1 vs. Model2, Figure2b), although its
effect size (slope 0.056) is small (compared with effect
size of non- significant animal data). Hence, for plants,
selection tends to favour increased rates of life- course
diversification. This said, there is substantial variation
in the correlation between the integrated sensitivities
among the species (Figure2). The model that allows for
one correlation per species, i.e. the model including the
interaction between species and the integrated sensitiv-
ity of entropy (Table1: Model 8) is better supported than
models that are restricted to main effects or additive-
only effects (Model 1, 2, 6 and 7). Similarly, there is sig-
nificant variation in correlations among matrix models
(i.e. slopes differ among correlations estimated for each
matrix population model separately) (Figure2) as Model
4 (Tabl e 1) that accounts for the interaction between the
matrix model ID and the correlation is better supported
than models that only fit main effects or additive effects
(Tab l e1: Model 2, 3 and 5). There is also significant vari-
ation among correlations per matrix within species, as
Model 4 which fits for each matrix within a species 1
correlation is better supported than Model 8 that fits 1
correlation per species (Tabl e 1: Model 8). These findings
suggest that selection differs among species, i.e. favour-
ing higher rates of diversification in life courses in some
species while selecting against such diversity in others.
In addition, variation in correlations among matrices
but within species (Model 8 vs. Model 4) is significant
and cannot be simply reduced to the species level. This
finding shows that differences in selection are observed
among populations of the same species or the same
population in different years. These patterns of variance
within and among species hold for both animal and plant
data. These patterns are also robust when the analyses
are limited to only matrix models with growth rates ≤1.5
(FigureS4), as one might be concerned that very fast-
growing populations might drive our results on variance
among species or populations. Fig ure2 also suggests
that many correlations show slopes close to 0, these cor-
relations with shallow slopes do not show reduced re-
sidual variances compared to those with steeper slopes,
i.e. slopes that are potentially under stronger selection
(steeper slopes) do not have increased or reduced resid-
ual variance around their regression lines (FigureS5).
We investigated the effect on the correlation between
sensitivities by using several possible grouping variables,
including age (at least one age class in addition to stage
structure), matrix dimension (number of stages), phy-
lum, organism types (e.g. algae, fungi and annual for
plants) or ontogeny. We found (Tab l e1, FigureS1) that
TABLE 1 Model selection among competing models based on animal and plant matrix population models evaluating the correlation
betwe en integrated sensitivities with respect to entropy (response variable) and integrate d sensitivit ies with respect to population grow th
lambda (explanatory var iable) and various covariates.
Model# Parameters
Animals Plants
df AIC ∆AIC df A IC ∆AIC
1 Interc ept only model 2 −720 6 .0 122.0 2−514 ,160. 2 16,74 0. 3
2SensEntr ~ Sen sLambd a 3−72 06. 8 121.2 3−514,47 7.0 16,423.5
3SensEntr ~ Sen sLambd a + MatrixI D 39 −7169.5 158.5 2146 −515, 011.4 15, 88 9.1
4SensEntr ~ Sen sLambd a × MatrixI D 75 −7328 .0 0.0 4289 −530,900.5 0.0
5SensEntr ~ Matr ixID 38 −7166. 0 162 .0 2145 −514, 832.1 16 ,0 68.4
6SensEntr ~ Spe cies 12 −7203.3 124 .7 263 −517,217.2 13,683.3
7SensEntr ~ Sen sLambd a + Specie s 13 −72 04. 5 123.5 264 −517,418 .6 13 ,481.9
8SensEntr ~ Sen sLambd a × Specie s 23 −7241. 3 86 .7 525 −5 25 ,791. 6 5108 .9
9SensEntr ~ Sen sLambd a × Speci es + Matri xID 49 −72 07.1 120.9 2407 −523,6 44.7 7255.8
