# Offspring number, size, and survival: State-dependent optimization of litter size in a long-lived capital breeder

Marwan Naciri, Jon Aars, Magnus Andersen, Marie-anne Blanchet, Andrew E Derocher, Marlène Gamelon, Øystein Wiig, Sarah Cubaynes

► To cite this version:

Marwan Naciri, Jon Aars, Magnus Andersen, Marie-anne Blanchet, Andrew E Derocher, et al. Offspring number, size, and survival: State-dependent optimization of litter size in a long-lived capital breeder. Ecosphere, 2025, 16 (8), pp.e70370. 10.1002/ecs2.70370. hal-05221458

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ARTICLE

Animal Ecology

# Offspring number, size, and survival: State-dependent optimization of litter size in a long-lived capital breeder

Marwan Naciri<fnref n="1" /> ID | Jon Aars<fnref n="2" /> | Magnus Andersen<fnref n="2" /> | Marie-Anne Blanchet<fnref n="2" /> |
Andrew E. Derocher<fnref n="3" /> ID | Marlène Gamelon<fnref n="4" /> ID | Øystein Wiig<fnref n="5" /> ID |
Sarah Cubaynes<fnref n="6" />

**Correspondence**

Marwan Naciri
Email: marwan.naciri@ens-lyon.fr

**Funding information**

Agence Nationale de la Recherche
(MathKinD), Grant/Award Number:
ANR-18-CE02-0011; Norwegian Polar
Institute

**Handling Editor:** Matthew A. Mumma

**Abstract**

Seventy-five years ago, Lack proposed that there should be an optimal clutch size shared by all individuals in a population and favored by natural selection, which maximizes the number of recruited offspring. While some studies support this “common optimum” hypothesis, others have shown that the optimal clutch size depends on maternal state (e.g., age and body condition, “state-dependent optimization”). These contrasting results suggest that the degree of state dependency might itself depend on the ecological context (e.g., capital vs. income breeding). Furthermore, almost all these studies were conducted on fast-living species and have ignored interdependencies among variables such as maternal age and condition. Here, we test whether females share a common optimal litter size or have a state-dependent optimal litter size in a slow-paced capital breeder, the polar bear (*Ursus maritimus*). To do so, we assess the influences of (1) maternal state on litter size, (2) maternal state and litter size on cub mass, and (3) maternal state, litter size, and cub mass on cub survival, using path analysis to account for interdependencies among variables and capture-recapture modeling to estimate cub survival. We use 34 years (1992–2025) of individual-based data from a polar bear subpopulation in the Svalbard region of Norway. In accordance with the state-dependent optimization hypothesis, litter size varied with maternal age and size. Middle-aged and sized females had the highest probability of having twins. Old and large females more often had triplet litters. Cub mass decreased

This is an open access article under the terms of the [Creative Commons Attribution License](https://creativecommons.org/licenses/by/4.0/), which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2025 The Author(s). *Ecosphere* published by Wiley Periodicals LLC on behalf of The Ecological Society of America.

<footnote n="1">¹CEFE, Univ Montpellier, CNRS, EPHE, IRD, Montpellier, France</footnote>

<footnote n="2">²Norwegian Polar Institute, FRAM Centre, Tromsø, Norway</footnote>

<footnote n="3">³Department of Biological Sciences, University of Alberta, Edmonton, Alberta, Canada</footnote>

<footnote n="4">⁴Laboratoire de Biométrie et Biologie Évolutive, CNRS, Unité Mixte de Recherche (UMR) 5558, Université Lyon 1, Université de Lyon, Villeurbanne, France</footnote>

<footnote n="5">⁵Natural History Museum, Oslo, Norway</footnote>

<footnote n="6">⁶CEFE, Univ. Montpellier, CNRS, EPHE-PSL University, IRD, Montpellier, France</footnote>
with increasing litter size, whereas litter mass increased, meaning females with a large litter allocated more resources to reproduction. Cub mass in turn strongly predicted cub survival, indicating that offspring survival was traded against offspring number. In middle-aged and old females, productivity—the number of cubs surviving their first year—increased with litter size, in accordance with the state-dependent optimization hypothesis. Overall, our results are consistent with a state-dependent optimization of litter size in this large, slow-paced mammal producing small litters. State dependence of litter size may be more likely in capital breeders as the amount of energy available for reproduction is known in advance.

KEYWORDS

age, body size, capture-recapture, life-history trade-offs, offspring mass, offspring survival, path analysis, polar bear, size-number trade-off, *Ursus maritimus*

INTRODUCTION

Why do all females in a population not produce the largest
possible number of offspring? Lack (1947) first investigated
this question more than 75 years ago in birds. He proposed
that “the average clutch-size is ultimately determined by
the average maximum number of young which the parents
can successfully raise” (Lack, 1947, p. 319). Since parents
have limited resources they can allocate to reproduction, if
the number of offspring increases, the share of parental
resources allocated to each offspring decreases. This
principle results in a negative relationship between off-
spring number and offspring size termed the “offspring
size-number trade-off.” Given the size-dependent
nature of offspring survival in most vertebrates (Ronget
et al., 2018), this size-number trade-off typically trans-
lates to a trade-off between offspring number and off-
spring survival. Parents are expected to balance the
number and size/survival of offspring to maximize off-
spring recruitment (Lloyd, 1987; Morris, 1998; Smith &
Fretwell, 1974).

Initially, Lack implicitly posited that this optimal
clutch size would be common to all individuals (“com-
mon optimum” hypothesis). While some studies support
this hypothesis (Eriksen et al., 2023; Gaillard et al., 2014),
substantial temporal variation and inter-individual dif-
ferences in observed clutch sizes in many populations led
researchers to propose that the optimal clutch size
depends on the state of the mother (“state-dependent
life-histories”; McNamara & Houston, 1996, Morris, 1998,
also acknowledged by Lack, 1948). Under this view,
the optimal clutch size (i.e., the clutch size that maxi-
mizes the number of recruited offspring) depends on
female state (e.g., age, body mass, habitat, experience)
such that each female has her own optimal clutch size

at a given reproductive event (“state-dependent optimi-
zation” hypothesis). This hypothesis has received
empirical support (Berger et al., 2015; Morris, 1996,
1998; Perrins & Moss, 1975; Pettifor et al., 1988; Risch
et al., 1995, 2007). For instance, in white-footed mice
(*Peromyscus leucopus*), large females achieved the same
offspring survival rates with litters of 6 and 7 as small
females with litters of 5, that is, large females had a
higher optimal litter size compared to small females
(Morris, 1996).

