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Downloaded from on: Apr 13, 2022Copepod life strategy and population viability in response to prey timing andtemperature: Testing a new model across latitude, time, and the size spectrumBanas, Neil S.; Møller, Eva Friis; Nielsen, Torkel Gissel; Eisner, Lisa B.Published in:Frontiers in Marine ScienceLink to article, DOI:doi: 10.3389/fmars.2016.00225Publication date:2016Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Banas, N. S., Møller, E. F., Nielsen, T. G., & Eisner, L. B. (2016). Copepod life strategy and population viabilityin response to prey timing and temperature: Testing a new model across latitude, time, and the size spectrum.Frontiers in Marine Science, 3, [225]. 10.3389/fmars.2016.00225General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyrightowners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portalIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

ORIGINAL RESEARCHpublished: 15 November 2016doi: 10.3389/fmars.2016.00225Copepod Life Strategy andPopulation Viability in Response toPrey Timing and Temperature:Testing a New Model across Latitude,Time, and the Size SpectrumNeil S. Banas 1*, Eva F. Møller 2 , Torkel G. Nielsen 3 and Lisa B. Eisner 41Department of Mathematics and Statistics, University of Strathclyde, Glasgow, UK, 2 Department of Bioscience, ArcticResearch Center, Aarhus University, Roskilde, Denmark, 3 Section for Ocean Ecology and Climate, National Institute ofAquatic Resources, Technical University of Denmark, Charlottenlund, Denmark, 4 NOAA Fisheries, Alaska Fisheries ScienceCenter, Seattle, WA, USAEdited by:Dag Lorents Aksnes,University of Bergen, NorwayReviewed by:Øyvind Fiksen,University of Bergen, NorwayNicholas R. Record,Bigelow Laboratory for OceanSciences, USA*Correspondence:Neil S. [email protected] section:This article was submitted toMarine Ecosystem Ecology,a section of the journalFrontiers in Marine ScienceReceived: 03 June 2016Accepted: 27 October 2016Published: 15 November 2016Citation:Banas NS, Møller EF, Nielsen TG andEisner LB (2016) Copepod LifeStrategy and Population Viability inResponse to Prey Timing andTemperature: Testing a New Modelacross Latitude, Time, and the SizeSpectrum. Front. Mar. Sci. 3:225.doi: 10.3389/fmars.2016.00225A new model (“Coltrane”: Copepod Life-history Traits and Adaptation to NovelEnvironments) describes environmental controls on copepod populations via (1)phenology and life history and (2) temperature and energy budgets in a unified framework.The model tracks a cohort of copepods spawned on a given date using a set of coupledequations for structural and reserve biomass, developmental stage, and survivorship,similar to many other individual-based models. It then analyzes a family of cases varyingspawning date over the year to produce population-level results, and families of casesvarying one or more traits to produce community-level results. In an idealized globalscale testbed, the model correctly predicts life strategies in large Calanus spp. rangingfrom multiple generations per year to multiple years per generation. In a Bering Seatestbed, the model replicates the dramatic variability in the abundance of Calanusglacialis/marshallae observed between warm and cold years of the 2000s, and indicatesthat prey phenology linked to sea ice is a more important driver than temperature per se.In a Disko Bay, West Greenland testbed, the model predicts the viability of a spectrum oflarge-copepod strategies from income breeders with a adult size 100 µgC reproducingonce per year through capital breeders with an adult size 1000 µgC with a multiple-yearlife cycle. This spectrum corresponds closely to the observed life histories and physiologyof local populations of Calanus finmarchicus, C. glacialis, and Calanus hyperboreus.Together, these complementary initial experiments demonstrate that many patterns incopepod community composition and productivity can be predicted from only a fewkey constraints on the individual energy budget: the total energy available in a givenenvironment per year; the energy and time required to build an adult body; the metabolicand predation penalties for taking too long to reproduce; and the size and temperaturedependence of the vital rates involved.Keywords: zooplankton, copepod, life history, diversity, biogeography, modeling, community ecology, ArcticFrontiers in Marine Science www.frontiersin.org1November 2016 Volume 3 Article 225

