# Evolution of stochastic demography with life history tradeoffs in density-dependent age-structured populations

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Contributed by Russell Lande, September 11, 2017 (sent for review June 14, 2017; reviewed by Robert Frekleton and Nigel G. Yoccoz)

## Significance

We derive a univariate approximation for the growth of a density-dependent age-structured population in a fluctuating environment. Its accuracy is demonstrated by comparison with simulations of the age-structured model under assumptions applicable to many vertebrate populations. This facilitates extension to age-structured populations of recent theory on the evolution of stochastic population dynamics, showing that evolution tends to maximize a simple function of three key population parameters: the maximum population growth rate and carrying capacity in the average environment,

## Abstract

We analyze the stochastic demography and evolution of a density-dependent age- (or stage-) structured population in a fluctuating environment. A positive linear combination of age classes (e.g., weighted by body mass) is assumed to act as the single variable of population size, N, exerting density dependence on age-specific vital rates through an increasing function of population size. The environment fluctuates in a stationary distribution with no autocorrelation. We show by analysis and simulation of age structure, under assumptions often met by vertebrate populations, that the stochastic dynamics of population size can be accurately approximated by a univariate model governed by three key demographic parameters: the intrinsic rate of increase and carrying capacity in the average environment,

Classical population genetic models assuming a constant population size or density in a constant environment (1⇓⇓⇓–5) show that evolution maximizes the mean fitness of individuals in the population. The ecological meaning of this result is that when population dynamics is density-dependent, but selection is not, evolution in a constant environment maximizes the intrinsic rate of population increase,

Density-dependent selection happens when the relative fitness of genotypes differs in response to change in population density. MacArthur (5) showed that density-dependent selection in a constant environment maximizes the equilibrium population size or carrying capacity, K. This result for density-dependent selection in a constant environment was extended to an age-structured population by Charlesworth (7) for the case where all density dependence in the life history is exerted by a single age or stage class (e.g., adults), showing that evolution maximizes the size of that class.

Contrasting conclusions for density-independent and density-dependent selection in a constant environment led to the conceptual theory of r- and K-selection, proposing a constraint or trade-off between these variables within populations, such that K-selection prevails in stable populations in nearly constant environments, while r-selection prevails in highly variable populations in strongly fluctuating environments (8⇓⇓⇓⇓–13). The dichotomous classification of species being either r- or K-selected lost popularity when more cognizance was taken that r and K are emergent properties of age-structured populations, with complex trade-offs between survival and fecundity among age classes. Furthermore, in every fluctuating population r- or K-selection prevails, respectively, when population density is low or high, corresponding on average to an increasing or decreasing population (7, 14).

Initial analytical investigations of evolution in fluctuating environments for populations without age structure were made by Dempster (15) for single-locus haploid or asexual inheritance, and by Haldane and Jayakar (16) for a single diallelic locus in an obligately sexual, randomly mating diploid population. They modeled random environmental fluctuations with a stationary distribution of fitnesses for each genotype, implicitly assuming density-independent selection, and found a simple condition that determines the final evolutionary outcome. For a haploid or asexual population, the genotype with the highest geometric mean fitness eventually becomes fixed. For a diallelic locus in a diploid randomly mating population, if a homozygote has the highest geometric mean fitness it eventually becomes fixed, but if the heterozygote has the highest mean fitness a genetic polymorphism is maintained by fluctuating selection (neglecting fixation by genetic drift in a finite population). These conclusions are especially interesting because the genotype with the highest geometric mean fitness may not be that with the highest arithmetic mean fitness, if the latter occasionally experiences very low fitness; thus, genotypes that are well-buffered against catastrophic loss of fitness in extreme environments may have a decisive long-term evolutionary advantage. For polygenic characters, fluctuating density-independent selection generally does not act to maintain genetic polymorphism except possibly at a single locus (17).

