Is individual heterogeneity in growth rates relevant to population dynamics of long‐lived reptiles?

Abstract

Many populations show pronounced individual heterogeneity in traits such as somatic growth rates, but the relevance of this heterogeneity to population dynamics remains unclear. Individual heterogeneity may be particularly relevant to long‐lived organisms for which vital rates (survival and reproduction) increase with adult growth, as subtle differences in growth rates can have major fitness consequences. Previous analysis of data for snapping turtles (Chelydra serpentina) in Algonquin Park, Canada, from 1972 to 2012 showed that individual heterogeneity in growth rates and size‐specific reproductive rates of adult females led to eightfold variation in lifetime reproductive output. Here, we test whether this individual heterogeneity affects population dynamics by comparing projections from alternative integrated population models (IPMs) where: (1) vital rates of adult females increase with size and there is individual heterogeneity in their adult growth and reproduction parameters as well as their ages at maturity; (2) vital rates increase with size but there is no individual heterogeneity; or (3) vital rates are assumed equal among adult females. The three IPMs all integrated component models for growth, reproduction, and survival, incorporated random annual variation in rates, and used data augmentation to model unobserved individuals including future recruits. The data augmentation approach allowed the individual heterogeneity in parameters to be extrapolated from observed to unobserved individuals under Model 1. Models 1 and 2 produced similar annual estimates of recruitment, mortality, and abundance from 1972 to 2012 and similar projections for the next 10 years. Those projections had wide prediction intervals (5% increase to 73% decline) due to annual variation in rates but were consistent with the 59% decline estimated based on new data collected from 2012 to 2022. The projected decline reflected predicted decreases in survival and recruitment due to a decrease in the average body size of adult females. Consequently, Model 3 gave more optimistic projections that were inconsistent with the observed decline. The results therefore showed that the size composition of adult females, and therefore their overall somatic growth rate, was important to the dynamics of the population. However, the results also indicated that the pronounced individual heterogeneity in growth rates observed was irrelevant to population dynamics.

INTRODUCTION

Population ecology is a fundamental aspect of ecological theory that is essential for informing conservation actions. Conservation decisions are often based on projected growth or decline of populations in response to management actions (Williams et al. 2002) or environmental change (Palmer et al., 2017). Population projections are especially important for long‐lived species, for which declines may be difficult to both detect and reverse, making it necessary to project decades into the future to guide current management (Spencer & Janzen, 2010). The models used to make projections must therefore reflect current theory on population dynamics.

Theory on population dynamics includes factors limiting population growth, the role of environmental and demographic stochasticity, and the role of population composition (Turchin, 2001). It has long been recognized that sex ratios and age structures of populations may shift over time and that failure to account for these shifts in composition will lead to biased projections (Lefkovitch, 1965; Leslie, 1945). Consequently, population models typically categorize individuals into discrete sex and age (or stage) classes and incorporate class‐specific estimates of vital rates, that is, survival and reproduction (Caswell, 2001).

It is now evident, however, that there may be pronounced variation among individuals that is not explained by sex, age, or environmental stochasticity and that some of this variation reflects individual differences in life histories (Cam et al., 2016; Plard, Fay, et al., 2019; Sorel et al., 2024). It is well known that failure to account for individual heterogeneity leads to misinterpretation of state dependence in vital rates. For example, apparent increases in survival probability with age may disappear when individual heterogeneity in frailty is modeled (Armstrong et al., 2021; Cam et al., 2016; Wienke, 2010). However, the relevance of individual heterogeneity to population dynamics is unclear.

Similar to the effects of sex and age structure, shifts in the composition of individuals in a population are expected to have some effect on vital rates and therefore population dynamics. However, the hypothesized effects of individual heterogeneity on populations are complex, with many possible effects acting simultaneously and potentially canceling out (Kendall & Fox, 2002; Stover et al., 2012; Vindenes et al., 2008). In addition, it is unclear whether such effects are likely to be a significant driver of population dynamics in relation to other sources of variation such as demographic and environmental stochasticity. It is therefore important to test the significance of individual heterogeneity on population dynamics in real‐world systems where strong heterogeneity in vital rates has been demonstrated (Forsythe, 2025). Experimental manipulation of individual heterogeneity is only likely to be possible in laboratory systems involving organisms with rapid life histories (e.g., Cressler et al., 2017). However, the effects of individual heterogeneity on natural populations can be tested by fitting models that do or do not incorporate individual heterogeneity and comparing the capacities of these models to interpret past population dynamics and project future trends.