Most studies on the optimization of clutch or litter
size have focused on fast-living birds and mammals.
Slow-living species typically produce fewer offspring than
fast-living species (many of the former even producing a
single offspring per reproductive attempt; De Magalhães &
Costa, 2009). In this regard, carnivores constitute an inter-
esting group, with some slow-living species producing lit-
ters up to 6 (Wilson & Mittermeier, 2009). However, many
carnivores are elusive, live at low density, and are difficult
to capture. These characteristics, along with the slow pace
of life of large carnivores, have hindered the acquisition of
large amounts of data on reproduction of females in differ-
ent states, largely preventing the investigation of the opti-
mization of litter size in such species (Table 1). One recent
study on the optimization of litter size in Eurasian lynx
(*Lynx lynx*) found a single optimal litter size across female
states and environments (Gaillard et al., 2014). This result
suggests that species with a slow life-history pace and
small litters (between 1 and 4 in Eurasian lynx) may fit
the common optimum hypothesis, perhaps because chang-
ing their litter size by 1 offspring represents a larger rela-
tive change than it does in species producing larger litters.

One further consideration is that state variables are
often colinear and affect several reproductive traits simul-
taneously, resulting in multiple direct and indirect
<table><caption>TABLE 1 The optimization of litter size in studies on large carnivores.</caption><thead><tr><td>Study</td><td>Species</td><td>Range of litter sizes</td><td>Optimization strategy</td></tr></thead><tbody><tr><td>Packer and Pusey (1995)<sup>a</sup></td><td>Lion (Panthera leo)</td><td>1–6</td><td>Frequency dependent</td></tr><tr><td>White (2005)</td><td>Spotted hyena (Crocuta crocuta)</td><td>1–2</td><td>State dependent<sup>b</sup></td></tr><tr><td>McNutt and Silk (2008)</td><td>African wild dog (Lycaon pictus)</td><td>3–16</td><td>State dependent<sup>b</sup></td></tr><tr><td>Gaillard et al. (2014)</td><td>Eurasian lynx (Lynx lynx)</td><td>1–4</td><td>Common optimum</td></tr><tr><td>Marneweck et al. (2019)</td><td>African wild dog (Lycaon pictus)</td><td>1–14</td><td>State dependent<sup>b</sup></td></tr></tbody></table>

Note: We conducted the literature search on Web of Science in November 2023 using the query provided in Appendix S1. We included studies if they (1) were conducted in a wild large carnivore, (2) tested the effect of maternal state on litter size, and (3) tested the effect of litter size on cub survival, or the direction of the effect could be deduced.

For instance, maternal age may affect litter size both directly and through maternal mass (Derocher & Wiig, 2002; Gamelon et al., 2013; Nussey et al., 2011). Similarly, maternal mass may simultaneously have a positive direct effect on offspring mass (Ronget et al., 2018) and a negative indirect effect through litter size (heavier females generally have larger litters, and offspring in larger litter are generally lighter; Lim et al., 2014). Thus, studying the determinants and consequences of variation in litter size requires disentangling direct effects from indirect effects. Yet, to date, most studies have ignored interdependencies among relevant variables, precluding a rigorous understanding of the optimization of litter size. Path analysis, a type of multivariate analysis, however, is well suited to answer questions about complex and multi-factorial systems (Grace, 2006).

Here, we use the polar bear (*Ursus maritimus*) as a case study to investigate the optimization of litter size in a large carnivore. Polar bears are at the slow end of the slow-fast continuum. Females give birth to 1–3 cubs in den while fasting over winter, relying on extensive fat stores (Amstrup, 2003). They are thus capital breeders during maternity denning (Atkinson & Ramsay, 1995), although they act as income breeders for the remaining two years of maternal care, hunting seals on the sea ice (Amstrup, 2003).

Under the state-dependent optimization hypothesis, maternal state determines litter size. In many species, middle-aged females have higher reproductive output than young and old females. Indeed, young females may preferentially allocate resources to growth and survival (Forslund & Pärt, 1995; Pianka & Parker, 1975) and may have fewer resources overall (Desrochers, 1992; Forslund & Pärt, 1995). Old females often have fewer resources due to somatic senescence (Derocher et al., 1992; Derocher & Stirling, 1994; MacNulty, Smith, Vucetich, et al., 2009; Nussey et al., 2011). Thus, we predict that litter size varies with maternal age, with an increase from young age to

middle age, followed by a senescent decline, as reported by Folio et al. (2019). Besides age, structural size may influence litter size. Large structural size may allow females to take larger prey and to better defend their kills against conspecifics (MacNulty, Smith, Mech, & Eberly, 2009; Rode et al., 2020) resulting in more resources to allocate to reproduction (Beauplet & Guinet, 2007). Furthermore, large females are more efficient in the use of their stored energy reserves (Lindstedt & Boyce, 1985), which may be critical for fasting. We therefore predict that litter size increases with maternal body length, as reported by Folio et al. (2019). Under the common optimum hypothesis, maternal state does not determine litter size; thus, our alternative prediction is that maternal age and length have no effect on litter size.

If, as posited by the state-dependent hypothesis, females in a good state have large litters, then, although cub mass should decrease with litter size (offspring size-number trade-off), litter mass should increase with litter size. Indeed, otherwise, the benefits of starting with more numerous offspring would be largely offset by the cost of having smaller offspring with reduced survival prospects (Lloyd, 1987). Under the common optimum hypothesis, cub mass is also expected to decrease with litter size, but litter mass may not necessarily increase with litter size. Regardless of the mode of optimization of litter size, we predict that cub survival is strongly positively related to cub mass (Derocher & Stirling, 1996; Ramsay & Stirling, 1988; Rode et al., 2020; Ronget et al., 2018). Furthermore, in accordance with theoretical expectations of diminishing fitness benefits of increased offspring mass beyond a certain point, cub survival should exhibit a roughly sigmoidal relationship with cub mass (Lloyd, 1987).

Finally, with regard to the productivity of each litter size, the state-dependent optimization hypothesis posits that females with a large litter can raise more offspring than females with a small litter. Thus, productivity is

<footnote><sup>a</sup>For unknown reasons, this study was not retrieved by our literature search, but it was included nonetheless due to its relevance.</footnote>

<footnote><sup>b</sup>The optimization of litter size was not explicitly studied, but the findings of the study provide support for a mode of optimization.</footnote>
predicted to increase with litter size. By contrast, under the common optimum hypothesis, productivity is predicted to peak at the optimal litter size, which is smaller than the maximum. We analyze 35 years of capture data from the Svalbard area (Norwegian Arctic, Barents Sea) to assess in a single integrated analysis (1) whether maternal state (age and body size) influences litter size, (2) how maternal state and litter size affect offspring mass, (3) how maternal state, litter size, and offspring mass affect offspring first-year survival, and (4) the productivity of each litter size.

# METHODS

## Study species and data collection

The polar bear is a long-lived species that depends on sea ice for hunting its main prey—seals, for traveling, and for mating (Stirling, 2011). In Svalbard, females have their first litter at age 5 or 6 years (Wiig, 1998). Mating occurs in spring, but implantation is delayed until autumn (Amstrup, 2003). Following implantation, pregnant females enter a maternity den, usually dug in snow. They give birth to very altricial cubs in their den in November–January and emerge with them in early spring (Amstrup, 2003). Over the denning period (up to 7 months), reproducing females fast and rely on stored energy to meet the nutritional demands of maintenance, gestation, and lactation (Ramsay & Stirling, 1988). Over this period, females thus act as capital breeders, losing an average of 40% of their autumn body mass (Atkinson & Ramsay, 1995). Litter size in spring varies between one and three cubs, but most pregnant females carry two fetuses (De Velde et al., 2003; Lønø, 1970), and the most frequent litter size in spring is two (Amstrup, 2003). Cubs are entirely reliant on their mother during their first 15 months and remain with her until age 2.5 years (Ramsay & Stirling, 1988). Mortality is higher in cubs of-the-year (hereafter “cubs”) than in yearlings and two-year-old cubs (Amstrup & Durner, 1995; Lunn et al., 2016), and causes of death include starvation, hypothermia, and infanticide and cannibalism by males (Derocher & Stirling, 1996; McGeachy et al., 2024).