A common misinterpretation of these classical results on fluctuating selection still pervades the evolutionary and ecological literature, which is that the geometric mean fitness of a genotype in a fluctuating environment measures the selection on it. This interpretation is erroneous because fitness is supposed to predict short-term changes in gene frequencies over time spans on the order of one generation, whereas the geometric mean fitness only predicts the final outcome of selection. For density-independent selection in a fluctuating environment, the correct measure of the expected relative fitness of a genotype is its Malthusian fitness (the intrinsic rate of increase in the average environment) minus the covariance of its fitness with that of the population (17⇓–19). A similar definition of expected mean fitness of a genotype also applies with density-dependent selection, with the Malthusian fitness conditioned on the current population density (20, 21).

In a constant environment, assuming constant genotypic fitnesses and approximate linkage equilibrium among loci, the log of mean fitness in a population,

In a fluctuating environment with a stationary distribution of environmental states and no autocorrelation, Lande (17, 18) showed that the long-run growth rate of the population,

Density-dependent selection in a fluctuating environment presents the additional complication that stochastic evolution is coupled with the stochastic dynamics of population size. Lande et al. (20) and Engen et al. (21) analyzed density-dependent selection in a fluctuating environment for populations without age structure, respectively both for asexual inheritance and for quantitative (polygenic) characters in an obligately sexual population. Neglecting genetic drift and demographic stochasticity due to small population size, they showed that evolution tends to maximize the expected density dependence averaged over the distribution of population size, E[*g*(*N*)]. The density-dependence function

Explicit age or stage structure is essential for any theory of selection, inheritance, and evolution of life histories. Here we extend the recent results on the evolution of stochastic demography and life-history trade-offs (20, 21) by assuming that all density dependence in the life history is exerted through a single positive linear combination of age classes, defined as the population size or density. The age-specific vital rates are all affected by the same possibly nonlinear function of population density but may differ in their sensitivity to population density and environmental stochasticity. We obtain parallel results for asexual inheritance and polygenic inheritance of quantitative traits.

## Stochastic Demography of Population Size

We analyze the stochastic density-dependent demography of an age- or stage-structured population that is genetically monomorphic, as in classical demographic theory neglecting genetic variation and evolution, to derive an approximation for the expected growth rate of the log of population size, N, as a function of its current size.

The dynamics of the age- (or stage-) vector n are

For most natural populations, and particularly for vertebrates with adult body weights above 1 kg and discrete annual intervals of reproduction, realized intrinsic rates of increase are usually below about 0.1 per year (34). This applies even for many species with potentially high fecundity, due to high (density-independent and nonselective) mortality of prereproductive life stages in the average environment (35, 36). Most species are therefore located in the region of parameter space with nonchaotic dynamics and smoothly damped responses to stochastic perturbations (37, 38).

The deterministic model in the average environment therefore is assumed to have a stable equilibrium at *ij*th element of the deterministic projection matrix

The expansion of population growth rate in Eq. **2a** is linear in the function **2a** can be extrapolated from **2a** yields

The overall strength of density dependence in the life history is defined by the elasticity of λ to a small change in population density,

In the stochastic model the environmental variance in population growth rate near the deterministic equilibrium is

The long-run growth rate of the population given the current population size is

The continuous-time diffusion approximation to Eq. **3** has infinitesimal mean and variance *Appendix*),**1a** and **1b**) were compared with analytical formulas for the diffusion approximation (*Appendix*) for two different avian life histories representing a typical raptor and passerine species. Standard matrix projection methods were used for stochastic simulations of a stage-structured population composed of a number of juvenile (prereproductive) and immature age classes and a final stage class including all adults of a certain age or older. Assuming a prebreeding census, with the first age class being one-year-olds, the annual fecundities include a multiplicative factor of the probability of survival during the first year of life (30). Vital rates and their density dependence and environmental sensitivity can change with age until the final stage at which they become independent of age, neglecting senescence in the oldest individuals which usually are so rare as to be of little demographic importance (44⇓–46). The vital rates of the youngest age classes are assumed to have both the highest environmental stochasticity (47⇓⇓⇓⇓–52) and to suffer the strongest density dependence (53, 54), which is exerted by a positive linear combination of stage classes with individuals weighted in proportion to their utilization of a limiting resource, such that weights generally increase with age and/or body size (*Supporting Information*).