10 SensEntr ~ Sen sLambd a × AgeStructure 5−72 05 .2 122.8 5−514, 541.0 16,359.5
11 SensEntr ~ Sen sLambd a + AgeStructure 4−72 07. 0 121.0 4−514,48 4. 8 16,415.7
12 SensEntr ~ Sen sLambd a × MatrixDimension 5−72 04.4 123.6 5514 ,536.1 1,045,436.6
13 SensEntr ~ Sen sLambd a + MatrixDimension 4−72 06. 3 121.7 4514, 475.0 1,045,375.5
14 SensEntr ~ Sen sLambd a × Phylum 9−720 4 .1 123.9 13 −514 ,79 3.7 16 ,10 6. 8
15 SensEntr ~ Sen sLambd a + Phylum 6−7201.6 126.4 8−514, 58 7.3 16 ,313.2
16 SensEntr ~ Sen sLambd a × OrganismType 11 −72 02 .1 125.9 21 −515,133.9 15,766.6
17 SensEntr ~ Sen sLambd a + OrganismType 7−72 01. 3 126.7 12 −514,491.7 16,408.8
18 SensEntr ~ Sen sLambd a × Ontoge ny 5−7202.8 125.2 5−514,555.5 16 ,345.0
19 SensEntr ~ Sen sLambd a + Ontoge ny 4−7 20 4 .8 123.2 4−514,476.0 16,424.5
Note: MatrixID=Data bas ed on ea ch Population Matrix Mod el evalu ated w ith respec t to the Popu lation Matri x Model, Ag eStru cture=Yes/No distinc tion
whethe r the Popu lation Matrix Mo del included at least s ome age classes in addition t o the sta ge str uctu re, Matr ixD imen sion=Numb er of stages in the matr ix
model. Fu rther deta ils on c ovariates ca n be obtained from the d ata sou rce COMA DRE & COMPADRE dat a base. Boldfa ced mo dels ar e best suppor ted mod els,
grey fond models a re lea st supported models , non- b oldfac ed bla ck models are pa rtly e qual ly well supported. ∆AIC compa re to the best overall supported mod el.
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545
STEINER and TULJAPURKAR
in animals these variables do not play an important role,
while in plants they do account for a small amount of
variability. Still, compared to the variance among spe-
cies and within species, these grouping variables are of
little importance. The number of stages per matrix (di-
mension) could potentially affect our findings because
we found an interaction among matrices of different
dimensions and integrated sensitivity with respect to
lambda for plant species, but there was no general trend
with increasing matrix dimension towards or against se-
lection for variance in life courses, suggesting no system-
atic bias regarding matrix dimension (FigureS1).
We further asked whether high or low diversity in life
courses (population entropy) is associated with high or
low fitness (population growth rates). Note here, we eval-
uate population entropy and lambda for the total pop-
ulation, i.e. one value for each matrix, not as above, a
measure at the matrix element level (integrated sensitivi-
ties measures). Figure3 shows this relationship between
entropy and lambda (see Tabl e 2 for model comparison).
We did not find any simple relationship between popu-
lation entropy and fitness for animals as a null model
with an intercept only was equally well supported than a
model that fitted a correlation between lambda and en-
tropy (Table2, Model 1 vs. 2, Fig ure3a). For plants, how-
ever, there was some tendency that matrices with higher
rates of diversification had lower fitness as the model
that fitted a correlation between lambda and entropy
was better supported than a simple intercept- only null
model (Table2 Model 1 vs. 2, Figure3b, slope −0.42).
One necessary caution is that these results are largely
driven by biologically questionable and extremely high
values of population growth rates (see also FigureS2).
Note, we tested that our findings on integrated sensitiv-
ities are not driven by these very fast- growing popula-
tions (FigureS4). Overall, there is significant variation
in both population entropy and population growth
rate but no clear correlation between the two variables.
Matrix dimension explains some additional variance in
the relationship between entropy and lambda, although
species differences are much more important in explain-
ing variance than matrix dimension (Ta b le2).