Polar bears (n = 2317 captures of 1561 individuals) were immobilized from a helicopter and live-captured in the Svalbard area (Norway, 76–81° N, 10–34° E) between 1992 and 2025. The captures were conducted annually (except 2020) between late March and early May (spring), when the majority of denning females had emerged from dens (Andersen et al., 2012; Hansson & Thomassen, 1983; Wiig, 1998). Immobilization followed procedures described in Stirling et al. (1989). Individuals captured for the first time were marked with uniquely numbered ear tags on

both ears. Some adults were fitted with satellite-tracking collars. For adults and subadults captured for the first time, age was estimated by counting layer groups in an extracted vestigial premolar tooth, with reasonably high accuracy (Calvert & Ramsay, 1998; Christensen-Dalsgaard et al., 2010). For cubs and yearlings, age was determined based on size and dentition. Litter size for females with cubs was recorded at capture. By this time, reproducing females may have lost cubs. The analysis we present thus pertains to litter size a few months after birth, not at birth. Body length, from the tip of the nose to the last tail vertebra, was recorded as a measure of structural size. The girth of bears under the armpits was measured. Like body mass, this metric increases with both body condition and structural size (Appendix S2: Figure S1). Because only 40% of adult females were weighed, we use girth of adults rather than body mass in our analysis. Almost all (95%) cubs were weighed to the nearest 0.25 kg.

Satellite tracking has revealed that female bears encountered in the Svalbard area are a mixture of resident bears—who remain close to Svalbard shores year-round— and pelagic bears—who follow the sea ice as it advances and recedes, occasionally coming to Svalbard, notably to den (Mauritzen et al., 2001). Pelagic bears roam over greater distances and have higher energy expenditure than resident bears (Blanchet et al., 2020), which may affect their state and their reproductive rates. Available data indicate that females are consistent in their space-use strategy (resident or pelagic) and daughters adopt the same space-use strategy as their mother (Brun et al., 2021), with no records of exceptions to these rules. Therefore, we assumed that the space-use strategy of independent daugh- ters was the same as their mother’s (if known). As they are less often in Svalbard, pelagic females are less often re-sighted than resident females. Space-use by males is largely unknown since they cannot be fitted with collars due to their neck being wider than their head.

## Hierarchical path analysis capture-recapture model

To assess our predictions under the common optimum and state-dependent optimization hypotheses, we built a hierarchical model combining path analysis and capture-recapture modeling (PA-CR model) in a Bayesian framework (Cubaynes et al., 2012; Gimenez et al., 2012). Path analysis is used to model networks of interrelated variables as it allows to use multiple predictors and response variables and allows variables to simultaneously be predictor and response variables (Grace, 2006). Path analysis allowed us to (1) account for the interdepend-encies between maternal state variables and the
reproductive traits of interest, namely, litter size, cub mass, and cub survival, and (2) discriminate between direct and indirect effects of variables. We used capture-recapture (CR) modeling within our path model to estimate cub survival because imperfect detection precluded the direct observation of the fate of cubs. We selected the structure of the PA-CR (Figure 1) and the variables to include based on (1) knowledge gained from earlier studies on polar bears (from our population Cubaynes et al., 2021; Folio et al., 2019; Naciri et al., 2022 and others, Derocher & Stirling, 1996, 1998; Lunn et al., 2016) and (2) expectations derived from life-history theory and literature (Berger et al., 2015; Gaillard et al., 2014; Risch et al., 2007).

In practice, we specified the submodel for each modeled variable as a standard (generalized) linear mixed model (or a standard capture-recapture model in the case of cub survival). Grouping all submodels into a single Bayesian model allows us to account for the covariances

between variables, which are used to tease apart direct effects and indirect effects (Grace, 2006). Parameters are estimated in an integrated manner, with a full propagation of the uncertainty.

## Modeling litter size

We assessed whether litter size in spring was dependent on maternal state. We considered two state variables as predictors: maternal age and maternal body length (hereafter “length”). Maternal length varied with maternal age; thus, it was itself modeled as a function of maternal age (Appendix S3). Here and in the whole PA-CR model, we divided maternal age into three groups (young, 5–9 years; middle aged, 10–15 years; and old, ≥16 years) to (1) ease the implementation and interpretation of interactions between maternal age and other variables

<figure><img src="image_2.png" /><figcaption>**FIGURE 1** Directed acyclic graph of the hierarchical path analysis capture-recapture model for Svalbard polar bears showing the tested relationships between maternal state, litter size, cub mass, and cub survival. Arrows represent the effects included in the model. Maternal age, maternal space-use strategy, litter size, and cub sex were coded as categorical variables. Cub survival was estimated within the path model using a capture-recapture submodel.</figcaption></figure>
and (2) be consistent with a previous study on reproductive rates in Svalbard polar bears (Naciri et al., 2022). The number of age groups and cut-offs between them were chosen to minimize sample size issues, to be biologically meaningful, and to match observed patterns in a simple GAM with maternal age as the predictor (Appendix S2: Figure S3).

Following Folio et al. (2019), we modeled litter size as a categorical variable with three possible outcomes (1, 2, or 3 cubs) using a multinomial logistic regression. The outcome litter size = 1 was used as the reference outcome. Two logistic regressions were fitted simultaneously for each of the remaining outcomes (litter size = 2 or 3) against the reference outcome:

$$ \log\left(\frac{\pi_{2,i,t}}{\pi_{1,i,t}}\right) = \beta_1 \times \text{mAgeY}_{i,t} + \beta_2 \times \text{mAgeM}_{i,t} + \beta_3 \\ \times \text{mAgeO}_{i,t} + \beta_4 \times \text{mL}_{i,t} + \beta_5 \times \text{mL}_{i,t}^2 + \beta_6 \\ \times \text{mAgeY}_{i,t} \times \text{DOY}_{i,t} + \beta_7 \times \text{mSpUnk}_i + \beta_8 \\ \times \text{mSpPel}_i + \eta_t $$

$$ \log\left(\frac{\pi_{3,i,t}}{\pi_{1,i,t}}\right) = \beta_9 \times \text{mAgeY}_{i,t} + \beta_{10} \times \text{mAgeM}_{i,t} + \beta_{11} \\ \times \text{mAgeO}_{i,t} + \beta_{12} \times \text{mL}_{i,t}, $$

where $\pi_{1,i,t}$, $\pi_{2,i,t}$, and $\pi_{3,i,t}$ are the probabilities of each outcome for female i in year t; parameters $\beta$ are the effects of maternal age, maternal length (mL), capture date (DOY), and maternal space-use strategy on litter size; we refer to maternal-level variables with the prefix $m$; mAgeY$_{i,t}$, mAgeM$_{i,t}$, and mAgeO$_{i,t}$ take the value 1 if the mother i in year t is young, middle aged, or old, respectively, and 0 otherwise; mSpUnk$_i$ takes the value 1 if the space-use strategy is unavailable for mother i and 0 otherwise, while mSpPel$_i$ takes the value 1 if female i is pelagic and 0 otherwise. $\eta_t$ is the year random effect, where $\eta_t \sim N(0, \sigma_\eta^2)$.