Fig. 1 and Fig. 2 compare the mean and SD of *Appendix*). Excluding estimation error variance, most vertebrate populations with long time series have small or moderate coefficients of variation (30).

## Evolution of Stochastic Demography

### Asexual Inheritance and Evolution.

For a genetically monomorphic population, such as a single asexual genotype, averaging both sides of Eq. **3** through time it can be seen that the expected long-run growth rate, averaged over the stationary distribution of population size, must equal zero, **2b** we find**5b** (*Appendix*).

For competition between two asexual genotypes, the final outcome of evolution is determined by the long-run growth rates of the genotypes, so it is sufficient to analyze only the invasion of a rare genotype into a population of a common resident genotype. Following Lande et al. (20), consider a resident genotype with population dynamics parameters **5a**, is positive,**5a**, or for the θ-logistic model that in Eq. **5b**, as in Lande et al. (20).

### Inheritance and Evolution of Quantitative Characters.

For populations with various combinations of age structure, density dependence of population dynamics and selection, fluctuating environments, and for different mechanisms of inheritance, evolution has been shown to depend on the gradient of a suitable mean fitness function with respect to the evolutionary variables. For selection in a fluctuating environment with a stationary distribution of environmental states and no autocorrelation, the expected evolution depends on the gradient of the long-run (or expected) growth rate of the population, conditioned on its current genetic state, and on its current population density if selection is density-dependent (17, 18, 20, 21, 55, 56). We analyze the evolution of a vector of mean phenotypes,

#### Expected Population Growth Rate.

For a population without age structure, in a fluctuating environment with a stationary distribution of environmental states and no autocorrelation, the mean fitness function governing both the stochastic demography and evolution is the long-run growth rate, conditioned on the current population density if selection is density-dependent (17⇓⇓⇓–21). Here we generalize this to age-structured populations using reproductive value weighting and then apply the gradient operator to derive the expected evolution of the mean phenotype.

The characters of individuals are assumed to be measured on scales such that their phenotypic and additive genetic (breeding) values follow multivariate normal distributions with constant variances and covariances independent of the mean phenotypes and population density (57⇓–59). The vector of individual phenotypes,

Each phenotype z has a matrix of vital rates of survival and reproduction

The deterministic projection matrix for the population in the average environment,

For a small intrinsic rate of increase in the average environment, Eq. **8a** can be extrapolated from **8a** yields**4a** and **4b**.

#### Expected Evolution of the Mean Phenotype.

Having reduced the age-structured population dynamics to the one-dimensional variable of population density, previous analyses (18, 21, 56) indicate that the expected evolutionary change in the mean phenotype vector per unit time, *i*th element **1**) differs from the generation time for inheritance (7, 57, 60). [The continuous-time models of Lande (18) and Engen et al. (21) implicitly use the time unit of generations.]

Again, we assume weak selection so that evolution is slow compared with population fluctuations, average both sides of Eq. **10** over the stationary distribution of population sizes, substitute for **9a**, and rearrange to yield**9a**, or for the θ-logistic model that in Eq. **9b**, as in Engen et al. (21). This result now allows interpretation of life-history trade-offs in r- and K-selection for age- and stage-structured populations.