DISCUSSION
We show that across animal populations there is no clear
selective force that acts towards or against increased or
decreased rates of diversification in life courses, whereas
for a large collection of plants, there is weak selection fa-
vouring diversification in life courses. Given that there is
selection for higher rates of diversification in life courses,
one might expect that populations with higher entropy
would also have higher fitness, but in apparent contrast
to this expectation, we find that plant populations (or
species) with high rates of life- course diversification
(high entropy) tend to have lower fitness than popula-
tions (or species) that show low rates of diversification.
However, the actual rate of diversification (population
entropy) and the selective force on that rate (integrated
sensitivities of population entropy) reveal two different
things. For instance, a population might have a low rate
of diversification, but there might be a strong selective
force of increasing that rate, or a population might have
a high rate of diversification and there might be only a
weak force of increasing or reducing that rate. The in-
tegrated sensitivity analyses investigate selective forces
on the diversification processes within a population
(Steiner & Tuljapurkar, 2020), whereas the population
entropy quantifies the current rate of diversification
(Tuljapurkar et al.,2009). The sensitivity analyses, there-
fore, focus on within- population selective processes,
whereas entropy and population growth are best used
for among- population comparison.
FIGUR E 3 Relationship betwee n population growth lambda (fitness) and population entropy, the rate of diversif ication, for animal (a) and
plant (b) popu lation models. Each dat a point repres ents one m atri x model. Colors depict di fferent dimensions of the matrix model. Populations
that showed extremely low or hig h lambda are not plotted for better i llustration. The full dataset, i ncluding the extreme values of lambda is
plotted in Figu reS2 and the model selection of Ta bl e2 is also based on the full data set.
0.25
0.50
0.75
0.51.0 1.52.0
Lambda
Entropy
MatrixDimension
3
4
5
6
8
10
(a)
0.00
0.25
0.50
0.75
1.00
0246
Lambda
Entropy
MatrixDimension
3
4
5
6
7
8
9
10
11
12
13
16
24
(b)
546
|
LIFE - COU RSE DIV ERSITY
Our finding of substantial variation in selective forces
on the rate of diversification, as well as substantial vari-
ation in the rate of diversification, might be of greater
interest than the small positive selective trend favour-
ing diversification for plant species. The interpretation
of the animal models remains challenging, as the spe-
cies for which non- sparse stage- structured population
models are available is biased towards specific types of
slow- growing marine organisms with often many off-
spring. These substantial levels of variability in selective
forces and rates of diversification might have three dif-
ferent biological origins or meanings: first, they might
indicate substantial (developmental) noise that leads
to the observed variability in life courses and selection
for or against diversification in life courses (Balázsi
et al., 2 011); second, it might indicate f luctuating se-
lection or high levels of phenotypic plasticity driven
by variable environmental conditions (Gillespie, 1975;
Philadelphia,1973); or third, it might indicate large num-
bers of distinct adaptive life courses that show similar fit-
ness but might, for instance, fill different niches, or solve
life- history trade- offs in many different ways that lead to
similar fitness (Hernández- Pacheco & Steiner,2 017). In
quantitative genetic terms, these options would relate to,
respectively, undetermined residual variation, gene- by-
environmental variation or additive variation.
If one assumes that noise explains the variability, it is
suggested that selection might not act very strongly on
this noise, as otherwise, the variability should be selected
against and variability should collapse (Fisher, 1930;
Haldane,1927, 1932; Wright,1931). Such neutral, or close
to neutral, arguments have been used in the past to ex-
plain life- course variability but are often met with scep-
ticism (Steiner & Tuljapurkar,2012). Our results might
indicate that selective forces on rates of diversification
in life courses are not generally weak, but partly go in
opposing directions, i.e. selecting for diversification in
some populations or species and against in others. This
interpretation is also supported by our finding that
residual variance is not related to the force of selection
(FigureS5). Conflicting findings as we reveal are also
found commonly in other fields, for instance, in quan-
titative genetic studies (Charlesworth,2015; Flatt, 2020;
Johnson & Barton,2005).