The logistic regression of “2 cubs versus 1 cub” comprised an effect of maternal age and maternal length (linear and quadratic effect), in line with past findings (Folio et al., 2019). It also comprised an effect of the capture date (as day of the year, with 1 being January 1) to account for cub loss over the sampling period, for young females only (litter size is independent of the capture date for older females). We further included space-use strategy because preliminary analyses indicated that pelagic females had twins more often than resident females. Finally, we included a year random effect to account for interannual variation in litter size.

The logistic regression of “3 cubs versus 1 cub” comprised an effect of maternal age and of maternal length as in Folio et al. (2019). We did not add an effect of

maternal space-use strategy or a year random effect considering the low sample size ($n = 8$ triplet litters).

Here and in all model subparts, we did not include a maternal identity random effect as the number of captures per mother was very low (mean = 1.2, range = 1–4). The sample size for this model subpart was 254 spring captures of mothers with a litter-of-the-year.

## Modeling cub mass

Our second objective was to assess how cub and litter mass were related to litter size and maternal state. To do so, we modeled the (log-transformed) mass of cub j of mother i in year t ($M_{j,t}$) as a Gaussian variable using the following linear mixed model:

$$ M_{j,t} = \text{LS1}_{i,t} \times (\delta_1 \times \text{mAgeY}_{i,t} + \delta_2 \times \text{mAgeM}_{i,t} \\ + \delta_3 \times \text{mAgeO}_{i,t} + \delta_4 \times \text{mL}_{i,t} \\ + \delta_5 \times \text{mG}_{i,t} + \delta_6 \times \text{DOY}_{i,t} + \delta_7 \times \text{Sex}_j \\ + \delta_8 \times \text{mSpUnk}_i + \delta_9 \times \text{mSpPel}_i) \\ + \text{LS2}_{i,t} \times (\delta_{10} \times \text{mAgeY}_{i,t} + \delta_{11} \times \text{mAgeM}_{i,t} \\ + \delta_{12} \times \text{mAgeO}_{i,t}) + \text{LS3}_{i,t} \times (\delta_{13} \times \text{mAgeY}_{i,t} \\ + \delta_{14} \times \text{mAgeM}_{i,t} + \delta_{15} \times \text{mAgeO}_{i,t}) \\ + (\text{LS2}_{i,t} + \text{LS3}_{i,t}) \times (\delta_{16} \times \text{mL}_{i,t} + \delta_{17} \times \text{mG}_{i,t} \\ + \delta_{18} \times \text{DOY}_{i,t} + \delta_{19} \times \text{Sex}_j \\ + \delta_{20} \times \text{mSpUnk}_i + \delta_{21} \times \text{mSpPel}_i) + \nu_t + \varepsilon_{j,t} $$

where parameters $\delta$ are the effects of litter size, maternal age, maternal length, maternal girth (mG), capture date, cub sex (Sex), and maternal space-use strategy on cub mass. $\text{LS1}_{i,t}$, $\text{LS2}_{i,t}$, and $\text{LS3}_{i,t}$ take the value 1 if the size of the litter of female i in year t (to which cub j belongs) is 1, 2, and 3, respectively, and 0 otherwise; $\text{Sex}_j$ takes the value 1 if cub j is a male and 0 for a female cub. $\nu_t$ is the year random effect, where $\nu_t \sim N(0, \sigma_\nu^2)$. $\varepsilon_{j,t}$ is the error term, where $\varepsilon_{j,t} \sim N(0, \sigma_\varepsilon^2)$.

We included the effect of litter size (1, 2, or 3) as a categorical variable to characterize the offspring size-number trade-off. We considered the effect of maternal age and length because these traits may affect cub mass directly (i.e., not only through litter size), as found in brown bears (*U. arctos*) and other species (Ronget et al., 2018; Van de Walle et al., 2021). Females in better body condition often have larger cubs (Rode et al., 2020; Ronget et al., 2018). Direct measures of maternal body condition were not available in our study, so we included an effect of maternal girth. Maternal girth was not independent of maternal age, length, and litter size, so it was itself modeled as a function of these variables (Appendix S3). Importantly, including an effect of maternal length on both
maternal girth and cub mass means that the effect of girth on cub mass is tested after accounting for the contribution of length to girth (Grace, 2006). Therefore, we tested the effect of length-corrected girth, which is highly related to body condition (Appendix S2: Figure S1C), on cub mass. We also reproduced our analysis using the body condition index from Cattet et al. (2002) and obtained very similar results (Appendix S4). We included the effect of capture date to account for growth over the sampling period (Lønø, 1970, p. 81) and an effect of cub sex because male cubs are slightly heavier than female cubs (Derocher & Stirling, 1998; Ramsay & Stirling, 1988). We also included an effect of maternal space-use strategy because pelagic females appear to have lighter cubs than resident females. Because pelagic individuals are often captured only once, their survival is biased low (see below). Ignoring the effect of maternal space-use strategy on cub mass and cub survival would thus lead to underestimating the survival of small cubs and thus to overestimating the steepness of the relationship between cub mass and cub survival.

We included the effect of maternal age in interaction with litter size to allow for a litter-size specific effect of maternal age, and thus, for an age-specific strength of the offspring size-number trade-off. Further, we estimated the effect of maternal length, maternal girth, maternal space-use strategy, capture date, and cub sex independently for singletons and for twins and triplets pooled together. We included a year random effect to account for interannual variation in cub mass. For this subpart, the sample size was 428 cubs (from the 254 litters modeled in the previous subparts).

We estimated the proportion of variance in log-transformed cub mass (var(M), which is equal to 1 as all Gaussian response variables were scaled) explained by the model using the formula $R^2 = 1 - \text{var}(\kappa)/\text{var}(M)$, with var($\kappa$) the mean value of the posterior distribution of the residual variance of (log-transformed) cub mass (Gelman & Pardoe, 2006; Nakagawa & Schielzeth, 2013).

## Modeling cub survival

Our third objective was to examine how cub survival was related to litter size, cub mass, and maternal state. Detection of individuals during capture sessions was imperfect, meaning that some individuals that were alive and in the study area were not detected. For instance, a bear marked as a cub may have gone undetected in all subsequent capture sessions even if it was alive for many years. Thus, survival could not be directly measured. By modeling the probability of recapturing (or missing) individuals, CR modeling allows us to estimate survival rates while accounting for imperfect detection (Lebreton et al., 1992).