Incorporating observed heterogeneity in vital rates into population models is technically challenging, both because the modeled heterogeneity needs to be extrapolated from observed to unobserved individuals and because other sources of variation need to be modeled simultaneously. The advent of integrated population models (IPMs) fitted using Markov chain Monte Carlo (MCMC) techniques (Plard, Fay, et al., 2019; Schaub & Abadi, 2011) has provided a major advance for addressing such challenges, not only because IPMs allow multiple data types to be analyzed in a unified framework but also because IPMs allow multiple sources of variation to be quantified and for the associated uncertainties to be propagated into population projections (McGowan et al., 2011). Plard, Turek, et al. (2019) combined an integral projection model (Easterling et al., 2000; Ellner et al., 2016) with an IPM to incorporate individual heterogeneity in a trait (egg laying date) when projecting population dynamics of barn swallows (Hirundo rustica) in Switzerland. Under this approach, which they termed integrated integral projection model (IPM²), laying date was modeled as a continuous statistical distribution that was extrapolated to the wider population. Armstrong et al. (2021) developed an alternative approach in which individual heterogeneity in multiple traits was incorporated into an IPM using data augmentation, and the model was used to project dynamics of a small hihi (Notiomystis cincta) population in New Zealand. Data augmentation involves adding hypothetical unobserved individuals to a data set. Although data augmentation is normally used to estimate the number of unobserved individuals present over the course of a study (Royle et al., 2007; Royle & Converse, 2014; Royle & Dorazio, 2008), it can also be extended to population projections. The data augmentation approach is more cumbersome than integral projection, in that traits are assigned to each hypothetical future individual rather than described by a distribution. However, the data augmentation approach is flexible and intuitive, as the MCMC sampling used to model individual heterogeneity in observed individuals is extrapolated to unobserved individuals in both the past and future.

One form of individual heterogeneity that could be particularly relevant to population dynamics is variation in somatic growth rates. Variation in juvenile growth rates leads to differences in age and/or size at maturity, and in species that continue to grow as adults (indeterminate growth), size differences may become more pronounced over individuals' lifetimes. Such heterogeneity may be particularly relevant to long‐lived ectotherms in which survival and reproduction increase with size, as even subtle differences in growth can have major lifetime fitness consequences. Previous analysis of data from a snapping turtle (Chelydra serpentina) population in Algonquin Provincial Park, Canada, from 1972 to 2012 showed that individual heterogeneity in growth rates and size‐specific reproduction rates of adult females led to eightfold variation in expected reproductive output after maturity (Armstrong et al., 2018). Here, we extend this modeling using data augmentation to test whether this individual heterogeneity affects predicted population dynamics. We specifically compare projections from alternative IPMs, where: (1) survival and reproduction rates of adult females are modeled as a function of body size, and there is individual heterogeneity in adult growth and reproduction parameters as well as age at maturity; (2) survival and reproduction rates of adult females are modeled as a function of body size, but there is no individual heterogeneity in parameters; or (3) survival and reproduction rates are assumed equal among adult females, that is, there is no size dependence. We compare model outputs in terms of annual recruitment, mortality, and egg production from 1972 to 2012 and population projections from 2012 to 2022, and we compare the projections from 2012 to 2022 to the abundance estimates obtained from subsequent survey data.

METHODS

Species and study population

North American snapping turtles are omnivorous predators and scavengers that live in lakes, ponds, and slow‐moving rivers (Steyermark et al., 2008). Adult females produce eggs annually in spring or early summer and bury them in sandy soil or gravel near water (Congdon et al., 2008). The study area surrounds the Algonquin Wildlife Research Station (AWRS; 45°35′ N, 78°30′ W) in Algonquin Provincial Park, Ontario, Canada, near the climatic northern edge of the species' range. We specifically focused on the population within the North Madawaska River catchment, which covers a ca. 100‐km² area surrounding the research station (see Appendix S1 for map and further details). Many females cross a highway to nest, so the population is subject to road mortality (Keevil et al., 2023) as well as winter mortality, especially from river otters (Lontra canadensis) (Keevil et al., 2018). The population declined by ca. 50% during a period of high otter predation in the late 1980s and showed no evidence of recovery over the subsequent 23 years (Brooks et al., 1991; Keevil et al., 2018).

Female snapping turtles in the Algonquin Park population predictably reach maturity (start nesting) when their midline carapace length reaches 24 cm, but their age at maturity is expected to be highly variable due to heterogeneity in juvenile growth rates (Armstrong & Brooks, 2013). Annual survival and reproductive rates of adult females increase with size as they continue to grow after maturity, and individual heterogeneity in their growth rates is estimated to result in fourfold variation in lifetime reproductive outputs (Armstrong et al., 2018). This increases to eightfold variation when individual heterogeneity in size‐specific reproductive rates is also accounted for (Armstrong et al., 2018).