## Discussion

Evolutionary maximization of the expected mean fitness in Eqs. **5** and **9** describes the expectation of a stochastic process. For asexual inheritance this is realized by eventual fixation of the genotype with the highest value of the function, with large excursions and repeated reversals of gene frequency evolution (20). With sexual reproduction and polygenic inheritance of quantitative traits, in a model without age structure, Engen et al. (21) performed numerical simulations of the joint evolution of mean phenotype and population size. After an initial stochastic increase, sample paths of the expected mean fitness function in Eq. **9b** fluctuated in a narrow range around the maximum, despite substantial fluctuations in population size. Deterministic approximations for the growth rate of an age- or stage-structured population using reproductive values, as in Eqs. **2a** and **8a**, have been shown quite accurate when the perturbations multiplied by their sensitivity coefficients are small (29, 30). Evidence from a variety of species shows that the age- or stage-specific vital rates subject to higher temporal variance have smaller sensitivity coefficients (47⇓⇓–50, 61), including both environmental stochasticity and density dependence operating through fluctuating population size. Recent studies separating these components of variance in vital rates indicate that the youngest and oldest age classes, which typically have the lowest sensitivity coefficients, are most subject to both environmental stochasticity and fluctuating density dependence (51, 53, 54).

### Reproductive Value Weighting of Individuals.

For populations lacking age structure, with discrete nonoverlapping generations, the geometric mean fitness of a genotype through time determines the final outcome of selection, regardless of temporal autocorrelation in the fitnesses (62). Environmental stochasticity in an age-structured population causes transient fluctuations in age structure (associated with complex eigenvalues of the projection matrix) even when the environment is not autocorrelated; this creates temporal autocorrelation in the genetic and phenotypic composition of a population that complicates the measurement and analysis of phenotypic evolution. These complications can be largely overcome by weighting individuals of different ages by their reproductive values, as first suggested by Fisher (1, 2). Reproductive value weighting is often used in life-history theory (7, 14, 57, 63, 64) and has been shown numerically to act as an efficient filter for minimizing autocorrelated fluctuations in population size and gene frequency caused by transient dynamics of population size, genetic composition, selection, and evolution (7, 14, 55, 65⇓⇓–68). In the estimation of population parameters, particularly the expectation and environmental variance of population growth rate, reproductive value weighting, such as that in Eqs. **2a** and **2c** and **8a** and **8c**, removes the bias caused by transient fluctuations in age structure (68, 69). Our simulations show that this approach also provides accurate results for the dynamics of total size of density-dependent age-structured populations in a stochastic environment, provided that the fluctuations of population size around carrying capacity are not very large (Figs. 1 and 2). The present theory of life history evolution in density-dependent age-structured populations in a fluctuating environment is therefore likely to have good accuracy, given the basic assumptions of a small intrinsic rate of increase and weak selection.

### Life History Trade-Offs and Evolution.

The expected mean fitness function that is maximized by long-term evolution in a fluctuating density-dependent population (Eqs. **5** and **9**) involves three major components that are key parameters in stochastic population dynamics and life history: the intrinsic rate of increase in the average environment, * ^{T}* indicates matrix transpose), such that

All populations fluctuate and are thus subject to alternating periods of r- and K-selection, when the population is at low or high density (7, 14, 57). For θ-logistic density dependence (Eqs. **5b** and **9b**), it is especially clear that the resolution of the genetic trade-off between r- and K-selection depends on the magnitude of environmental stochasticity,

Three considerations may thus help to explain the limited genetic and comparative evidence supporting a simple trade-off between r- and K-selection (12, 71, 72). First is the difficulty of measuring genetic variance and genetic correlations between r and K within populations (12). Second, as with any major components of fitness, the environmental components of

Regardless of the complexity of genetic trade-offs among the three key population dynamics parameters, the simplest expression of our results for the θ-logistic form of density dependence is that evolution in a fluctuating environment maximizes **5b** and **9b**, but also holds for any positive value of θ, as illustrated in Figs. 1 and 2.

