Evolution of Stochastic Demography with Life History Tradeoffs in Density

Home , Density dependence, Population size, Theoretical ecology

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POPULATION INAUGURAL ARTICLE BIOLOGY 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, lnw ¯ = r, as a function of gene frequencies, describes an adaptive topography that governs both short-term and long-term evolution of gene frequencies by natural selection (3, 4, 22, 23). A similar adaptive topography has been derived for the evolution of mean phenotypes in quantitative characters (24, 25). The gradient of the adaptive topography (or selection gradient) at any given point, ∇r, is a vector with elements that are the partial derivatives of the log mean fitness in the population with respect to gene frequencies or mean phenotypes. The gradient dynamics imply that evolution in a constant environment is a hill-climbing process, continually increasing the mean fitness of the population until a local maximum in mean fitness, an adaptive peak, is achieved. In a fluctuating environment with a stationary distribution of environmental states and no autocorrelation, Lande (17, 18) showed 2 that the long-run growth rate of the population, r − σe /2, provides an unchanging adaptive topography governing the expected evolution. From the theory of stochastic demography, the long-run growth rate of a density-independent population gives the expected 2 (or asymptotic) rate of increase of ln N , where r is the expected growth rate in the average environment and σe is the environmental variance in growth rate (26–30). 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 g(N ) may often be nonlinear, as suggested by evidence from a variety of species (31, 32). θ θ 2 θ For the θ-logistic model of density dependence, g(N ) = N (33), evolution maximizes E[N ] = [1 − σe /(2r)]K . This simple result generalizes to stochastic environments MacArthur’s (5) finding that density-dependent selection in a constant environment maximizes the equilibrium population size, K . In a stochastic environment, genetic trade-offs can constrain adaptive evolution toward 2 increasing r and K and decreasing σe . This incorporates the trade-offs most often considered in the literature on r- and K -selection, by including the environmental variance in population growth rate. 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

n(t + 1) = L(N , t)n(t), [1a]

where the density-dependent projection matrix L(N , t) has nonnegative elements, the vital rates of age- or stage-speciﬁc annual survival and reproduction

Lij (N , t) = lij (g(N )) + ij (t) [1b]

with stochastic terms having mean E[ij (t)] = 0 and covariance Ciji0j 0 = E[ij (t)i0j 0 (t)]. Elements lij (g(N )) of the expected pro- P jection matrix l(g(N )) involve a density-dependent function g(N ), where N = i bi ni is a linear combination of the age or stage classes with nonnegative coefficients bi . In some cases it will be convenient to set the largest coefficient to 1, so that the contributions of all age classes to the population density are scaled to this unit. If all coefficients are unity then N is the total population size. All density dependence in the life history is therefore exerted by a single linear combination of age classes, weighted, for example, by body mass and/or metabolic rate, governing the utilization, and competition for, a limiting resource, acting through g(N ). Neglecting