If one assumes that fluctuating environments or
similar extrinsic variation causes vital rates to differ
among matrices and leads to highly diverse life courses
(Gillespie, 1985; Philadelphia, 1973), we might assume
that a large fraction of variability would be explained by
among matrix models within species, and less so among
species. Model selection indicates that among- species
variation is substantially greater compared to variability
among matrices within species. Hence, variability among
populations or time (years), or conditions (environments)
within species contribute less to variability in life courses
than variability among species. These arguments align
with findings that phenotypic plasticity might not be in
general adaptive (Acasuso- Rivero et al.,2019). The meta-
analysis we did might not be ideal for such within- species
evaluation, as the average number of matrices per spe-
cies (3.4 for animals and 8.2 for plants) is not very large,
but our analysis still provides more general insights com-
pared to studies focusing on single- model species for
which rich data exist (Flatt,2020).
If one assumes that diversity i n life courses is produced
because many life courses are equally fit (Hernández-
Pacheco & Steiner,2017; Nevado et al.,2019), we would
be challenged to explain the strong selective patterns
against diversification that is observed for some popu-
lations and species. Under such an assumption, the opti-
mal number of distinct life courses would need to differ
substantially among species or populations. Also, from
more detailed analyses of systems, certain life courses,
or genotypes, that are commonly observed seem to have
low fitness (Flatt, 2020; Steiner et al., 2021), suggesting
that not all life- course variability might be adaptive. In
addition, different solutions to life- history trade- offs
that could generate life- course diversity would frequently
TABLE 2 Model selection among competing models based on animal and plant matrix models evaluating the cor relation between
population entropy (response variable) and population growth, lambd a (explanatory var iable), as wel l as matrix dimension and species
comparison.
Model# Parameters
Animals Plants
Slope df AIC ∆AIC Slope df AIC ∆AIC
1 Lambda ~ Intercept only model 2 50.19 49.72 228 39.36 1255.66
2 Lambda ~ Entr 0.46 350.54 50.07 − 0.42 3277 9.3 0 1195.6
3 Lambda ~ Entr + Matri xDim 452.02 51. 55 42780.43 1196.73
4 Lambda ~ Entr × Matri xDim 547.25 46.78 52754.89 1171.19
5 Lambda ~ MatrixD im 352.16 51.69 32835.28 1251.58
6 Lambda ~ Specie s 12 19.16 18.6 9 263 1625.00 41. 3
7 Lambda ~ Entr + Spec ies 13 0.47 0264 162 4. 83 41.13
8 Lambda ~ Entr × Spec ies 20 3.52 3.05 456 15 83.70 0
Note: MatrixDi m=Numbe r of stage s of the matri x population mod el the entropy and l ambda w as est imated from. Boldfac ed models are b est sup port ed mode ls,
grey fond models a re lea st supported models , non- b oldfac ed bla ck models are pa rtly e qual ly well supported. ∆AIC compa re to the best overall supported mod el.
|
547
STEINER and TULJAPURKAR
be comprised within a single- matrix model, potentially
contributing to patterns that resemble noise and, there-
fore, could explain the maintenance of noisy signals.