We built a Cormack–Jolly–Seber (CJS) CR model to estimate survival rates (Lebreton et al., 1992). Goodness-of-fit tests of the CJS model, performed using the package R2ucare (Gimenez et al., 2018), indicated a transience effect ($\chi^2_{29} = 144.45, p < 0.01$), that is, a lower expectation of recapture of individuals captured for the first time as compared to other individuals (Pradel et al., 1997), most likely caused by pelagic individuals that are never recaptured, and strong differences in survival between cubs and older bears. We included an additive effect of space-use strategy on apparent survival and recapture probabilities, and we estimated age-specific apparent survivals to account for this transience effect.

We modeled apparent survival (hereafter “survival”) of individual $k$ in year $t$ ($\phi_{k,t}$) as age dependent. Based on studies on other polar bear populations, we considered the following age groups: 0 year (cubs), 1–2 years, 3–4 years, 5–19 years, and $\ge$ 20 years (Bromaghin et al., 2021; Lunn et al., 2016). Although we were only interested in cub survival (i.e., first-year survival), we had to model the whole lifetime because a CR model jointly estimates all survival rates. Below, we describe how we modeled survival for cubs only (see Appendix S3 for survival modeling for bears >0 year).

We modeled $\phi_{k,t}$ for cub $k$ belonging to mother $i$ over the interval $t$ to $t + 1$ using the following logistic regression embedded in the CR part of the PA-CR model:

$$ \text{Logit}(\phi_{k,t}) = \text{Age0}_{k,t} \times (\theta_1 \times \text{mAgeY}_{i,t} + \theta_2 \times \text{mAgeM}_{i,t} \\ + \theta_3 \times \text{mAge0}_{i,t} + \theta_4 \times M_{k,t} \\ + \theta_5 \times (\text{LS2}_{i,t} + \text{LS3}_{i,t}) + \theta_6 \times \text{mL}_{i,t} \\ + \theta_7 \times \text{mG}_{i,t} + \theta_8 \times \text{DOY}_{k,t} \\ + \theta_9 \times \text{Sex}_k + \theta_{10} \times \text{mSpUnk}_i \\ + \theta_{11} \times \text{mSpPel}_i), $$

where parameters $\theta$ are the effects of cub mass, litter size, maternal length, capture date, and maternal space-use strategy on cub survival. We modeled $\phi_{k,t}$ as a function of (log-transformed) cub mass, litter size, maternal age, maternal length, maternal (length-corrected) girth, cub sex, capture date, and maternal space-use strategy. We included the effect of cub mass to determine the effect of the offspring size-number trade-off on survival. We considered direct effects of litter size, maternal age, maternal length, and maternal (length-corrected) girth on cub survival to assess (1) cub-mass-independent effects of these variables (e.g., older, more experienced females may avoid infanticidal males and swimming better; Freitas et al., 2012) and (2) effects of these variables that are mediated by cub mass after capture. For instance, the growth rate of cubs of mothers in good spring body condition may be higher after capture, resulting in higher
survival. We included the effect of sex because survival may vary by sex in a mass-independent way (e.g., Hall et al., 2001) and because sex may affect the trajectory of cub mass after capture, and thereby, cub survival (Derocher & Stirling, 1998). We included the capture date to account for viability selection (cubs still alive late in the sampling period may not be a random sample of the cubs that were alive at the beginning of the sampling period). We controlled for maternal space-use strategy to (1) account for differences in energy expenditure, prey availability, swimming frequency, and other factors which may affect cub survival, and (2) capture the potential downward bias in survival estimates caused by transience of pelagic females and females with unknown space-use. Although cub survival is expected to be sensitive to environmental conditions (Gaillard et al., 1998; Pfister, 1998), we could not include a year random effect on cub survival because the variance of the random effect failed to converge. We modeled recapture probability as a function of sex, space-use strategy, age, and year (Appendix S3).

In this subpart, we analyzed 2317 captures of 1561 individuals, including the 428 captures of cubs analyzed in the cub mass subpart of the model.

## Predicting productivity

Using our PA-CR model, we predicted productivity, defined as the number of cubs surviving their first year per litter, for each litter size, given the observed characteristics (age, length, and girth) of females who have a given litter size. We considered all direct and indirect effects. Because we used all the samples from the posterior distribution of each parameter when making predictions, the uncertainty in parameter estimates was fully propagated to our final predictions.

## Implementation

We ran our model in R (R Core Team, 2022) and Rstudio (RStudio Team, 2022) in a Bayesian framework using the NIMBLE package (de Valpine et al., 2024). We used a marginalized likelihood of the capture-recapture model to speed up computation (Yackulic et al., 2020). All continuous variables were scaled (mean = 0; SD = 1) over the 254 captures of females with cubs to ease the comparison of coefficients and the interpretation of estimated SDs. We used vague, uninformative priors for all parameters ($N(0, 5)$ for coefficients of linear regression, $N(0, 1.5)$ for coefficients of logistic regressions, and $U(0, 10)$ for SDs). Parameters were estimated by running three

Markov chain Monte Carlo (MCMC) chains with differing, randomly drawn, initialization for 25,000 iterations (the first 10,000 of which were discarded as burn-in). We retained one out of every 10 iterations, yielding 1500 samples of each posterior distribution per chain and 3000 in total. We assessed convergence visually by inspecting the trace plots and by ensuring that the $\hat{R}$ diagnostic had a value lower than 1.1 for all parameters (Gelman & Rubin, 1992).

We report the mean of the posterior distributions with the 95% credible intervals in brackets. When relevant, we also report the proportion of the posterior distribution of a parameter that is of the median's sign, called the probability of direction ($p_d$; Makowski et al., 2019). The $p_d$ can be understood as the probability that the parameter is strictly positive (if the median is positive) or negative (if the median is negative). For a ratio, the $p_d$ corresponds to the proportion of the posterior distribution that is on the same side of 1 as the median of the posterior. We refer to $p_d$ values between 0.80 and 0.90 as "low evidence," between 0.90 and 0.95 as "moderate evidence," between 0.95 and 0.99 as "strong evidence," and $p_d$-values >0.99 as "very strong evidence," as recommended to reflect the weight of evidence contained in our data (Muff et al., 2022). Figures were constructed using ggplot2 (Wickham, 2016).

## RESULTS

The model converged, and all parameters could be estimated, as indicated by adequate mixing and all $\hat{R}$ values are <1.01. The posteriors of all the parameters estimated by the PA-CR model along with their definition are presented in Appendix S3: Figure S1 and Table S1.

### Litter size

There were 88 singleton litters, 158 twin litters, and 8 triplet litters in our sample. Mean litter size in spring was 1.69 (SD = 0.53).