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Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 g-tutr iuain hwarmral odfi oteoedmninldfuinapoiainfrsalvle of values small for approximation diffusion al. one-dimensional et Lande the to fit good remarkably a show simulations increase age-structure generally of weights that values such environ- resource, different combination limiting highest linear a the positive of Information ). both a utilization (Supporting by have their size exerted to to body is assumed proportion be and/or which in to are age 54), weighted as (53, classes with rare individuals dependence age the so with density youngest are until classes strongest the usually age stage the which with of suffer of individuals to change prebreeding rates oldest and can vital a the (47–52) sensitivity The in Assuming stochasticity environmental survival senescence (44–46). mental older. of and neglecting importance probability dependence or age, demographic the density of age of little their independent certain factor become and of a multiplicative they rates which a of Vital at include (30). adults (prere- stage fecundities life all final juvenile annual of including the of year one-year-olds, number class first being matrix a the stage class during Standard of final age composed species. first a population passerine the and stage-structured with and classes census, a raptor age of typical immature simulations a and stochastic representing productive) for histories used life were avian methods different (Eqs. projection two model for age-structured (Appendix) full approximation the of simulations Stochastic stationary the size, population small at of effects distribution positive, Allee stationary is (Appendix), and increase the of stochasticity for rate approximation demographic intrinsic diffusion Neglecting stochastic the (31). when Engen exists distribution and Diserud by ticity the in hc a eue ofidtesainr itiuinof distribution stationary the find to used be can which The (26–30). classes) stage or age of combination linear therefore, positive is, any time discrete (or in size size population population log of of dynamics log stochastic the the for of approximation increase one-dimensional of rate expected the that assume We 43). (26–30, 33). (29, dependence density nonlinear with structure) age (lacking −∂ population one-dimensional a for form standard a is This that so eigenvalue dominant r where the For 41). 40, (29, perturbation small a Eq. Introduction), the in (discussed environment average the in increase of rate intrinsic from small extrapolated be a For dependence. density of where 2) Thus, (29). r smoothly and of dynamics neighborhood nonchaotic with space with eigenvalue parameter species dominant of many region for average the 38). even the (37, in in applies perturbations located stages This stochastic life therefore (34). to prereproductive are year responses of species damped mortality per Most nonselective) 0.1 36). and about (density-independent (35, below high environment usually to due are fecundity, increase high of potentially rates intrinsic realized reproduction, matrix projection expected the of density elements or nonzero size that population such small at effects Allee 0 0 , i.1adFg opr h enadS of SD and mean the compare 2 Fig. and 1 Fig. h otnostm ifso prxmto oEq. to approximation diffusion continuous-time The is size population current the given population the of rate growth long-run The of elasticity the by defined is history life the in dependence density of strength overall The Eq. in rate growth population of expansion The ntesohsi oe h niomna ainei ouaingot aena h eemnsi qiiru is equilibrium deterministic the near rate growth population