Data sets

Encounter histories, clutch sizes, and carapace lengths of adult females were obtained by patrolling most of the likely nesting habitat in the North Madawaska River catchment several times per day throughout the nesting season (usually late May to late June) from 1972 onward. Other sites in the surrounding landscape were also monitored, and we used the data on growth and reproductive output from these other sites but not the encounter histories. We captured turtles by hand, immediately if they were not nesting and otherwise at the completion of nesting. We individually marked turtles on first capture by wiring a numbered aluminum tag to the rear edge of the carapace and notching the edges of the carapace (Loncke & Obbard, 1977), measured midline carapace length to the nearest 1 mm using calipers on each capture (method D in Iverson & Lewis, 2018), and recorded clutch sizes by temporarily excavating nests. We permanently marked hatchlings by toe‐clip in some years, allowing ages to be known for females that survived to maturity and began nesting in the survey area (see below). All procedures conformed to the guidelines of the Canadian Council on Animal Care and were approved by the Animal Care Committees of the University of Guelph, Laurentian University, and University of Toronto.

We arranged the data from 1972 to 2012 into a set of parallel matrices where rows corresponded to individual females and columns to years. The encounter histories matrix indicated whether each turtle was encountered (1) or not (0) each year, whereas the clutch size and carapace length matrices consisted of the clutch sizes or carapace lengths in the years they were recorded, with missing values (NA) entered otherwise. To facilitate the fitting of the Jolly–Seber component of the IPMs (see below), we also generated a “notyetdead” matrix (Link & Barker, 2010), indicating whether each female was known to be not yet dead (1), known to be dead (0), or her status was unknown (NA) each year. To model the effect of nest site on detection probability, we also generated a site matrix indicating whether females were encountered at Sasajewun Lake (1), Amoyoa Swamp (2), or another site (3), or NA if not encountered. This site matrix allowed us to account for the higher detectability of females nesting at Sasajewun Lake due to its accessible location within the AWRS property and to account for the fact that Amoyoa Swamp was surveyed only in some years due to its inaccessibility.

We fitted the population models to all data from the 223 females captured in the North Madawaska River catchment from 1972 to 2012, in combination with somatic growth and clutch‐size data from 114 females from the surrounding landscape. We used the encounter histories from the 89 females captured in the North Madawaska River catchment from 2012 to 2022 to obtain population abundance estimates to compare to the projections from the models. Both data sets are available in Armstrong et al. (2025).

Modeling

The IPMs used in this study extended Armstrong et al.'s (2018) analysis which integrated hierarchical versions of a Bertalanffy growth model, a linear model for size‐dependent reproductive output, and a Cormack–Jolly–Seber model for size‐dependent survival (see below). The key extensions were (a) incorporating a Jolly–Seber model (Jolly, 1965; Seber, 1965) rather than a Cormack–Jolly–Seber model so that annual abundance (number of adult females) and recruitment (number of newly matured females) could be estimated as well as survival; (b) incorporating data on ages at maturity for known‐aged females so that recruitment could be modeled as a function of past egg production; (c) using data augmentation (Link & Barker, 2010) to extend the hierarchical modeling of individual heterogeneity to unobserved females; and (d) extending this modeling to future recruits to project population dynamics 40 years into the future. To assess the sensitivity of projections to individual heterogeneity and size dependence, we created three alternative IPMs. The full model (Model 1) incorporated the individual heterogeneity in adult growth rates and size‐specific reproduction rates detected by Armstrong et al. (2018) and also incorporated heterogeneity in age at maturity to reflect heterogeneity in juvenile growth rates. We first reduced this model by removing all forms of individual heterogeneity (“no individual heterogeneity,” Model 2) and then further reduced it by removing size dependence in survival and reproductive output (“no size dependence,” Model 3). We also used a separate Jolly–Seber model to estimate population abundance from the new data collected from 2012 to 2022. Code for each of these models is available in Armstrong et al. (2025).

The growth, reproduction, and survival functions used in the full IPM (Model 1) were

$$E\left(L_{\mathit{ij}}\right)=a_i-\left(a_i-L_{\mathit{ij}-1}\right)\times \exp \left(-k_{\mathit{ij}}/a_i\right),$$ (1)

$$E\left({\mathit{CS}}_{\mathit{ij}}\right)={\mathrm{\alpha }\mathit{CS}}_{\mathit{ij}}+\mathrm{\beta }\mathit{LCS}\times \left(L_{\mathit{ij}-1}-24\right),$$ (2)

and

$$\text{logit}\left({\varnothing }_{\mathit{ij}}\right)=\mathrm{\alpha }{\varnothing }_j+\mathrm{\beta }L\varnothing \times \left(L_{\mathit{ij}-1}-24\right),$$ (3)