Litter size varied with maternal age (Figure 2A) with middle-aged females having a higher probability of having twins than young females (difference: 0.12 [-0.03; 0.27], moderate evidence, $p_d$ = 0.93) and old females (difference: 0.20 [0.02; 0.37], strong evidence, $p_d$ = 0.99). There was low evidence that the probability of having twins differed between young and old females (difference: 0.08 [-0.11; 0.27], $p_d$ = 0.81). Females that did not have twins mostly had a singleton. Old females had a higher propensity to have triplets than young ($p_d$ = 0.97) and middle-aged ($p_d$ = 0.97) females, although they were uncommon across all age classes.
<figure><img src="image_3.png" /><figcaption>**FIGURE 2** Predicted direct influences of maternal (A) age and (B) length on the probability of having a singleton, twin, or triplet litter in Svalbard polar bears. (A) Maternal length and capture date (as day of the year, with 1 being January 1 of each year) were held at their mean value. The mean (open circles), 50% credible interval (thick lines), and 95% credible interval (thin lines) are represented. The sample size in each combination of litter size and maternal age is indicated. (B) Capture date was held at its mean value. The mean (solid lines) and the 95% credible interval (shaded areas) are represented.</figcaption></figure>

There was evidence that litter size varied with maternal length (Figure 2B), with females of intermediate length (194 cm) having a higher chance of having twins (quadratic coefficient for the logistic regression “2 cubs versus 1”: -0.24 [-0.45; -0.03], $p_d = 0.99$). Females <194 cm had higher chances of having a singleton, while females >194 cm had higher chances of having a singleton, but also triplets. Maternal length itself depended on maternal age (Appendix S3).

There was strong evidence that a later capture date was associated with a lower probability of having twins rather than a singleton for young females ($p_d = 0.97$). There was also evidence that pelagic females and females with unknown space-use had higher chances of having

twins rather than a singleton, compared to resident females ($p_d = 0.96$ and $p_d > 0.99$, respectively). Interannual variation was relatively low (Appendix S3: Figure S3B).

## Cub mass

Cub mass ranged between 3.00 and 31.25 kg. Our model provided very strong evidence that cub mass decreased with litter size (Figure 3A; Appendix S3: Figure S4A). Singletons were 1.20 [1.10; 1.31] times heavier than twins ($p_d = 1$), and 1.46 [1.27; 1.68] times heavier than triplets ($p_d = 1$). Twins were themselves 1.22 [1.08; 1.39] times heavier than triplets ($p_d > 0.99$). By contrast, there was
<figure><img src="image_4.png" /><figcaption>**FIGURE 3** Predicted direct influences of litter size and maternal characteristics on cub mass in Svalbard polar bears. (A) Predicted direct effect of litter size and maternal age on the mass of cubs. The mean (open circles), 50% credible interval (thick lines), and 95% credible interval (thin lines) are represented. Open circles in the background correspond to observations corrected for the effect of other variables and for the year random effect. The sample size (number of cubs) in each combination of litter size and maternal age is indicated. Predictions are made for female cubs of a resident female. Note that this leads to underestimating the mass difference between singleton cubs on the one hand and twin and triplet cubs on the other, since male cubs were heavier than female cubs, but in singleton litters only. Maternal length, maternal girth, and capture date (as day of the year, with 1 being January 1 of each year) were held at their mean values. Note that this means we do not account for the fact that maternal length and girth vary with litter size. (B) Predicted direct effect of maternal length on cub mass for a female cub belonging to a middle-aged resident female. (C) Predicted direct effect of maternal girth on cub mass for a female cub belonging to a middle-aged resident female. In (B) and (C), the mean (solid, dashed, or dotted line) and the 95% credible interval (shaded area) are represented. Points in the background correspond to observations corrected for the effect of other variables and for the year random effect. Continuous variables other than (B) maternal length or (C) maternal girth were held at their mean values.</figcaption></figure>

strong evidence that the mass of the whole litter increased with litter size (Appendix S3: Figure S4B), with twin litters being 1.67 [1.52; 1.82] times heavier than singleton litters ($p_d = 1$) and triplet litters being 1.23 [1.08; 1.39] times heavier than twin litters ($p_d > 0.99$).

Among singletons, there was very strong evidence that cub mass varied with maternal age, with middle-aged females having heavier singletons than young and old females ($p_d = 1$ and $p_d > 0.99$; Figure 3A). Similarly, middle-aged females had heavier twins than young
females ($p_d$ > 0.99; Figure 3A) and possibly old females ($p_d$ = 0.80). There was some evidence that the weight of cubs of young and old females differed in singleton and twin litters ($p_d$ = 0.81 and $p_d$ = 0.90). Because the weight of cubs (both singletons and twins) differed depending on maternal age, the mass difference between singletons and twins (i.e., the steepness of the size-number trade-off) also depended on maternal age. The relative mass difference was greater in middle-aged females than in young females ($p_d$ = 0.99) and in old females ($p_d$ = 0.97). Among triplets, our model provided evidence that cub mass varied with maternal age (Figure 3A), but the sample size was low for young and middle-aged females.

There was evidence that cub mass was positively related to maternal length in both singleton ($p_d$ = 0.94) and larger litters ($p_d$ > 0.99; Figure 3B). Maternal (length-corrected) girth and capture date were also positively associated with cub mass (all $p_d$ > 0.99; Figure 3C). Maternal girth itself depended on maternal age, maternal length, litter size, capture date, and year (Appendix S3). There was also strong evidence that male cubs were heavier than female cubs in singleton litters (ratio: 1.15 [1.02; 1.28], $p_d$ = 1; Appendix S3: Figure S6) but not in larger litters ($p_d$ = 0.66). Finally, there was very strong evidence that cubs of pelagic females and females with unknown space-use were lighter than cubs of resident females (all $p_d$ ≥ 0.99). Interannual variation in cub mass was substantial (Appendix S3: Figure S2B). Our model explained 38% of the variation in cub mass.

## Cub survival

Recapture probability was variable among years (Appendix S3: Figure S2B) but the mean remained low overall: 0.25 [0.19; 0.32] for resident females and their dependent offspring, 0.06 [0.04, 0.09] for females with unknown space-use and their dependent offspring, and 0.05 [0.03; 0.08] for pelagic females and their dependent offspring.

There was strong evidence of a marked positive effect of cub mass on cub survival ($p_d$ = 0.97; Figure 4A). There was also strong evidence for a direct effect of maternal age on cub survival (Figure 4B), with cubs of middle-aged females having a higher probability of survival than cubs of young females (difference: 0.36 [0.11; 0.61], $p_d$ > 0.99) and cubs of old females (difference: 0.27 [0.00; 0.54], $p_d$ = 0.97). There was strong evidence for a positive effect of maternal girth (corrected for length) on cub survival ($p_d$ = 0.96). There was little evidence for an effect (independent of cub mass) of maternal length, maternal girth, capture date, sex, and litter size on cub survival (all $p_d$ ≤ 0.80). Our model estimated a lower survival for cubs of pelagic females and females with an unknown space-use strategy ($p_d$ = 0.88 and 0.96, respectively), but this was likely a result of transience. Overall, most traits included in our PA-CR model influenced several other traits, which resulted in a complex web of direct and indirect effects (Figure 5).