Banas et al.Copepod Life History Strategy Modeling1. INTRODUCTIONcopepods phytoplankton and protist prey: Mackas et al., 2012).This logic provides a route by which the energetics of fish,seabird, and mammal foraging are tied to temperature andphytoplankton phenology via the tradeoffs governing copepodlife history.There is likely a gap, then, between the focus of conventionaloceanographic plankton models—total productivity byfunctional group—and the copepod traits of greatest importanceto predators. A number of dynamical-modeling studies haveattempted to fill this gap by modeling the copepods species byspecies in relation to climate forcing, often in an individualbased-model (IBM) framework (Miller et al., 2002; Ji et al.,2012; Maar et al., 2013; Wilson et al., 2016). There are two keylimitations to the species-by-species approach, however. First, itis difficult to see how it can scale or generalize to the communitylevel, given that our empirical information on the physiologyand life history of the copepods is a patchwork, and realisticallywill always remain so. Second, it does not address the questionof adaptation, either on the individual or species level. Asindividuals make use of their phenotypic plasticity in behavior,physiology, and life cycle, and as natural selection acts onexisting species and subpopulations, it is likely that shifts in thebiogeography of copepod traits such as size, lipid content, and lifehistory pattern will not move in lockstep with the biogeographyof existing species (Barton et al., 2013). Indeed, subpopulationsof individual copepod species display so much life-history andphysiological diversity (Heath et al., 2004; Daase et al., 2013)that it is not clear what the basic units of a general species-basedmodel would even be. Observations of hybridization amongspecies (Parent et al., 2015) only underscore this problem.This paper presents a proof-of-concept for a trait-based, asopposed to species-based, copepod IBM, intended for eventualuse in problems linking planktivores to climate and environmenton global or regional scales. Record et al. (2013) presented acopepod community IBM in which explicit competition via agenetic algorithm was used to pick community assemblages outof a trait-based metacommunity along a latitudinal gradient. Thatstudy was concerned mainly with the emergent behavior of a verycomplex model system (predation-structured competition alongwith the interacting effects of six variable traits). In contrast, wehave included as few explicitly variable traits as possible, guidedby a strategic set of heuristic and quantitative comparisonswith data (Figure 1). The balance point we have sought in thisphase of work is the lightest-weight representation of diversityand plasticity that allows the model to (1) generate a realisticlandscape of competitors in a single environment, (2) correctlypredict fitness fluctuations in one population as a function ofhabitat, and (3) give sensible results over a wide biogeographicrange.The first of these criteria, captured by a Disko Bay, WestGreenland model experiment (Figure 1, Section 3.4) is central tothe goal of eventually allowing climate-to-copepod model studiesto replace hand-picked sets of fixed types with a trait continuum.The second and third criteria (captured by a Bering Sea hindcastexperiment and an heuristic, idealized biogeographic experiment:Figure 1, Sections 3.2, 3.3) provide complementary constraintson the parameterization of individual energetics, and helpCalanoid copepods occupy a crucial position in marine foodwebs, the dominant mesozooplankton in many temperate andpolar systems, important to packaging of microbial productionin a form accessible to higher predators. They also representthe point at which biogeochemical processes, and numericalapproaches like NPZ (nutrient–phytoplankton–zooplankton)models, start to be significantly modulated by life-history andbehavioral constraints. The population- and community-levelresponse of copepods to environmental change (temperature,prey availability, seasonality) thus forms a crucial filter lyingbetween the biogeochemical impacts of climate change onprimary production patterns and the food-web impacts thatfollow.Across many scales in many systems, the response offish, seabirds, and marine mammals to climate change hasbeen observed, or hypothesized, to follow copepod communitycomposition more closely than it follows total copepod or totalzooplankton production. Examples include interannual variationin pollock recruitment in the Eastern Bering Sea (Coyle et al.,2011; Eisner et al., 2014), interdecadal fluctuations in salmonmarine survival across the Northeast Pacific (Mantua et al., 1997;Hooff and Peterson, 2006; Burke et al., 2013), and long-termtrends in forage fish and seabird abundance in the North Sea(Beaugrand and Kirby, 2010; MacDonald et al., 2015). These casescan be all be schematized as following the “junk food” hypothesis(Österblom et al., 2008) in which the crucial axis of variationis not between high and low total prey productivity, but ratherbetween high and low relative abundance of large, lipid-rich preytaxa.Calanoid copepods range in adult body size by more than twoorders of magnitude, from 10 to 1000 µg C. Lipid