where $E\left(L_{\mathit{ij}}\right)$, $E\left({\mathit{CS}}_{\mathit{ij}}\right)$, and ${\varnothing }_{\mathit{ij}}$ are the expected carapace length, expected clutch size, and annual survival probability, respectively, for adult female i in year j; $a_i$ and $k_{\mathit{ij}}$ are her growth parameters; ${\mathrm{\alpha }\mathit{CS}}_{\mathit{ij}}$ and $\mathrm{\alpha }{\varnothing }_j$ are her expected clutch size and survival probability if she were 24 cm in carapace length; and $\mathrm{\beta }\mathit{LCS}$ and $\mathrm{\beta }L\varnothing$ are the effects of a 1‐cm increase in carapace length on clutch size and survival probability, respectively. The parameters $a_i$, $k_{\mathit{ij}}$, ${\mathrm{\alpha }\mathit{CS}}_{\mathit{ij}}$, and $\mathrm{\alpha }{\varnothing }_j$ were subject to individual and/or annual random effects that were taken to be normally or log‐normally distributed (see Armstrong et al., 2018 for further explanation). $L_{\mathit{ij}}$ and ${\mathit{CS}}_{\mathit{ij}}$ were taken to have normal and Poisson error distributions, respectively. Detection probability was modeled as a logit‐linear function of site and year, with year treated as a random effect. We modeled missing site values by sampling each female's site (Sasajewun, Amoyoa, or other) from a multinomial distribution each year, with uniform prior probabilities for her initial site and subsequent transition probabilities.

Model 1 was reduced to Model 2 by removing the individual heterogeneity from these functions (i.e., $a_i$, $k_{\mathit{ij}}$, and ${\mathrm{\alpha }\mathit{CS}}_{\mathit{ij}}$ became $a$, $k_j$, and ${\mathrm{\alpha }\mathit{CS}}_j$) and from age at maturity (see below). Model 2 was reduced to Model 3 by removing the terms for size dependency (i.e., $\mathrm{\beta }\mathit{LCS}$ and $\mathrm{\beta }L\varnothing$ were set to 0). There was no individual heterogeneity in the size‐specific survival parameter ($\mathrm{\alpha }{\varnothing }_j$) under any model due to lack of evidence for this form of heterogeneity (Armstrong et al., 2018). Similarly, there were no trade‐offs between individual random effects (e.g., faster growing females having lower size‐specific survival rates) due to lack of evidence for them (Armstrong et al., 2018). Annual random effects were included in all functions due to strong evidence of annual variation that needed to be accounted for in population projections.

The Jolly–Seber component of the IPMs was modified from Link and Barker's (2010: 257–258) formulation, in which individuals' times of entry and death are modeled to allow annual recruitment to be estimated (Crosbie & Manly, 1985; Schwarz & Arnason, 1996), and data augmentation is used to estimate abundances of unobserved individuals. The data augmentation involved adding rows of zeros to the encounter histories matrix to represent hypothetical unobserved adult females and modeling their probabilities of having existed. Rows of missing values (NA) were added to the parallel matrices of clutch size and carapace length data, and these rows were used to model the somatic growth and reproduction of the hypothetical females. Individual random effects for growth and reproduction parameters were assigned to these hypothetical females based on the hierarchical modeling of data for the observed females, and a 24‐cm carapace length (i.e., size at maturity) assigned at entry. Consequently, growth, reproduction, and survival could be integrated for both observed and unobserved females, allowing annual estimates of abundance, egg production, and deaths that account for individual heterogeneity in parameters as well as size dependence in rates. We initially added 30 hypothetical unobserved females to the 1972–2012 data set (the number is expected to be low due to high detection probability) and then progressively increased this number to check that results were insensitive to these increases, and therefore, the augmentation was sufficient. The Jolly–Seber model used to estimate population abundance from 2012 to 2022 had a similar formulation but with survival and detection functions including only annual random effects.

We generated projections by modeling the future growth, reproduction, and survival of these females using the same processes as during the study period, and we further extended this modeling to hypothetical future recruits, which were also assigned individual growth and reproduction parameters and an initial carapace length of 24 cm. We set the number of hypothetical female recruits to 20 each year based on preliminary modeling and assigned each of these an order number, meaning that an individual would only recruit if the number of female recruits equaled or exceeded that number. The number of female recruits in each future year was a binomial sample from the potential number based on past egg production, with the probability of an egg generating a female recruit (prec) estimated based on the annual recruitment and egg production from 1972 to 2012. For Model 1, the individual heterogeneity in age at maturity (and therefore juvenile growth rate) was sampled by allocating the eggs produced each year to different recruitment years (i.e., the year they potentially reach maturity) based on a multinomial distribution for age at maturity (AM). The total number of potential recruits each year was calculated by summing the numbers allocated from all cohorts of eggs. For Models 2 and 3 (without individual heterogeneity), we considered the potential number of recruits in year j to be ${\text{eggs}}_{j-\mathrm{AM}}$, where ${\text{eggs}}_j$ is the total number of eggs produced in a year and AM is the mean age at maturity.