<figure><img src="image_5.png" /><figcaption>**FIGURE 4** Predicted direct influences of (A) cub mass and (B) maternal age on cub survival in Svalbard polar bears. (A) The mean (solid lines) and the 95% credible interval (shaded areas) are represented. (B) The mean (open circles), 50% credible interval (thick lines), and 95% credible interval (thin lines) are represented. The sample size (number of cubs) in each maternal age group is indicated. In (A) and (B), the predictions are made for female twins. Maternal length, maternal girth, and capture date (as day of the year, with 1 being January 1 of each year) were set at their mean values. In (B), cub mass was set at its mean value (12.8 kg), along with maternal length, maternal girth, and capture date.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**FIGURE 5** Directed acyclic graph showing the direction and level of evidence supporting the tested relationships between maternal state, litter size, cub mass, and cub survival for Svalbard polar bears, as estimated by the hierarchical path analysis capture-recapture model. The posterior distributions of all fixed effects and variances estimated by our model are available in Appendix S3: Figure S1. Maternal age, maternal space-use strategy, litter size, and cub sex were coded as categorical variables. “Quadratic” relationship may either mean an actual quadratic effect or that the three levels of a categorical variable produce a concave/convex pattern. For maternal space-use strategy, the level of reference was “resident”; thus, the arrows emanating from this cell indicate the effect of not being a resident female. For the effect of maternal age on cub mass, the “quadratic” effect pertains to singleton and twin litters, while for the rare triplet litters, cub mass increased with maternal age.</figcaption></figure>

## Productivity depending on litter size

There was evidence that productivity (i.e., the number of cubs reaching their second spring, i.e., 15 months old) increased with litter size, having accounted for the differences in age, length, and girth between females with a singleton, females with twins, and females with triplets ($p_d$ > 0.99 for twin litters vs. singleton litters and for

triplet litters vs. twin litters; Appendix S3: Figure S7). However, the effect of litter size on productivity interacted with maternal age (Figure 6). For middle-aged and old females, there was strong evidence that twin litters were more productive than singleton litters (difference: 0.44 [0.08; 0.77] more cubs per litter reaching their second spring, $p_d$ > 0.99 for middle-aged females, and 0.28 [-0.04; 0.65], $p_d$ = 0.95 for old females). For young
<figure><img src="image_7.png" /><figcaption>**FIGURE 6** Predicted number of cubs that reach their second spring depending on litter size and maternal age in Svalbard polar bears. These predictions were made for a resident female using values of maternal girth and cub mass predicted by the path analysis capture-recapture model for each combination of maternal age and litter size, assuming a cub sex ratio at unity. Maternal length was set to its mean value (having corrected for the effect of space-use strategy and the cohort random effect) within each combination of maternal age and litter size. The capture date was set at its mean value. The mean (open circles), 50% credible interval (thick lines), and 95% credible interval (thin lines) of our predictions are represented. The sample size (number of litters) in each combination of litter size and maternal age is indicated.</figcaption></figure>

females, there was weaker evidence that twin litters were more productive than singleton litters ($p_d = 0.87$). Our model further suggested that the productivity of triplet litters was higher than that of twin litters in both middle-aged and old females ($p_d = 0.97$ and 1, respectively).

## DISCUSSION

Using the polar bear as a case study, we investigated the optimization of litter size in a species with small litters by testing two competing hypotheses, the “common optimum” hypothesis and the “state-dependent optimization” hypothesis. We went further by analyzing offspring mass, the main mediator of the trade-off between offspring number and survival. Our path model allowed us to perform a single analysis accounting for dependencies among state variables and disentangling direct from indirect effects (Figure 5). Imperfect detection is a major issue in many studies of wild populations. One novelty in our study is that we used capture-recapture modeling within our path model to estimate cub survival and relate it to cub mass, litter size, and maternal state. This approach allowed us to rigorously assess the optimization of litter size. Below, we discuss how maternal state influences litter size and cub mass and how all these variables influence cub survival. This allows us to discuss how

female polar bears optimize their litter size. We then outline some limitations of our study.

## State dependence of litter size

During maternity denning, polar bears are capital breeders (Atkinson & Ramsay, 1995). Thus, a female's state before den entry and during denning is indicative of her ability to care for a given number of offspring until den emergence. Accordingly, body condition in autumn, before den entrance, is a key determinant of litter size (Molnár et al., 2011). No measure of autumn condition was available in our population as captures took place in spring. Nonetheless, the maternal state variables available in our study proved to be determinants of litter size, though they potentially acted through body condition.

In accordance with studies on the same population (Folio et al., 2019; Naciri et al., 2022) and on Hudson Bay polar bears (Derocher & Stirling, 1994), we found that middle-aged females had the highest probability of having twins in spring. Derocher et al. (1992) reported that young and old pregnant females are lighter than middle-aged females in early autumn, before den entry. Given the positive effect of body condition at den entry on litter size (Molnár et al., 2011), it is reasonable to assume that much of the effect of age on litter size that
we document is mediated by body condition at den entry. Other, condition-independent factors may also influence litter size in spring, such as lower quality of care from young females (Snyder et al., 2016).

Despite the limited number of triplet litters in our data and the resulting uncertainty in parameter estimates, we detected a positive effect of size and of being old on the probability of having triplets. These results are consistent with previous findings (Lim et al., 2014; Ramsay & Stirling, 1988), suggesting that our estimates regarding triplets were reliable. Although triplets were rare, one female was encountered thrice with different triplets each time (including the only yearling triplets recorded in our population) and another was encountered twice with different triplets each time. This apparent repeatability of litter size suggests that some high-quality females are able to consistently produce triplets, whereas average females hardly ever do.

Contrary to what Gaillard et al. (2014) suggested, even in species producing small litter sizes, females seem able to adjust their litter size to their state, despite the fact that increasing their litter size by 1 offspring represents a much larger relative increase than it does in species producing larger litters. We note that the state dependence of litter size may be more likely in capital breeders—which can predict their future state based on their current state—than in income breeders.

## Cub and litter mass

As predicted and in accordance with many studies (in polar bears: Derocher & Stirling, 1998; Ramsay & Stirling, 1988; see Lim et al., 2014 for a meta-analysis), offspring number was traded against offspring size in polar bears, with singletons, twins, and triplets weighing a mean of 15.1, 12.6, and 10.2 kg, respectively. Given the positive relationship between cub mass and cub survival (see below), this means that female polar bears trade offspring number against offspring survival. By contrast, litter mass increased with litter size, as predicted and previously reported (Ramsay & Stirling, 1988), from 15.1 kg for a singleton litter to 30.5 kg for a triplet litter. This indicates that although fewer resources were allocated per offspring in larger litters, more resources were allocated to the litter overall. This positive relationship between litter size and litter mass is consistent with state dependence of litter size: if females who have more resources to allocate have larger litters, larger litters are expected to be heavier than small litters.

Alongside litter size, maternal age was a major determinant of cub (and litter) mass, with young and old females having lighter cubs than middle-aged females.