content islikewise quite variable (Kattner and Hagen, 2009), even amongcongeneric species in a single environment (Swalethorp et al.,2011). Many but not all species enter a seasonal period ofdiapause in deep water, in which they do not feed and basalmetabolism is reduced to 1/4 of what it is during active periods(Maps et al., 2014). Reproductive strategies include both incomebreeding (egg production fueled by ingestion of fresh prey duringphytoplankton blooms) and capital breeding (egg productionfueled by stored lipids in winter), as well as hybrids betweenthe two strategies (Hirche and Kattner, 1993; Daase et al., 2013).Generation lengths vary from several weeks to several years.These life-history traits (generation length, diapause,reproductive strategy, and annual routine more generally)constitute the mechanistic link between environment and thequality of the copepod community as prey (i.e., body size andcomposition). Lipid storage, coupled to diapause in deep water, isa strategy for surviving the winter in environments where winterforaging is not cost-effective energetically; and just as important,it provides energetic free scope for optimizing reproductivetiming relative to prey availability (Falk-Petersen et al., 2009;Varpe et al., 2009). Lipid storage is tied to climate via temperature(which determines the rate at which an animal burns throughits reserves during winter and rates of ingestion, growth, anddevelopment year-round) and phenology (i.e., timing of theFrontiers in Marine Science www.frontiersin.org2November 2016 Volume 3 Article 225

Banas et al.Copepod Life History Strategy ModelingFIGURE 1 (A) Locations of model testbeds. The “global” model experiment spans a gradient from approximately Ice Station Sheba to Newport, Oregon, andbeyond. This experiment and the Bering Sea and Disko Bay testbeds constitute (B) a complementary set examining variation in space, time, and size called the “potential” or ϕ model and the fuller version the“egg/reserve” or ER model.The ϕ and ER models take different approaches to generatingpopulation-level results from this cohort model, as explainedin detail below (Section 2.4). In both cases, the logic changesfrom the simple forward time-integration at the cohort level:one runs the cohort model for all possible spawning datest0 , retroactively determines which spawning dates would proveoptimal or sustainable, and considers the cohort time series fromthose t0 values, appropriately weighted, to constitute the modelsolution (Section 2.4). The biological logic here is similar to thebackwards-in-time dynamical optimization method frequentlyused in studies of optimal annual routines (Houston et al., 1993;Varpe et al., 2007), although our solving method is quite differentand less exact. This is a compromise with the eventual goalof coupling Coltrane to oceanographic models as a spatiallyexplicit IBM.Communities are generated in Coltrane 1.0 simply by runningfamilies of cases of the population-level model that vary oneor more traits. Treating coexisting populations as uncoupledvastly simplifies the interpretation of the landscape of viablestrategies in a given environment, or the fundamental niche ofa particular trait combination, our primary modes of analysis.At the same time, it tightly restricts our choices regarding theformulation of predation mortality. In reality, coupling throughshared predators can rival bottom-up effects as a determinant ofcommunity structure (Holt et al., 1994; Chesson, 2000; Recordet al., 2013, 2014), and we expect that many potential applicationsof this model would require that this be better represented. Inthe present study, we have taken the minimalist, incrementalistapproach of imposing the simplest possible form of predationmortality—a linear function, with scalings that closely mirror thegrowth and development functions (Section 2.2)—and restrictingdistinguish the effects of temperature and prey seasonality. As wewill show, these initial experiments suggest a general hypothesis:that the viability of the calanoid community, at least near its highlatitude limit, is much more sensitive to prey abundance andphenology than to temperature.2. MODEL DESCRIPTION2.1. General ApproachThe model introduced here is “Coltrane” (Copepod Life-historyTraits and Adaptation to New Environments) version 1.0. Matlabsource code is available at overview of the model structure is shown in Figure 2.Like many individual-based models, Coltrane represents thetime-evolution of one cohort of a clonal population, all bearingthe same traits and spawned on the same date t0 , with a setof ODEs. The state variables describing a cohort are relativedevelopmental stage D, where D 0 represents a newlyspawned egg and D 1 an adult; survivorship N, thefraction of initially spawned individuals that remain after someamount of cumulative predation