### Phenotypic Plasticity.

Phenotypic plasticity is described by the norm of reaction giving the expected phenotype of a genotype or population as a function of the environment in which it develops (73). Unmeasured microenvironmental variation among individuals affecting their phenotypic development and developmental noise contributes to the incomplete heritability of quantitative characters (58, 59). Measurable macroenvironmental effects such as temperature and precipitation often occur on a regional scale, producing parallel effects on the phenotypic development of all individuals in a population. Models of the evolution of phenotypic plasticity show that its evolution depends on the environmental predictability over developmental time lag between critical stages environmental influence on development and phenotypic selection (74⇓–76).

The present model is based on discrete-time annual censuses at the reproductive season, assuming random environmental fluctuations with no autocorrelation between years. Critical development events occurring within years allow adaptive plasticity to evolve based on short-term environmental predictability within years. Simple models of phenotypic evolution in fluctuating environments with linear norms of reaction indicate that the main effect of adaptive plasticity is to reduce the intensity of fluctuating selection (18). The main conclusions of the present models should therefore still be applicable with a constant level of plasticity.

### Limitations of the Model.

Our results are based on several assumptions, and relaxing them presents opportunities for future investigation.

#### Autocorrelated environments and complex life cycles.

In populations with no age structure the long-run growth does not depend on whether the environment is autocorrelated, but with age structure environmental autocorrelation does affect the long-run growth rate (30, 43, 77). Life history evolution in autocorrelated and cyclical environments is not well understood despite much biological attention especially for seasonal environments. Adaptation to such environments often involves complex life cycles, including morphological transformation and seasonal migration, with a major change in habitat, ecology, and density regulation at different life history stages (78⇓⇓–81). In such populations density dependence may not operate through a single variable of population size or a positive linear combination of age classes (51).

#### Sexual dimorphism.

In species with separate sexes, males and females may differ substantially in morphology and life history. This produces a disparity between ecological and evolutionary definitions of mean fitness. Nevertheless, it is generally possible to extend the basic results for monoecious populations to those with sexual dimorphism (e.g., refs. 60 and 82). Sexual selection typically causes selection on one sex (usually males) to be frequency-dependent, violating an assumption necessary for the evolutionary maximization of mean fitness (23, 82, 83). Frequency-dependent selection also arises for asymmetric density-dependent resource competition (84).

#### Genetic variance in the form of density dependence.

Genetic variance in the strength of density dependence on age-specific vital rates is allowed in the present theory but not in the functional form of density dependence. In the θ-logistic model genetic variance in θ, in combination with heritable variation in other population parameters, creates frequency-dependent selection which produces complications that have prevented any general conclusions about what evolution maximizes in stochastic density-dependent populations (85⇓–87). For asexual inheritance, with other population parameters constant, ref. 20 showed that θ evolves to the interval

#### Delayed density dependence.

Although chaotic population dynamics is precluded by realistic intrinsic rates of increase in the average environment, long time lags in density dependence interacting with environmental stochasticity can create a tendency for sustained oscillations. Multiyear time lags can arise from the life history of a population with high adult annual survival and/or high age at first reproduction (29). Multigenerational time lags can occur because of density-dependent interspecific interactions, especially predation (88).

## Appendix

The stationary distribution of population size relative to carrying capacity, **4b** and **8b**.

The moments can be used to derive the mean

## SI Text

In simulations of the full age-structured model we assume that elements of the projection matrix are linear in

Stochasticity is introduced by replacing

## Raptor Life History

The annual stage projection matrix for a raptor life history is

## Passerine Life History

The annual stage projection matrix for a passerine life history is

## Acknowledgments

We thank the reviewers for critical comments. This work was supported by a Center of Excellence grant from the research Council of Norway (SFF-III Project 223257) and the Norwegian University of Science and Technology.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: r.lande{at}imperial.ac.uk.

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2015.

Author contributions: R.L. designed research; R.L. and S.E. developed the models; S.E. and B.-E.S. performed simulations; and R.L., S.E., and B.-E.S. wrote the paper.

Reviewers: R.F., University of Sheffield; and N.G.Y., University of Tromso.

Conflict of interest statement: The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1710679114/-/DCSupplemental.

Published under the PNAS license.

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