Maternal length and length-corrected girth in spring influenced cub mass too, with longer and thicker females having heavier cubs. Studies in polar bears and brown bears have related litter mass in spring to body condition at den entry (Atkinson & Ramsay, 1995; Robbins et al., 2012; Rode et al., 2020). Here too, the effects of maternal age and maternal girth in spring likely reflect body condition at den entry. Maternal length positively affected cub mass in spring likely because, for a given body condition, large females have larger energy stores and use them more efficiently than small females (Lindstedt & Boyce, 1985). Taken together, the results discussed so far indicate that there is variation in how much energy females allocate to reproduction, as measured by litter mass, and that females who allocate more energy overall tend to have more offspring.

A lot of variability in cub mass remained unexplained. Unpublished birth dates (inferred using unpublished temperature and activity data obtained via satellite telemetry collars) indicate that in our population, births are spread over almost two months, inevitably leading to mass differences at capture in spring. In polar bears, the implantation of embryos after the diapause is thought to be triggered by photoperiod (Spady et al., 2007), but it is possible that other factors such as body condition in late summer can modulate the timing of implantation, as found in European badgers (*Meles meles*) and brown bears (Robbins et al., 2012; Woodroffe, 1995). In this case, the higher mass of cubs of middle-aged female polar bears could be at least partly due to earlier birth. Regardless, the implications in terms of cub survival of being large in spring are likely similar irrespective of whether the cub has grown fast or was born earlier. By giving birth early, females in good body condition would ensure that their cubs have grown sufficiently inside the den before being exposed to the outside environment, increasing their survival prospects.

## Cub survival

As predicted and in accordance with Derocher and Stirling (1996) and studies in other species (Ronget et al., 2018), we found that cub survival increased with cub mass. Moreover, as predicted by the theory (Lloyd, 1987), the relationship between cub mass and survival was sigmoid, meaning that beyond a certain cub mass, it becomes more profitable for a female to increase litter size than to allocate more energy per cub. Nonetheless, most of the observed values of cub mass were in the range where the curve relating cub mass to cub survival is steep, suggesting strong positive selection on increased cub mass.

We also found that maternal age strongly influences cub survival with a peak in midlife, as predicted by
theoretical models (Moorad & Nussey, 2016), and recorded in other large carnivores (Payne et al., 2024; Stahler et al., 2013). If, as discussed above, maternal age reflects body condition, the age effect on cub survival may reflect the greater ability of middle-aged females to acquire food and provision their offspring relative to young and old females. This would suggest that the “direct” effect of maternal age is mediated by cub mass after capture. This greater food acquisition interpretation fits well with the fact that improved maternal body condition at capture is consistently found to increase offspring survival in polar bears (this study, Ramsay & Stirling, 1988; Rode et al., 2020) and other species (Ronget et al., 2018).

In a study on cub survival in the same population using a multievent capture-recapture model, Naciri et al. (2022) failed to detect a decline in cub survival from middle age to old age. However, they only used data up to 2019. When we repeated their analysis using data up to 2025, we detected a decline in cub survival in late life, especially for twins. It is unclear why the results from Naciri et al. (2022) vary so substantially depending on whether data from 2020 to 2025 are included. By contrast, our model produces the same age-related pattern in cub survival regardless of whether data from 2020 to 2025 are included (Appendix S3: Figure S8).

## Productivity depending on litter size

The last question addressed by our analysis is whether females who allocated a lot of resources to reproduction had more numerous surviving cubs, that is, if they were “right” to have larger litters with smaller cubs rather than smaller litters with larger cubs. We found that, across all age groups, productivity increased with litter size (in support of the state-dependent optimization hypothesis, Appendix S3: Figure S7). When considering age classes, the same pattern was evident in both middle-aged and old females, but not in young females. The latter negative result, along with the limited number of triplet litters in our dataset, means that our results cannot decisively support our prediction of increased productivity with increased litter size.

Young females tended to lose a cub over the sampling period, that is, shortly after den emergence (Folio et al., 2019 and this study). This cub loss suggests that they are unable to adequately adjust their litter size before substantial investment into their offspring and instead suffer litter reduction at den emergence after several months of costly lactation. This inability of young females to produce the optimal litter size given their state, perhaps caused by a lack of breeding experience, may explain why we had difficulties detecting an increase

in productivity with increasing litter size in young females.

## Limitations

Gaining a complete understanding of the optimization of litter size would require studying the full period of maternal care (and beyond). However, doing so requires knowing offspring mass, litter size, and maternal state for all yearlings of our dataset, which was not the case for those individuals captured as cubs but not recaptured as yearlings. We therefore restricted our analysis to first-year cub survival. Regardless, we believe the most critical period of the reproductive cycle for mothers is the denning period (particularly following parturition), which requires enormous amounts of energy stores (Atkinson & Ramsay, 1995), and the cubs' first year, when they are highly dependent on milk (Archer et al., 2023; Arnould & Ramsay, 1994).

A limitation of this study is that we considered that the optimization of litter size occurs within each breeding attempt. However, high reproductive effort in the current reproduction tends to reduce future reproductive success in long-lived species (Hamel et al., 2010). Consequently, the optimal litter size for an individual at a given breeding attempt is not necessarily that which maximizes the productivity of this breeding attempt but that which maximizes productivity over the individual's whole life (Charnov & Krebs, 1974). As polar bears are long-lived, jeopardizing future reproduction prospects by over-investing in current reproduction may strongly impact lifetime reproductive success. Unfortunately, our data are too sparse to test this hypothesis.

## Conclusions

Our findings of state dependence of litter size and of increasing litter mass with increasing litter size are consistent with a state-dependent adjustment of litter size in a capital breeder. We further found some support for increasing productivity with increasing litter size. Our results are thus consistent with a state-dependent optimization of litter size in this large carnivore with small litters.

## AUTHOR CONTRIBUTIONS

Jon Aars, Andrew E. Derocher, Magnus Andersen, and Øystein Wiig conducted the fieldwork and curated the data. Marwan Naciri and Sarah Cubaynes designed the study with insight from Marlène Gamelon, Jon Aars, and Marie-Anne Blanchet. Marwan Naciri analyzed the data
and drafted the manuscript. All authors contributed to the manuscript.

## ACKNOWLEDGMENTS

We are grateful to Thor Larsen for initiating the monitoring and to the many people who have been involved in the monitoring in different ways over the years. We thank Statoil, WWF, and Polar Bears International for support.

## CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

## DATA AVAILABILITY STATEMENT

Data and code (Naciri et al., 2025) are available on Zenodo: https://doi.org/10.5281/zenodo.15870511.

## ORCID

Marwan Naciri https://orcid.org/0000-0003-1489-9363

Andrew E. Derocher https://orcid.org/0000-0002-1104-7774

Marlène Gamelon https://orcid.org/0000-0002-9433-2369

Øystein Wiig https://orcid.org/0000-0003-0395-5251

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**How to cite this article:** Naciri, Marwan, Jon Aars, Magnus Andersen, Marie-Anne Blanchet, Andrew E. Derocher, Marlène Gamelon, Øystein Wiig, and Sarah Cubaynes. 2025. "Offspring Number, Size, and Survival: State-Dependent Optimization of Litter Size in a Long-Lived Capital Breeder." *Ecosphere* 16(8): e70370. https://doi.org/10.1002/ecs2.70370