in variance environmental the model stochastic the In h eemnsi oe nteaeaeevrnetteeoei sue ohv tbeeulbimat equilibrium stable a have to assumed is therefore environment average the in model deterministic The of intervals annual discrete and kg 1 above weights body adult with vertebrates for particularly and populations, natural most For = K ln r and , λ/∂ heeeto h eemnsi rjcinmatrix projection deterministic the of element ijth sthen is (0), ξ γ P (t ij ) = i ln σ u sanral itiue admvral ihmean with variable random distributed normally a is r −∂ e 2 0 i N 1 = r eie rmtefl arxpoeto oe o h g structure. age the for model projection matrix full the from derived are sa motn aaee oenn ouainflcutosaon h qiiru raeaesz 2,42). (29, size average or equilibrium the around fluctuations population governing parameter important an is l vlae at evaluated ij g /∂ and r (K 0 θ g ln iheape ftepoaiiydsrbto of distribution probability the of examples with along , = λ g λ(g ) vlae at evaluated P (N h rtodrepninof expansion first-order the γ 1 = g i (N (K v ) i r epcieytesal g itiuinadrpoutv auseautdat evaluated values reproductive and distribution age stable the respectively are near u )) N n usiuigti nEq. in this substituting and ), i 1 = fteepce rjcinmatrix projection expected the of = l g ij K h coefficient The . (K (g g hc ie h xetdrt frtr oteeulbimo vrg ouainsz following size population average or equilibrium the to return of rate expected the gives which , = (N ) to g ln )) (K g θ N eed ekyon weakly depends Defining 0. = (0) lgsi model, -logistic h ih n eteigenvectors left and right The ). (t r )=ln = 1) + (g σ g e 2a 2 (N r (N v r (g = N i slna ntefunction the in linear is ln = ) u )= )) (N /K iji X j 3 easm that assume we ln 0 steeatct ftepplto rwhrt ihrsett ml change small a to respect with rate growth population the of elasticity the is a nntsmlma n variance and mean infinitesimal has j )= )) ∂ N 0 N λ ntesmltoswt hs rmdfuinter sfntosof functions as theory diffusion from those with simulations the in N v X 2a ln a g (t i ij yields u /K or (N γ = λ/∂ + ) yields r j l g 1a v v (g 0 = N σ (N i i = ) [1 variance 0, n t oet,dpn nyon only depend moments, its and , u 0 e 2 (N u r l and ln P j /(2r 2) sdn o the for done as (29), (g − (g ) j γ 0 N ij l )) ij othat so E[ (N ij g (N [g θ (N eecmae ihaayia omlsfrtediffusion the for formulas analytical with compared were 1b) v 0 eraewt increasing with decrease ξ rf 0 .29adrf 39). ref. and 209 p. 30, (ref. g i (K with ). )) ij )) u 0 0 = (0) )/g (t j steslto of solution the as γ s − ) )ξ n oatcreainadtepplto parameters population the and autocorrelation no and 1, ij σ = g − > θ (K σ e 2 h eemnsi nrni aeo nrae eoe as denoted increase, of rate intrinsic deterministic the , i u (N 0 r e N 2 g j snal osatadidpnetof independent and constant nearly is 0 and 0 + /2 at )]. (N (t h vrl tegho est eedneis dependence density of strength overall the 0, ) /K − ahrthan rather )] N )], σ v r e 2 o w ausof values two for σ 0 = (g ftedtriitcpoeto arxwt the with matrix projection deterministic the of /2 e (N ξ θ (t lgsi oe ihevrnetlstochas- environmental with model -logistic > n that and )) ), o h stochastic the For 0. λ M λ(g − oasalcag npplto density, population in change small a to N ln σ g (K opeev h osbenonlinearity possible the preserve to N e 2 (N oeacrtl hsgvsthe gives this accurately More /2. g = o ausof values For 1. = )) (N θ ). r n n opst parameter composite one and (g ) a (N nrae ihincreasing with increases ntelgsi oe.The model. logistic the in )) N − NSEryEdition Early PNAS = σ N θ e 2 lgsi oe,the model, -logistic K /2 = K n normalized and and N ie ythe by given , . g V (N ln r N ) 0 | 2a nthe in = ≤ a f9 of 3 [2b] [2a] [2c] can θ [4] [3] σ 0.1 for N r e 2 0 ,