We generated the multinomial distribution for AM using data from 13 turtles who were marked as hatchlings and subsequently recruited into the population as adult females. Turtles hatched after 1992 were excluded so that the interval from hatching to the last year of monitoring was ≥30 years, which we considered the maximum plausible AM. For each of these 13 females, we determined the maximum AM based on the first year she was found to be nesting or gravid and determined minimum AMs by assuming that females were immature if they were (1) found non‐gravid early in the season; (2) estimated to have <1% probability of having entered the population as adult females based on a Jolly–Seber model, or (3) estimated to have <1% probability of having a carapace length ≥24 cm. We considered the 13 AMs to be missing values subject to these constraints, sampled these values from a log‐normal distribution, and sampled new rounded values from that distribution to obtain the multinomial distribution (code and data for generating the AM distribution are available in Armstrong et al., 2025).

We modeled the data using OpenBUGS (Spiegelhalter et al., 2014), which uses MCMC fitting, allowing multiple data types to be integrated and uncertainties and covariances to be propagated into population projections (Besbeas et al., 2005; Schaub & Abadi, 2011). We used uninformative priors for all parameters. We ran two MCMC chains for 100,000 iterations after a burn‐in of 50,000 iterations, checking for convergence using Brooks–Gelman Rhat statistics (Brooks & Gelman, 1998) and visual inspection of the chains.

RESULTS

The estimated growth, reproduction, survival, and detection parameters for adult females under the full IPM were similar to those from the original analysis (Armstrong et al., 2018), with pronounced individual heterogeneity in growth rate, asymptotic carapace length and size‐specific clutch sizes, and annual variation in growth rates, clutch sizes, survival probability, and detection probability (Table 1). Under these parameters, an average female increases from 24 to 30 cm in carapace length over 60 years, resulting in her expected clutch size increasing from 28 to 39 eggs and her annual survival probability increasing from 0.87 to 0.97. The minimum and maximum ages at maturity (AM) for the 13 known‐aged females ranged from 9 to 20 years and from 16 to 31 years, respectively, and modeling of those data gave a multinomial distribution ranging from 11 to 25 years with a mean of 17 years (Figure 1). The probability of an egg becoming a female recruit was estimated to be 0.0015 under all models (Table 1). This probability is equivalent to 3 out of 1000 eggs becoming an adult female if the primary sex ratio and survival are equal between the sexes. In the Algonquin Park population, in which the sex ratio at hatching is slightly female‐biased (0.64 female; Leivesley et al., 2022), this equates to ca. 2 out of 1000 female eggs surviving to maturity.

TABLE 1.

Estimated population parameters (SE) for female snapping turtles in Algonquin Park, Canada, under three different integrated population models fitted to growth, reproduction, and encounter data collected from 1972 to 2012.

Explanation	Parameter	Estimate (SE)
		Full model a	No individual heterogeneity b	No size dependence b
Mean asymptotic carapace length (CL)	$a_c$	31.710 (0.239)	32.080 (0.257)
Mean log growth rate	$\log \left(k_c\right)$	−0.226 (0.126)	−0.285 (0.117)
Among‐individual heterogeneity in asymptotic CL	SD ($\mathrm{\mu }a$)	0.368 (0.193)
Among‐individual heterogeneity in log growth rate	SD ($\mathrm{\mu }k$)	0.413 (0.048)
Among‐year variation in log growth rate	SD ($\mathrm{\mu }\mathit{tk}$)	0.558 (0.098)	0.578 (0.104)
Residual variation in CL	SD ($\mathrm{\epsilon }L$)	0.186 (0.003)	0.196 (0.003)
Mean intercept for clutch size c	${\mathrm{\alpha }\mathit{CS}}_c$	28.230 (0.719)	27.320 (0.379)	36.210 (0.273)
Effect of 1‐cm increase in CL on mean clutch size	$\mathrm{\beta }\mathit{LCS}$	1.710 (0.147)	1.899 (0.070)
Among‐individual heterogeneity in clutch size	SD ($\mathrm{\mu }\mathit{CS}$)	5.070 (0.298)
Among‐year variation in clutch size	SD ($\mathrm{\mu }\mathit{tCS}$)	0.373 (0.224)	0.967 (0.221)	1.383 (0.253)
Probability of an egg becoming a female recruit	$\text{prec}$	0.002 (0.0002)	0.002 (0.0002)	0.0015 (0.0002)
Intercept for logit annual survival probability c	$\mathrm{\alpha }\varnothing$	1.886 (0.295)	1.859 (0.291)	3.075 (0.233)
Effect of 1‐cm increase in CL on logit survival prob.	$\mathrm{\beta }L\varnothing$	0.287 (0.041)	0.288 (0.042)
Among‐year variation in logit survival probability	$\mathrm{SD}\left(\mathrm{\mu }t\varnothing \right)$	1.215 (0.234)	1.205 (0.232)	1.058 (0.230)
Intercept for logit detection probability at Sasajewun	$\mathrm{\alpha }P$	1.634 (0.200)	1.631 (0.198)	1.733 (0.165)
Difference in logit detection probability at other sites	$\mathrm{\beta }P$	−1.011 (0.154)	−1.004 (0.157)	−0.966 (0.154)
Among‐year variation in logit detection probability	$\mathrm{SD}\left(\mathrm{\mu }P\right)$	1.062 (0.144)	1.061 (0.146)	0.790 (0.121)