The potential explanations that help to understand
the selective forces on the rate of diversification of life
courses are not mutually exclusive and we do not have
means to quantify each contribution to the diversifica-
tion using the models in this study. More detailed stud-
ies that focus and explore selection on diversification
could help to better understand the influence of these
three factors (Flatt, 2020). Studies might include how
genes (or gene knockouts) influence the rate of diversifi-
cation, how experimental evolution studies in stochastic
environments differing in amplitude and autocorrelation
(noise colour and wavelength) would lead to the evolution
of different rates of diversification, how “heritability” of
distinct life- course strategies potentially determine life-
course diversification under different environmental
conditions or how trade- offs among life- history traits
maintain and generate diverse life courses. Quantitative
genetics studies have identif ied a similar lack of under-
standing of the maintenance and the evolution of vari-
ability (Charlesworth, 2 015; Johnson & Barton, 2005),
although with a focus on genetic explanations empha-
sizing mutation– selection balance being driven by few
strongly deleterious mutations (Charlesworth, 2 015;
Mul ler, 1950), or alternatively many polymorphic loci
that maintain variability (Dobzhansky,1955; Johnson &
Barton,2005). Such genetic variation interacts with neu-
tral and non- genetically determined processes that in-
fluence evolutionary processes and the pace of evolution
(Steiner & Tuljapurkar,2012). For that, a purely quanti-
tative genetic vision might be too short- sighted. Growing
literature emphasizing how noisy gene regulation might
scale and trigger cascading effects across levels of bio-
logical organization offers ways for more mechanistic
understanding (Elowitz et al.,2002; Robert et al.,2018).
Generally, we believe we are only beginning to under-
stand selection of processes that lead to the observed
variability in life courses (Flatt,2020).
AUTHOR CONTRIBUTIONS
Ulrich Karl Steiner and Shripad Tuljapurkar developed
the concept and theory. Ulrich Karl Steiner analysed the
data. Ulrich Karl Steiner wrote the f irst and final drafts
with subst antial contributions from Shripad Tuljapurkar.
ACKNO WLE DGE MENTS
We thank the referees and editors, Robin Snyder,
Mathias Franz and members of the AG Rolf, AG
Armitage and AG Steiner at the Freie Universität Berlin,
as well as members of the Tuljapurkar lab for discussions
and comments.
FUNDING INFORMATION
Deutsche Forschungsgemeinschaft, Grant/Award Number:
43 017 0797
PEER REVIEW
The peer review history for this article is available at
ht t ps: //pub l o n s .c o m / p u blo n /10.1111/e l e .14174.
DATA AVA ILABILITY STAT EME NT
R code and data for reproducibility are available on
Figshare https://doi.org/10.6084/m9.figsh are.21830 2 35.
v3. No new data were generated. This R code loads the
currently available data from the COMPADRE and
COMADRE Databases and is open access data under
the terms of the Creative Commons CC BY- SA 4.0 li-
cence. It can be accessed at https://www.compa dre- db.
org, for further details see h t t p s: //d o i.or g /10 .1111 /1365-
2745.12334 and ht t p s ://doi.o r g /10.1111/1365 - 265 6.12482 .
ORCI D
Ulrich Karl Steiner https://orcid.org/0000-0002-1778-5989
Shripad Tuljapurkar https://orcid.org/0000-0001-5549-4245
REFERENCES
Acasuso- Rivero, C., Murren, C.J., Schlichting, C.D. & Steiner,
U.K. (2019) Adaptive phenotypic plasticity for l ife- history and
less f itness- related traits. Proceedings of the Royal Society B:
Biological Sciences, 286(1904), 20190653.
Balázsi, G., Van Oudenaarden, A. & Collins, J.J. (2011) Cellular deci-
sion mak ing and biological noise: from microbes to mam mals.
Cell, 144, 910– 925.
Barton, N.H., Etheridge, A.M. & Véber, A. (2017) The infin itesimal
model: Definition, derivation, and i mplications. Theoretical
Population Biology, 118, 50– 73.
Bell, G. (2010) Fluctuating selection: The perpetu al renewal of adap-
tation i n variable environments. Philosophical Transactions of
the Royal Society, B: Biological Sciences, 365, 87– 97.
Burn ham, K.P. & Ander son, D.R. (Eds.). (2004) Model selection and
multimodel inference. New York, NY: Springer New York.
Caswell, H. (2001) Matrix population models: construction , analysis,
and interpretation. Natu ral Resource Modeling. Sunderland, MA:
Sinauer Associates.