POPULATION INAUGURAL ARTICLE BIOLOGY A B

2 Fig. 1. (A) Mean, E[N/K], and (B) SD, σN/K , as functions of the composite parameter a = σe /(2r0), in the stochastic θ-logistic model of density dependence for θ = 0.5 (red), 1 (black), and 2 (blue). (C) Stationary probability distribution φ(N/K) in the stochastic logistic model (θ = 1) for two values of a. Solid lines calculated from diffusion approximation (Appendix). Each point in A or B and each distribution in C gives statistics for a sample of 104 observations from a 6 stage-structured model of an avian raptor with age at ﬁrst breeding of 4 y and r0 = 0.04, simulated for 10 y and recorded once every 100 y (Supporting Information).

(Figs. 1 and 2) and remain close for values of a < 0.5, corresponding coefficients of variation in population size below one for the logistic model, CVN < 1 (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 2 size, must equal zero, E[r(g(N ))] − σe /2 = 0, and using Eq. 2b we find

Fig. 2. The same as Fig. 1 but with simulations for a passerine bird life history with age at ﬁrst breeding of 1 y and r0 = 0.1.

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(K 2r σ (¯ X r z eR 0 ij 2 0R γ I = ) (¯ ) (¯ z v z − − )[1 > L(¯ i ¯ )[g r (¯ σ σ g z g z (¯ z − (K z e e )u 2 2 (K (K , 1 /(2r /(2r , = N g j 0) r − R I (¯ (¯ (N 0I 8a , ) x z z ) t 2r sthen is )γ )) [1 0 0 σ )n + )/g )]g )]K a eetaoae from extrapolated be can eR − 2 0R ij − − (t e (¯ (K σ (K sdtrie ya diiegntccmoet(or component genetic additive an by determined is , z L(z g θ ) g eI 2 ) ¯ . l (N (N ij g ), /2 (¯ r (N (¯ (K , z 0 z R )], N θ ))]. (¯ , > )/g R z lgsi oe geswt Eq. with agrees model -logistic ¯ R l g , (¯ = ) )] ). t (N z 0, ) (K , rmEq. from g ihelements with γ )= )) (N I (¯ )] z )g a oiateigenvalue dominant has )), − E R (K othat so , r σ p 0R 5a eI (z 2 (¯ P z , suigwa eeto,so selection, weak Assuming /2. )l spositive, is , . n usiuigti nEq. in this substituting and )), g K ij (N R (z L and , P r ) , ij (¯ g z near = (z (N , g , G NSEryEdition Early PNAS σ N (N ))dz eR 2 g + , (K )=ln = )) t n aeinvader rare a and E = ) L(¯ ¯ 5b n stochastic a and h phenotype The . (¯ z z l , )) ij (Appendix). N (z λ(¯ to λ , , t (¯ z g g sused is ), z , 0 0. = (0) (N , g | g 5a (N (N f9 of 5 [5b] [8b] )+ )) [5a] [8a] or , [7] [6] )), ))

POPULATION INAUGURAL ARTICLE BIOLOGY In the stochastic model the environmental variance in population growth rate is 2 X ¯ ¯ σe (¯z) = vi (¯z)uj (¯z)vi0 (¯z)uj 0 (¯z)E[ξij (¯z, t)ξi0j 0 (¯z, t)]. [8c] iji0j 0 The stochastic demography of the population is therefore described by population parameters that are now functions of the mean 2 phenotype, r0(¯z), K (¯z), and σe (¯z). The expected (or long-run) growth rate of the population, conditional on the mean phenotype and 2 population density, is r(¯z, g(N )) − σe (¯z)/2. Neglecting demographic stochasticity and Allee effects, and provided that the stochastic 2 intrinsic rate of increase s(¯z) = r0(¯z) − σe (¯z)/2 is positive, a stationary distribution of population size exists. For a population ﬂuctuating around carrying capacity the expected growth rate through time equals zero, giving s(¯z) σ2(¯z) E[g(N )] = g(K (¯z)) = 1 − e g(K (¯z)) [9a] r0(¯z) 2r0(¯z) or, for the θ-logistic model, s(¯z) σ2(¯z) E[N θ] = [K (¯z)]θ = 1 − e [K (¯z)]θ, [9b] r0(¯z) 2r0(¯z) analogous to Eqs. 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, ∆¯z conditional on the mean phenotype and population density, is proportional to the additive genetic covariance matrix G times the gradient with respect to z¯ of the expected growth rate of the population as a function of z¯ and N , G g(N ) E[∆¯z|z¯, N ] = ∇ s(¯z) − r0(¯z) , [10] T g(K (¯z))

where ∇ is the gradient operator with respect to the mean phenotypes, a column vector with ith element ∂/∂z¯i . The generation time T is the average age of mothers of newborn individuals when the population is at the stable age distribution in the average environment (29, 30); it accounts for overlapping generations when the (annual) demographic time unit (Eq. 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 E[g(N )] using Eq. 9a, and rearrange to yield G r (¯z) E[∆¯z|z¯] = ∇s(¯z) − E[g(N )]∇ 0 T g(K (¯z)) G s(¯z) = s(¯z)∇ ln g(K (¯z)) . [11] T r0(¯z) The expected evolution therefore maximizes the function in Eq. 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