^a Under the full model, survival probability and reproductive output (clutch size) are size dependent, and individual females vary in their growth parameters, size‐specific reproductive rates, and ages at maturity.

^b Under the second model, there is no individual heterogeneity in parameters, and under the third model, there is also no size dependence.

^c For models with size‐dependent rates, intercepts refer to the expected rate for a female with 24‐cm carapace length—the size at which nesting typically begins.

FIGURE 1.

Age‐at‐maturity distribution for female snapping turtles in Algonquin Provincial Park, Canada. The distribution was derived by fitting a log‐normal distribution to data on minimum and maximum possible ages at maturity for 13 turtles that were marked as hatchlings from 1982 to 1992 and subsequently observed as reproductive females by 2022.

The three alternative IPMs (Models 1, 2, and 3) showed good convergence for all main and hyperparameters within 50,000 MCMC iterations (output provided in Armstrong et al., 2025). These parameters were also insensitive to additional data augmentation, remaining constant as we increased the number of unobserved females in the 1972–2012 data set from 30 to 100. However, the posterior distribution for the initial (1972) number of females widened with progressive augmentation, with consequent effects on estimated numbers of females, recruits, and eggs produced over the next three years, so these parameters were considered inestimable for 1972–1975 (Appendix S2). We therefore only report estimates of population abundance, deaths, and recruits from 1976 onward (Figures 2 and 3).

FIGURE 2.

Annual estimates of abundance and 40‐year projections for adult female snapping turtles in the North Madawaska River catchment of Algonquin Provincial Park, Canada, based on three models fitted to data collected from 1972 to 2012. Females are considered to reach maturity at 24‐cm carapace length, the size when nesting typically begins. Under the full model (Model 1), survival probability and reproductive output (clutch size) are size dependent, and individual females vary in their growth parameters and size‐specific reproductive output. Under Model 2, survival and reproduction rates are size dependent, but there is no individual heterogeneity. Under Model 3, there is no size dependence in rates. Model 1 also incorporates individual heterogeneity in age at maturity (Figure 1), whereas Models 2 and 3 assume all females recruit at 17 years. All models include random annual variation in rates. The colored dots and error bars for 1972–2012 show the medians and 95% credible intervals for abundance under each model, and the solid and dotted lines for 2012–2052 show the medians and 95% prediction intervals for the population projections. The black dots and error bars show median abundances and 95% credible intervals based on a Jolly–Seber model fitted to subsequent encounter data collected from 2012 to 2022.

FIGURE 3.

Annual estimates of recruitment, deaths, and egg production for adult female snapping turtles in the North Madawaska River catchment of Algonquin Provincial Park, Canada, under three different models fitted to growth, reproduction, and encounter data collected from 1972 to 2012. Females are considered to recruit at 24‐cm carapace length, the size when nesting typically begins. See Figure 2 for an explanation of the three models. Dots show medians of posterior distributions, and vertical bars show 95% credible intervals.

The full model (Model 1) and the model with individual heterogeneity removed (Model 2) gave similar estimates for the number of adult females in the catchment each year from 1972 to 2012, with both models indicating a drop from ca. 90 to ca. 55 females in the late 1980s then remaining near this level (Figure 2). Models 1 and 2 also gave similar projections, with 95% prediction intervals ranging from 17 to 68 adult females in 2022 and from 3 to 51 in 2052. Abundance estimates from the Jolly–Seber model fitted to new data from 2012 to 2022 suggested that the population initially increased then declined to ca. 22 adult females by 2022, consistent with the 95% prediction intervals under Models 1 and 2 (Figure 2). Models 1 and 2 also gave near identical estimates for the numbers of recruits, deaths, eggs laid each year from 1972 to 2012 (Figure 3), and both models indicated that the average carapace length of adult females increased from the mid‐1970s to mid‐1990s and then declined (Figure 4). Comparison of output for Models 1 and 2 also showed that the removal of individual heterogeneity generally had trivial effects on the other main parameters and hyperparameters (Table 1). The exception is that removal of individual heterogeneity in size‐specific reproductive output ($\mathrm{\mu }\mathit{CS}$) resulted in a substantially higher estimate for annual variation in reproductive output ($\mathrm{\mu }\mathit{tCS}$)—that is, individual heterogeneity was misinterpreted as additional environmental stochasticity.