mortality; structural biomassper individual S, and “potential” or “free scope” ϕ, whichrepresents all net energy gain not committed to structure, orequivalently, the combination of internal energy reserves andeggs already produced. Combining reserves and eggs into onepool in this way lets us cleanly separate results that depend onlyon the fundamental energy budget (gain from ingestion, lossto metabolism, and energy required to build somatic structure)from results that depend on particular assumptions about eggproduction (costs, cues, and strategies). An alternate form ofthe model explicitly divides ϕ into internal reserves R and eggproduction rate E: the simpler model without this distinction willFrontiers in Marine Science www.frontiersin.org3November 2016 Volume 3 Article 225

Banas et al.Copepod Life History Strategy ModelingFIGURE 2 Overview of the logic and execution of the two versions of the Coltrane 1.0 model. In the ϕ version, a cohort is represented by four state variables(D, S, ϕ, N) integrated forward in time over the cohort’s lifespan. These are used to calculate a time series of fitness F. Next, this calculation is repeated across a familyof spawning dates t0 , and optimal and viable t0 cases determined. This constitutes a population-level description, which then can be repeated across a range of traitvalues (u0 , relative development rate) to describe a size-variable metacommunity. In the ER version, at the cohort level, ϕ is replaced by the state variable R and a timeseries of egg production E. Across a family of t0 cases, a transition-matrix method is used to determine a stable annual pattern of relative egg production n(t0 ), whichis taken as the population-level prediction. A metacommunity is formed by varying two traits, u0 and the date at which egg production begins tegg .yearday t. The level of predation mortality (Section 2.2.4) mightalso be viewed as an environmental characteristic.the terms of analysis. In particular, we will describe modeloutput in terms of trait correlations, optimality, and viability,but not in terms of absolute copepod biomass or abundance.Likewise, while some plankton models resolve the process ofadaptation explicitly (Clark et al., 2013), we address it only inthe indirect sense of mapping the viable and optimal regionsof the strategy landscape. This approach is less mechanistic butalso helpfully agnostic about whether adaptation in the copepodsarises through individual plasticity, species composition shifts, ornatural selection per se.An environment in Coltrane 1.0 is defined byannual cycles of three variables, total concentration ofphytoplankton/microzooplankton prey P, surface temperatureT0 , and deep temperature Td . At present, these annual cycles areassumed to be perfectly repeatable, so that a “viable” strategy canbe defined as a set of traits that lead to annual egg productionabove the replacement rate, given P, T0 , and Td as functions ofFrontiers in Marine Science www.frontiersin.org2.2. Time Evolution of One Cohort2.2.1. Ontogenetic DevelopmentCalanoid copepods have a determinate developmental sequence,comprising the embryonic period, six naupliiar stages (N1–6),five copepodid stages (C1–5), and adulthood (C6). Similar toMaps et al. (2012), conversions between relative developmentalstage D and the actual 13-stage sequence have been done usingrelative stage durations for C. finmarchicus from Campbell et al.(2001), which appear to be appropriate for other Calanus spp.with the proviso that C5 duration is particularly variable andstrategy-dependent. Development in the model followsdD u, D 1dt4(1)November 2016 Volume 3 Article 225

Banas et al.Copepod Life History Strategy Modelingand ra is an assimilation efficiency. Ingestion follows a Kleiber’sLaw-like dependence on structural body mass S, with θ 0.7(Kleiber, 1932; Saiz and Calbet, 2007). I0 is specific ingestion rateat saturating prey concentration, T 0 C, and S 1 µg C. Thisis modulated by the activity switch a and prey saturation σ as inEquation (2), and a power-law temperature response for growthwhere developmental rate u is(2)u a qd σ u 0andT/10 Cqd Q dT a T0 (1 a) TdPσ Ks P(3)(4)qg QgT/10(5)a rb (1 rb )aǫ Potential ϕ is allowed to run modestly negative, to representconsumption of body structure during starvation conditions. Acohort is terminated by starvation if(6)(7)ϕ rstarv S2.2.4. Predation MortalityPredation mortality is assumed to have the same dependence ontemperature and body size as ingestion, metabolism, and net gain(Hirst and Kiørboe, 2002). Survivorship N is set to 1 initially anddecreases according tod(ln N) mdt(9)Frontiers in Marine Science is convenient to calculate the numerical solution using ln Nrather than N as the state variable, since values become extremelysmall). The mortality rate m iswhere ingestion I and metabolic loss M are given byM a rm qg