Charlesworth, B. (2015) Causes of natural variation in f itness: evi-
dence from stud ies of Drosophila populations. Proceedings of the
National Academy of Sciences of the United States of America,
112(6), 1662– 1669.
Coulson, T., Tu ljapurkar, S. & Childs, D.Z. (2010) Using evolutionary
demography to li nk li fe history theory, quantitative genetics and
population ecology. The Journal of Animal Ecology, 79, 1226 – 1240.
Coulson, T., Tu ljapurkar, S. & Step, T. (2008) The dynamics of a qua n-
titative trait in an age- structure d population living in a variable
environment. The American Naturalist, 172, 599– 612.
Crow, J.F. & K imura, M. (1970) An introduction to population genetics
theory. Minneapolis, MN: Burgess Publishing Company.
Demetrius, L. (1974) Demographic parameters and natu ral selection.
Proceedings of the National Academy of Sciences of the United
States of America, 71, 4645 – 4647.
Dobzha nsky, T. (1955) A review of some fundamental concepts and
problems of population genetics. Cold Spring Harbor Symposia
on Quantitative Biology, 20, 1– 15.
Elowitz , M.B., Levine, A.J., Sigg ia, E.D. & Swain, P.S. (2002) Stochastic
gene expression in a single cell. S c i e nce ( 8 0 - .), 297, 1183– 1186.
Endler, J.A. (1986) Natural selection in the wild. Monographs in
Population Biology 21. Pri nceton, NJ: Princeton University Press.
Finch, C. & Kirkwood, T.B. (2000) Cha nce, development, and aging.
Oxford: Oxford Univer sity Press.
Fisher, R. (1930) The genetical theory of natural selection. Oxford:
Clarendon.
548
|
LIFE - COU RSE DIV ERSITY
Flatt, T. (2020) Life- histor y evolution and the genetics of fitness com-
ponents in Drosophila melanogaster. Genetics, 214 (1), 3 – 48.
Gillespie, J.H. (1975) Natural selection for within- generation variance
in offspring number II. discrete haploid models. Genetics, 81,
403– 413.
Gillespie, J.H. (1985) The interaction of genetic drift and muta-
tion with selection in a flu ctuating env ironment. Theoretical
Population Biology, 27, 222– 237.
Haldane, J.B.S. (1927) A mathematical theor y of natural and a rti-
ficial selection, part V: Selection and mutation. Mathematical
Proceedings of the Cambridge Philosophical Society, 23, 838– 844.
Haldane, J.B.S. (1932) The ca uses of evolution. London, New York:
Longmans, Green and Co.
Hartl, D.J. & Clark, A.G. (2007) P rinciples of population ge netics.
Sunderland: Sinauer.
Herná ndez- Pacheco, R. & Steiner, U.K. (2017) Drivers of diversifi-
cation i n individual life course s. The American Naturalist, 190,
E132– E144.
Johnson, T. & Barton, N. (2005) Theoretical models of selection and
mutation on quantit ative traits. Philosophical Transactions of the
Royal Society, B: Biological Sciences, 360, 1411– 1425.
Jones, O.R., Barks, P., Stott, I., James, T.D., Levin, S., Petry, W.K.
et al. (2022) Rcompadre and Rage— Two R packages to facili-
tate the use of the COMPADRE and COMADRE d atabases and
calculation of life- histor y traits from matrix populat ion models.
Methods in Ecology and Evolution, 13, 770– 781.
Jouvet, L., Rodr íguez- Rojas, A. & Stei ner, U.K. (2018) Demographic
variabilit y and heterogeneity among indiv idual s with in and
among clonal bacteria strai ns. Oikos, 127, 728– 737.
Kimu ra, M. (1968) Evolutionary rate at the molecular level. Nature,
217, 624 – 626.
Melbour ne, B.A. & Hastings, A. (2008) Extinction risk depends
strongly on factors contributing to stochasticity. Nature, 454,
100 – 103.