FIGURE 4.

Annual estimates of average carapace length of adult female snapping turtles in the North Madawaska River catchment of Algonquin Provincial Park, Canada, based on growth and encounter data collected from 1972 to 2012. Estimates were obtained under models that do or do not include individual heterogeneity in growth and reproduction rates (see Figure 2). The averages include unobserved females interpreted to be present under each model, with the missing measurements for these females modeled based on the other data. Points show medians of posterior distributions, and vertical bars show 95% credible intervals.

The IPM with no size dependence in rates (Model 3) gave estimated numbers of births and recruits similar to those from the other models but gave higher estimates of egg production and population abundance from 1990 onward (Figure 2, Figure 3). The population projections from Model 3 were also more optimistic than those of Models 1 and 2, with 95% prediction intervals ranging from 27 to 87 adult females in 2022 under Model 3, which is inconsistent with the estimated population decline based on the new data collected (Figure 2).

DISCUSSION

The key results from this study were that population projections for snapping turtles in Algonquin Park were significantly affected by the exclusion of size‐dependent survival and reproductive rates in adult females but were unaffected by the exclusion of individual heterogeneity in growth parameters and size‐specific reproductive output. Despite this individual heterogeneity translating into pronounced variation in life expectancy and lifetime reproductive output (Armstrong et al., 2018), parameters measured at the population level (abundance, recruitment, deaths) were similar when the individual heterogeneity was modeled and when it was ignored, leading to similar population projections. The results therefore show that the pronounced variation in growth and reproductive parameters among adult females in this system is having a negligible effect on the population's dynamics.

The relevance of individual heterogeneity depends on the specific question being addressed. Our results indicate that individual heterogeneity is irrelevant to the dynamics of the North Madawaska snapping turtle population over the next 40 years, a time scale appropriate for management decisions. However, consideration of individual heterogeneity may result in a better mechanistic understanding of population processes. For example, failure to consider individual heterogeneity may exaggerate environmental stochasticity, as shown in our study, or distort state dependence in vital rates (Cam et al., 2016; Plard, Fay, et al., 2019). Such heterogeneity may reflect differences in the quality of territories or home ranges rather than the animals themselves. Although density‐dependent population dynamics may be predicted satisfactorily by standard models assuming that all individuals are affected equally by population density, at a mechanistic level, changes in vital rates may reflect the proportion of individuals able to occupy high‐quality territories (Both, 1998). Such density‐dependent dynamics were not considered in our study due to previous research showing no evidence of density‐dependent compensation in vital rates following the decline of the population in the late 1980s (Keevil et al., 2018).

Individual heterogeneity may also become more relevant over longer time frames where natural selection on heritable traits will affect the composition of populations (Vaupel & Yashin, 1985). The study of evolutionary processes is a key motivation for disentangling individual heterogeneity from stochasticity and state dependence in vital rates (Cam et al., 2016) and can also motivate incorporation of individual heterogeneity in integrated population models (Plard, Fay, et al., 2019; Plard, Turek, et al., 2019). These evolutionary processes can be relevant to conservation and management. For example, fisheries biologists have long recognized the importance of fishing‐induced evolution on population dynamics and therefore the need to account for it in models used for management (Giske, 1998; Thambithurai & Kuparinen, 2023). Such processes are not relevant to the North Madawaska population of snapping turtles over a 40‐year time frame, given that this equates to less than one generation, but may become relevant over hundreds of years and as the environment warms. We also suspect that the pronounced individual heterogeneity we observed in Algonquin Park snapping turtles is attributable to environmental variation among the different parts of the catchment that different females inhabit rather than to heritable variation. Low‐quality or stressful environments tend to enhance environmental sources of variation in the phenotype while dampening heritable variation (Charmantier & Garant, 2005; Rowinski & Rogell, 2017), and our study population is at the edge of the species' range where there are strong thermal constraints on reproduction (Edge et al., 2016) and probably somatic growth.