I0 S(15)where in this study rstarv 0.1. A convenient numericalimplementation of this scheme is to integrate S implicitly so thatit is guaranteed 0, and to integrate ϕ explicitly so that it isallowed to change sign, with no change of dynamics at ϕ 0.When G 0, the deficit is taken entirely from reserves: fs 0.Before the first feeding stage (D Df ) we assume G 0 forsimplicity. After feeding begins,θ 1(14)2.2.3. Starvationwhere G is net energy gain (ingestion minus metabolic losses).When net gain is positive, it is allocated between structure andpotential according to the factor fs , which commits net gainentirely to structure before a developmental point Ds , entirelyto potential during adulthood, and to a combination of them inbetween: D Ds 1,1 D,Dfs 1 D(8)s D 1s 0,D 1 rmG ra Iσwhen a 1. We have set rm 0.14 such that ǫ 0 whenP 1/4 Ks .The two energy stores S (structure) and ϕ (reserves/potential)followI a σ qg I0 Sθ 1(13)Note that in this formalism, gross growth efficiency ǫ becomes2.2.2. Energy Gain and LossG ra I M(12)which is parallel to that for development (qd ) but with adifferent Q10 . Q10 values have been found to vary among copepodspecies but Banas and Campbell (2016) argue that commonvalues derived from a fit across community-level data are moreappropriate for comparing species near their thermal optima. Weuse Qg 2.5 and Qd 3.0, as an approximation to the best-fitcomplex allometric curves reported by Forster et al. (2011).Energy loss to metabolism M follows the same temperatureand size scalings. The factor rm is the ratio of metabolismto ingestion when prey is saturating. Unlike development andingestion, which are assumed zero during diapause, M duringdiapause is nonzero but reduced to a basal fraction rb 1/4(Maps et al., 2014):All variables and parameters are defined in Table 1. Activity levela is, in this version of the model, a two-state switch calculated ateach time step, 1 during active feeding and 0 during diapause.The temperature-dependent factor qd describes a power-lawresponse with a Q10 of Qd , where temperature is assumed to beT0 during active feeding and Td during diapause. We use theQ10 functional form for convenience: the differences betweenthis and the leading alternatives (Belěhrádek, Arrhenius: Forsteret al., 2011; Record et al., 2012) appear to be small comparedwith interspecies differences in this study (Banas and Campbell,2016). Prey saturation σ is a simple Michaelis–Menten functionwith half-saturation Ks . The parameter u0 , the development ratecorrected to 0 C, was found by Banas and Campbell (2016) tobe the primary trait responsible for differences in adult body sizeamong Calanus spp. and other calanoids 50 µg C adult size,although not at a broader scale of diversity. It represents theaspect of development-rate variation that we interpret to be astrategy choice as opposed to a physiological or thermodynamicconstraint.dS fs GSdtdϕ (1 fs )GSdt C(10)m a qg Sθ 1 m0(11)5(17)November 2016 Volume 3 Article 225

Banas et al.Copepod Life History Strategy ModelingTABLE 1 Parameter values and other symbols used in the NMENTAL FORCINGPPrey concentrationmg chl m 3T0Surface temperature CTdDeep temperature CδtEffective duration of prey availability (global testbed)dδt′Width of P window (global testbed)dSTATE VARIABLESDRelative developmental stageNSurvivorshipRIndividual reserve biomassµgCSIndividual structural biomassµgCϕPotential reserves and egg productionµgCTRAITS AND FREE PARAMETERSDdiaStage at which diapause becomes possible0.49Stage C3DfStage of first feeding0.1Stage N3: Campbell et al., 2001DsStage at which lipid storage begins0.35Stage C1I0Specific ingestion at σ 1, T 0 C, S 1 µgC0.4 d 1Banas and Campbell, 2016KsHalf-saturation for ingestionSee Table 2m0Specific predation mortality at T 0 C, S 1 µgCSee Table 2QdQ10 for development3.0Forster et al., 2011QgQ10 for growth2.5Forster et al., 2011raFraction of ingestion assimilated0.67rbDiapause metabolism relative to active metabolism0.25Maps et al., 2014reaScaling constant for egg:adult size ratio0.013Kiørboe and Sabatini, 1995rmMetabolism relative to prey-saturated ingestion0.14rstarvFraction of S consumable under starvation conditions0.1rϕmaxUpper limit on ϕ/S used in diapause criterion1.5teggEarliest possible date of egg productionSee Table 2C. hyperboreus: Swalethorp et al., 2011t0Yearday of spawning0–365u0Development rate corrected to 0 CSee Table 2θAllometric exponent for vital rates0.7Saiz and Calbet, 2007θeaAllometric exponent for egg:adult size ratio0.62Kiørboe and Sabatini, 1995aActivity level0, 1a Variation of metabolism with arb , 1CdiaCoefficient arising in the diapause criterionETotal egg productionµgC d 1EcapCapital egg productionµgC d 1EincIncome egg productionµgC d 1FEgg fitnessOTHER QUANTITIESF1/2 , F1 , F2Maximum egg fitness at 1/2, 1, 2 generations per yearfsFraction of G allocated to SGNet energy gaind 1ISpecific ingestiond 1MSpecific metabolismd 1mSpecific predation mortalityd 1qdTemperature dependence of developmentqgTemperature dependence of growthuOntogenetic development rated 1WaAdult body sizeµgCWeEgg biomassµgCλPopulation growth rateyr 1σPrey saturationFrontiers in Marine Science www.frontiersin.org6November 2016 Volume 3 Article 225