Muller, H.J. (1950) Our load of mut ations. Amer ican Journa l of Human
Genetics, 2 , 111– 176.
Nevado, B., Wong, E.L.Y., Osborne, O.G. & Filatov, D.A. (2019)
Adaptive evolution is common in rapid evolutionar y radiations.
Current Biology, 29, 3081– 3086.e5.
Pfister, C.A. (1998) Patterns of variance in stage- str uctu red popu-
lations: Evolutionary predictions and ecological implications.
Proceedings of the National Academy of Sciences of the United
States of America, 95, 213– 218.
Philadelph ia, J.G. (1973) Polymorphism in Random Environments.
Theoretical Population Biology, 195, 193 – 195.
Robert, L., Ollion, J., Robert, J., Song, X., Matic, I. & Elez , M. (2018)
Mutation dy namics and f itne ss effects followed in single c ells.
Sci e n c e (8 0 - .), 359, 1283– 1286.
Salguero- Gómez, R., Jones, O.R., Archer, C.R., Bein, C., Buhr, H.,
Farack, C. et al. (2016) COMADRE: a global data base of animal
de mog raphy. The Journal of Animal Ecology, 85, 371– 384.
Snyder, R.E. & Ell ner, S.P. (2016) We happy few: Using str uctured
population models to identify the decisive events in the lives of
exceptional individuals. The American Naturalist, 188, E28– E45.
Snyder, R.E. & Ell ner, S.P. (2018) Pluck or luck: Does trait variation
or chance drive variation in l ifeti me reproductive succ ess? The
American Naturalist, 191, E90 – E107.
Snyder, R.E., Ellner, S.P. & Hooker, G. (2021) Time and chance:
using age partitioning to underst and how luck drives varia-
tion in reproduc tive suc cess. The American Naturalist, 197,
E110– E128.
Steiner, U.K., Lenart, A., Ni, M., Chen, P., Song, X., Taddei, F.
et al. (2019) Two stochastic proc esses shap e diverse senes-
cenc e patterns in a single- cell organis m. Evolution (N. Y).,
73, 847– 857.
Steiner, U.K. & Tuljapurkar, S. (2012) Neutral theor y for life h istor ies
and ind ividual variabil ity in f itness components. Proceedings of
the National Academy of Scie nces of the United States of America,
109, 4684– 4689.
Steiner, U.K. & Tuljapurkar, S. (2020) Drivers of diversity in i ndi-
vidua l life courses: Sensitivit y of the population entropy of a
Markov chai n. Theoretical Population Biology, 133, 159– 167.
Steiner, U.K., Tuljapurka r, S. & Roach, D.A. (2021) Quantifying the
effec t of genetic, env ironmenta l and individual demograph ic sto-
chastic variability for population dynamics in Plantago lanceo-
lata. Scientific Reports, 11, 23174.
Tuljapurkar, S., Stei ner, U.K. & Orzack, S.H. (2009) Dynamic hetero-
geneity in li fe histories. Ecology Letters, 12, 93– 106.
Tuljapurkar, S.D. (1982) Why use population entropy? It deter mines
the rate of convergence. Journal of Mathematical Biology, 13,
325– 33 7.
van Daalen, S.F. & Caswell, H. (2020) Varianc e as a li fe history out-
come: Sensitiv ity analysis of the contr ibutions of stochasticity
and heterogeneity. Ecological Modelling, 417, 108856.
Wright, S. (1931) Evolution in Mendelian populations. Genetics, 16,
97– 159.
SUPPORTING INFORMATION
Additional supporting information can be found online
in the Supporting Information section at the end of this
article.
How to cite this article: Steiner, U.K. &
Tuljapurkar, S. (2023) Adaption, neutrality and
life- course diversity. Ecology Letters, 26, 540 –548.
Available from: htt p s://doi.o rg /10.1111/el e .14174