The relevance of individual heterogeneity over short to medium time frames will also depend on the characteristics of the species and system. For example, the combination of high egg production and low probability of recruitment in snapping turtles (Congdon et al., 2008) may result in individual heterogeneity in reproductive output being overwhelmed by demographic stochasticity. In addition, the effects of all forms of individual heterogeneity may have been overwhelmed by environmental stochasticity due to the high annual variation observed in all rates, similar to how demographic stochasticity is often overwhelmed by environmental stochasticity (Lande, 1993). Given the expected complexity of the effects of individual heterogeneity (Forsythe, 2025; Kendall & Fox, 2002; Stover et al., 2012; Vindenes et al., 2008), it will be essential to test the significance of those effects on population dynamics in multiple systems.

Population models are theoretical expressions of the factors believed to be key drivers of the species and system (Beissinger & Westphal, 1998; White, 2000), but they are otherwise generally kept as simple as possible to maximize the precision of projections (García‐Díaz et al., 2019; Getz et al., 2018; Starfield, 1997). It is therefore notable that while individual heterogeneity was found to be irrelevant to population dynamics, this additional complexity caused only a minor reduction in the precision of abundance estimates and projections. Other key aspects of our models for Algonquin Park snapping turtles were our treatment of sex classes, environmental stochasticity, and size dependence in vital rates. Our models focused on adult female snapping turtles, with juvenile growth and survival inferred from egg production and recruitment, including a sample of known‐aged recruits used to model age of maturation. This focus partly reflects data constraints, as adult females are easiest to detect, but also reflects the fact that females are expected to be the limiting sex, as is the case with many wildlife populations (Caswell, 2001). However, female‐only models may be inappropriate in populations where vital rates are affected by fluctuations in sex ratio—for example, if reproduction rates are limited by male parental care in female‐biased populations or by sexual harassment in male‐biased populations (Gerber & White, 2014). We modeled environmental stochasticity through annual random effects that allowed annual variation to be distinguished from sampling variation (White, 2000) as well as from individual heterogeneity. This revealed pronounced annual variation in rates that needed to be accounted for both when interpreting past dynamics and making future projections, reinforcing the strong emphasis that has always been placed on environmental stochasticity in population viability analysis (Beissinger & Westphal, 1998; White, 2000).

In combination with annual variation, the key factor needed to capture the population's dynamics was size dependence in vital rates of adult females and therefore the growth of those females. Models ignoring stage dependence in rates are expected to give misleading predictions unless the population conforms to the stable stage distribution (Lefkovitch, 1965; Leslie, 1945), and such conformance is unlikely under pronounced environmental stochasticity. In Algonquin Park snapping turtles, the period of high mortality in the late 1980s was followed by a decline in average body size of adult females that would have resulted in lower survival and reproduction. Consequently, the model ignoring size dependence gave population projections that were more optimistic than the model that accounted for it and inconsistent with the population decline observed over the next decade. Size structure is traditionally modeled as discrete stages, with growth represented by transitional probabilities (Crouse et al., 1987), and such models continue to be applied to management of long‐lived reptiles (e.g., Crawford et al., 2014; Zimmer‐Shaffer et al., 2014). However, the state‐space modeling methods that we employed allow the more natural approach of treating size as a continuous variable that changes with somatic growth and modeling survival and reproduction as a function of size. More importantly, stage‐based models usually emphasize the major distinctions between juveniles, subadults, and adults, whereas our results illustrate the importance of subtle indeterminate growth, that is, the subtle growth of adult females after maturity. Although individual heterogeneity in growth rates may be irrelevant to population dynamics, at least in our system, our results show that the overall growth rate of adult females can be an important driver of population dynamics in long‐lived reptiles.

AUTHOR CONTRIBUTIONS

Ronald J. Brooks, Jacqueline D. Litzgus, and Njal Rollinson have led the Algonquin Park snapping turtle project over the last 50 years with major contributions from Matthew G. Keevil and Patrick D. Moldowan. Doug P. Armstrong conceived the ideas for the paper in collaboration with Ronald J. Brooks, Matthew G. Keevil, and Njal Rollinson; conducted the analysis; wrote the first draft; and made a minor contribution to the data collection. All authors contributed to the final manuscript, including Ronald J. Brooks, who was able to read the first draft before his death. The surviving authors dedicate this paper to his memory.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

Supporting information

Appendix S1.

Appendix S2.

Armstrong, Doug P. , Keevil Matthew G., Moldowan Patrick D., Rollinson Njal, Litzgus Jacqueline D., and Brooks Ronald J.. 2025. “Is Individual Heterogeneity in Growth Rates Relevant to Population Dynamics of Long‐Lived Reptiles?” Ecology 106(9): e70185. 10.1002/ecy.70185

DATA AVAILABILITY STATEMENT

Data and code (Armstrong et al., 2025) are available in Zenodo at https://doi.org/10.5281/zenodo.15875387.