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(57, between them increase trade-off between of to correlation components environmental phenotypic that the positive than fitness, a of produce components major may any populations with as Second, (12). populations r below. discussed plasticity, and reducing parameters these for between between correlations maximize trade-off genetic classical to this positive evolution However, by (20). obscured al. et or Lande modified by structure be age without inheritance asexual the for and variance environmental between For 57). 14, (7, density matrix covariance genetic additive a their using as selection; manifested to G are response expected them of among lack a trade-offs by G implies expressed is equilibrium fitness evolutionary of An components major 70). the (57, between two correlation other genetic the negative on conditional selection: directional decrease, consistent to under is fitness of components and - prtn nterslcingradient, ∇ selection their on operating , h rsn oe sbsdo iceetm nulcnue tterpoutv esn suigrno niomna fluctu- environmental random assuming season, reproductive the at censuses annual discrete-time on based is model present The of expression simplest the parameters, dynamics population key three the among trade-offs genetic of complexity the of Regardless between trade-off simple a supporting evidence comparative and genetic limited the explain to help thus may considerations Three of periods alternating to subject thus are and fluctuate populations All E [N K θ slcin(2 1 2.Frti h ifiut fmauiggntcvrac n eei orltosbetween correlations genetic and variance genetic measuring of difficulty the is First 72). 71, (12, -selection r 0. = ] K and - n h niomna variance, environmental the and , σ e 2 K n hrfr nraigteepce teso ouaina n ouainsz,i hog h vlto of evolution the through is size, population any at population a of fitness expected the increasing therefore and , K slcintoward -selection θ r slcindpnso h antd fevrnetlstochasticity, environmental of magnitude the on depends -selection lgsi omo est eednei hteouini utaigevrnetmxmzsE maximizes environment fluctuating a in evolution that is dependence density of form -logistic 0 nseiswt eaaesxs ae n eae a ifrsbtnilyi opooyadlf itr.This history. life and morphology in substantially differ may females and males sexes, separate with species In and hntpcpatct sdsrbdb h omo ecingvn h xetdpeoyeo eoyeor genotype a of phenotype expected the giving reaction of norm the by described is plasticity Phenotypic r θ [1 0 lgsi est eedne(Eqs. dependence density -logistic or K − u eut r ae nsvrlasmtos n eaigte rsnsopruiisfrfuture for opportunities presents them relaxing and assumptions, several on based are results Our . σ K r e 2 slcinpealn ne ihevrnetlvrac.Ti a lutae ynmrclsimulations numerical by illustrated was This variance. environmental high under prevailing -selection /(2r ota h oiattaeofbcmsta between that becomes trade-off dominant the that so 0 θ r )]K silsrtdi is n 2. and 1 Figs. in illustrated as , slcini htevrnetlsohsiiyrdcsteaeaesz fapplto eo its below population a of size average the reduces stochasticity environmental that is -selection h xetdma tesfnto hti aiie yln-emeouini fluctuating a in evolution long-term by maximized is that function fitness mean expected The r 5 0 and and θ ∇E a cu oetruhreducing through more occur may K novstremjrcmoet htaekyprmtr nsohsi population stochastic in parameters key are that components major three involves 9) [N n/rpstv eei orltosbtenteeand these between correlations genetic positive and/or , σ θ e 2 ] l mlctfntoso h rmr hntpccharacters phenotypic primary the of functions implicit all , (where eei ainei h tegho est eedneo g-pcfi ia rates vital age-specific on dependence density of strength the in variance Genetic nppltoswt oaesrcueteln-u rwhde o eedon depend not does growth long-run the structure age no with populations In ∇ 5b (∂/∂ = and r 0 and ,i sepcal la htterslto ftegntctrade-off genetic the of resolution the that clear especially is it 9b), r 0 K ∂/∂ , K h esnwyalre niomna ainesit the shifts variance environmental larger a why reason The . r lasslce oices,and increase, to selected always are r K and - σ ∂/∂σ , e 2 ahrta increasing than rather K slcin hntepplto sa o rhigh or low at is population the when -selection, e 2 θ ) T r ,a a ese rmEqs. from seen be can as 1, = σ 0 and σ h arigcpct nteaeaeenvi- average the in capacity carrying the , e 2 σ e 2 ne ihevrnetlstochasticity, environmental high Under . e and 2 with , T θ niae arxtasoe,sc that such transpose), matrix indicates lgsi oe eei ainein variance genetic model -logistic r 0 or K slcinpealn ne low under prevailing -selection K r σ 0 r e 2 ahrta h traditional the than rather , 0 and eei rd-f among trade-off genetic A . nipratmechanism important An . σ K r NSEryEdition Early PNAS e 2 and - z ¯ (¯ ihnado among and/or within z aho h major the of Each . σ [N ) e 2 sawy selected always is K a estronger be may θ r hsprovides This ]. 5b slcinmay -selection and and K but 9b, | within f9 of 7 θ ,