Banas et al.Copepod Life History Strategy Modelingconcentrations when predation is high. A third, temperaturedependent term has been neglected. The second, mortalitydependent term tends to produce unrealistic, rapid oscillationsin which the copepods briefly “top up” on prey and thenhide in a brief “diapause” to burn them. It is unclear whetherthis model behavior is a mathematical artifact—a limitation ofcombining actual lipid reserves and potential egg productioninto a single state variable—or whether it suggests that undersome conditions the optimal level of foraging is intermediatebetween full activity and none. Incorporating a more mechanistictreatment of optimal foraging (Visser and Fiksen, 2013) andallowing a to vary continuously would address this. In this study,we have eliminated the phenomenon by approximating Cdia assuch that that predation pressure relative to energy gain isencapsulated in a single parameter m0 . In practice m0 is a tuningparameter but we can solve for the value that would lead toan approximate equilibrium between growth and mortality. Thecondition1 d(NS) 0NS dt(18)is equivalent, by Equations (6) and (16), tom fs G(19)and with a 1 this becomesm0 (ra σ rm )fsI0h ϕ iCdia max 0, 1 min rϕmax ,S(20)Averaging fs over the maturation period 0 D 1 withDs 0.35, and assuming σ 2/3 on average for an organismthat has aligned its development with the productive season, givesm0 0.2 I0 . This is the default level of predation in the modelexcept where otherwise specified.where rϕmax Eggs and Potential EggsThe evolution equations above Equations (1), (6), (7), (16) specifythe development of one cohort in the ϕ model. If this modelis elaborated with an explicit scheme for calculating total eggproduction over time E(t), then it is possible to define R(t),individual storage/reserve biomass, and interpret R as a statevariable and ϕ as a derived quantity. The relationship betweenthe two is2.2.5. Activity Level and DiapauseModulation of activity level a has been treated as simplyas possible, using a “myopic” criterion that considers onlythe instantaneous energy budget, rather than an optimizationover the annual routine or lifetime (Sainmont et al., 2015).Furthermore, we treat a as a binary switch—diapause or fullforaging activity—although intermediate overwintering stateshave been sometimes observed, e.g., C. glacialis/marshallae onthe Eastern Bering Sea shelf in November (Campbell, personalcommunication). In the present model, we set a 0 ifD Ddia (the stage at which diapause first becomes possible)and the environment is such that total population energygaind(ϕ S)N (GS)NdtdR (1 fs )GS EdtZ tE(t ′ ) dt ′ϕ(t) R(t) rm (1 rb ) Cdia m0 rara I0(21)Einc G,G 0Ecap Emax Einc , D 0(22)(27)where Emax is a maximum egg production rate which we assumeto be equal to food-saturated assimilation:(23)Emax ra qg I0 Sθwhere Cdia 1 ϕ/S. The first term in Equation (23) canbe derived more simply by setting dG/da 0, a criterionbased on ingestion and metabolism alone. The second termadjusts this criterion by discouraging foraging at marginal preyFrontiers in Marine Science, ϕ tracks the reserves that an animal would have remainingif it had not previously started egg production. This is a usefulmetric for optimizing reproductive timing, as we will show(Section 2.4).Any explicit expression for E(t) allows Equation (25) toreplace Equation (7). In one model experiment below (Section3.4), we use the following scheme: E(t) is the sum of incomeegg production Einc and capital egg production Ecap , which are0 until maturity is reached (D 1) and an additional timingthreshhold has been passed (t tegg ). Past those threshholds,they are calculated ascan be rearranged to give a critical prey-saturation levelσcrit (25)t0would be higher under diapause. We can derive an expression forthe threshhold at which this occurs by maximizing populationenergy gain as a function of a. When d/da of GSN ispositive, active foraging a 1 is the optimal instantaneousstrategy and when it is negative, a 0 is optimal. Thethreshholdd(GSN) 0da(24)(28)Thus, the trait tegg determines whether egg production beginsimmediately upon maturation (if tegg is prior to the date onwhich D reaches 1) or after some additional delay. Instead

congeneric species in a single environment (Swalethorp et al., 2011). Many but not all species enter a seasonal period of . seabird, and mammal foraging are tied to temperature and . phase of work is the lightest-weight representation of diversity and pla