POPULATION INAUGURAL ARTICLE BIOLOGY (85–87). For asexual inheritance, with other population parameters constant, ref. 20 showed that θ evolves to the interval (1.781, 2) 2 corresponding to σe in the range (0, ∞). 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, N /K , for the continuous-time stochastic θ-logistic model without age structure, was derived using diffusion theory, neglecting demographic stochasticity and Allee effects at small population size, θ 1 (1/a−1)/θ N 1/a−2 φ(N /K ) = Γ[(1/a − 1)/θ] aθ K ( ) (N /K )θ × exp − aθ

2 where a = σe /(2r0) (31). This distribution exists only if 0 < a < 1, corresponding to a positive stochastic intrinsic rate of increase, 2 r0 − σe /2 > 0 and has moments Γ[(1/a − 1 + b)/θ] E[(N /K )b ] = (aθ)b/θ Γ[(1/a − 1)/θ] which when b = θ reduces to E[(N /K )θ] = 1 − a as in Eqs. 4b and 8b. 2 2 The moments can be used to derive the mean E[N /K ], variance σN /K , and squared coefﬁcient of variation CVN , as illustrated in Figs. 1 and 2: Γ[1/(aθ)] E[N /K ] = (aθ)1/θ, Γ[(1/a − 1)/θ] ( ) Γ[(1/a + 1)/θ] Γ[(1/aθ)] 2 σ2 = − (aθ)2/θ, N /K Γ[(1/a − 1)/θ] Γ[(1/a − 1)/θ]

2 2 σN /K Γ[(1/a + 1)/θ]Γ[(1/a − 1)/θ] CVN = = − 1. (E[N /K ])2 (Γ[1/(aθ)])2 For the standard logistic model with θ = 1 these become a E[N /K ] = 1 − a, σ2 = a(1 − a), CV2 = . N /K N 1 − a

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.

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Sexual 83. a in alternative (1980) in displacement of R times character maintenance Lande development and evolutionary and regulation The 82. sizes Population (1992) optimal (2012) NA on TD Moran rates Price growth R, 81. flexible Lande of EE, populations. effect Goldberg age-structured The density-independent (1996) 80. of C Hedenstr demography Wiklund T, the S, Alerstam Nylin on environment. O, 79. environments fluctuating Leimar a autoccorrelated PA, in temporally Abrams character of quantitative effect 78. labile the a Estimating assimilation. of (2013) genetic tolerance and al, environmental plasticity et and phenotypic plasticity S, of phenotypic shape. Engen evolution of norm by Evolution reaction 77. environment (2014) of extraordinary R Evolution an V. Lande plasticity. to phenotypic Adaptation 76. of (2009) genetics R The Lande (1993) SM 75. Scheiner S, for Gavrilets Evidence (1998) M 74. (2016) Pigliucci CD, E Schlichting Engen histories. 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POPULATION INAUGURAL ARTICLE BIOLOGY