Metabolic theory, life history and the distribution of a terrestrial ectotherm

Authors


Correspondence author. E-mail: mrke@unimelb.edu.au

Summary

1. Life histories, population dynamics and geographic range limits are fundamentally constrained by the way organisms acquire and allocate energy and matter. Metabolic theories provide general, parameter-sparse frameworks for understanding these constraints. However, they require the accurate estimation of body temperature which can be especially challenging in terrestrial environments.

2. Here, I integrate a metabolic theory (Dynamic Energy Budget theory, DEB) with a biophysical model for inferring field body temperatures and activity periods of terrestrial ectotherms and apply it to study life-history variation and geographic range limits in a widespread North American lizard, Sceloporus undulatus.

3. The model successfully predicted trait co-variation (size at maturity, maximum size, reproductive output and length-mass allometry) through changes in a single parameter. It also predicted seasonal and geographic variation in field growth rates, age at first reproduction, reproductive output and geographic range limits (via rmax estimates), all as a function of spatial climatic data. Although variation in age at maturity was mostly explained by climate, variation in annual reproduction was largely a product of local body size.

4. Dynamic Energy Budget metabolic theory is concluded to be a powerful and general means to mechanistically integrate the dynamics of growth and reproduction into niche models of ectotherms.

Introduction

Organisms take up energy and matter from their surroundings and use it to build and maintain their bodies and to produce offspring. The life history, distribution and abundance of species reflect the operation of these metabolic processes in the context of varying environments, especially temperature, water and food (Andrewartha & Birch 1954). The set of these environmental conditions and resources permitting population persistence represents the organism’s ‘fundamental niche’, setting the stage for understanding biotic interactions (excluding feeding) such as predation and competition, i.e., the ‘realized niche’ (Hutchinson 1957). Mechanistic niche models aim to capture these processes as a function of the interaction between traits and environmental gradients, especially climate, terrain and vegetation (Kearney & Porter 2009).

Recent progress in mechanistic niche modelling has happened rapidly along two different fronts that are yet to be united. On one hand, the field of biophysical ecology has been applied to predict body temperature (and also water loss) through space and time as a function of an organism’s biology (size, shape, colour, behaviour) and of the increasingly rich array of available spatial environmental data (Kearney & Porter 2004, 2009; Helmuth, Kingsolver & Carrington 2005; Gilman, Wethey & Helmuth 2006; Buckley 2008; Helmuth 2009; Kearney, Shine & Porter 2009). On the other hand, the parallel emergence of ‘metabolic theories in ecology’ has provided first-principles models of the processes of energy and matter uptake and its use for maintenance, development, growth and reproduction (Kooijman 1986, 2010; Nisbet et al. 2000; Brown et al. 2004; van der Meer 2006b). To date, the field of biophysical ecology has incorporated phenomenological, static (snapshot in time) energy budgets and empirical descriptions of physiological processes such as metabolic rates, growth rates and feeding rates (e.g. Grant & Porter 1992; Adolph & Porter 1993, 1996; Buckley 2008; Buckley et al. 2010; Kearney, Wintle & Porter 2010). In contrast, applications of metabolic theory to environmental gradients have been performed under the assumption that body temperature equals air or water temperature (e.g. Dillon, Wang & Huey 2010), which is unrealistic for many organisms.

In this study, I integrate a metabolic theory, the Dynamic Energy Budget (DEB) model, with the Niche Mapper system (US Patent 7,155,377B2; wpporter@wisc.edu) that predicts spatial and temporal variation in field body temperatures in thermoregulating organisms. DEB theory is unique among metabolic theories in capturing the metabolic process of an organism through its the entire life cycle as an explicit function of body temperature and food availability (Kooijman 1986, 2010). It is thus highly suited to integration with mechanistic niche models (Kearney et al. 2010). Niche Mapper models body temperature and thermoregulatory behaviour, including potential foraging time, as a function of macroclimatic and organismal data (Porter et al. 1973; Kearney & Porter 2009), providing the environmental input for the DEB model.

Here, I apply the integrated models to study the fundamental niche of the widespread North American lizard Sceloporus undulatus. This species complex (Reeder, Cole & Dessauer 2002) has been the subject of many field and laboratory studies of its ecophysiology (Niewiarowski & Waldschmidt 1992; Angilletta 1999, 2001a,b; Angilletta, Hill & Robson 2002), life history (Tinkle 1972, 1973; Tinkle & Ballinger 1972; Vinegar 1975; Ferguson, Bohlen & Woolley 1980; Ballinger, Droge & Jones 1981; Tinkle & Dunham 1986; Ferguson & Talent 1993; Niewiarowski 2001) and geographic range (Buckley 2008; Buckley et al. 2010). Detailed life-history studies at 11 widely dispersed localities across the geographic range of S. undulatus document broad variation in growth rate, size and age at maturity, egg and clutch size and clutch frequency (summarized in Tinkle & Dunham 1986). This wealth of information makes S. undulatus an excellent case for testing the approach.

In this study, I use the combined models of metabolism and behavioural thermoregulation, together with spatial climatic data, to predict geographic variation in growth rate, age at maturity and reproductive output in S. undulatus from first principles. I compare the effectiveness of the model and the relative number of parameters required with previous efforts to model the life-history variation in S. undulatus from an energetics perspective (Grant & Porter 1992; Adolph & Porter 1993, 1996). I also assess the implied patterns of life-history trait co-variation stemming from DEB theory with the observed patterns. Finally, I project this depiction of the niche of S. undulatus across North America to infer constraints on its range and compare these results with recent range predictions derived from more empirically based, static energy budgets (Buckley 2008, 2010).

Materials and methods

Dynamic Energy Budget Theory and Its Implementation for Sceloporus undulatus

Dynamic Energy Budget theory is described in detail in Kooijman (2010), with useful summaries by van der Meer (2006a) and Sousa, Domingos & Kooijman (2008). The parameters for the standard DEB model used in this study are provided in Table S1 (Supporting information) and excel spreadsheet implementing the model is available from the author on request. The author is also developing an r package (R Foundation for Statistical Computing, Vienna, Austria) that implements the integrated Niche Mapper and DEB model. Because it is not widely applied in the studies of animal energetics, I provide a brief summary of the Standard DEB model and associated theory here.

A key feature of DEB theory is the partitioning of mass into the abstract quantities of ‘structural volume’, V and ‘reserve’, E. The reserve, which may consist of, e.g., fat, carbohydrate and amino acids scattered across the body, is used and replenished and hence does not require maintenance. The structure is the ‘permanent’ biomass such as proteins and membranes and requires energy and matter for its maintenance (protein turnover and the maintenance of concentration gradients and ionic potentials) in direct proportion to structural volume. The rate of energy assimilation A is explicitly related to food density through a functional response curve A = f{Am}V2/3, where f is the scaled functional response (ranging from 0 to 1) and {Am} is the maximum assimilation rate per unit surface area (note that, in DEB theory notation, square and curly brackets denote volume-specific and surface area-specific terms, respectively). DEB theory follows the flows of both energy and matter and does not necessarily assume that energy per se is limiting. Development, growth and reproduction are predicted dynamically according to the κ-rule whereby a fixed (throughout ontogeny) fraction κ of the energy/matter in the reserves flows to growth and to somatic maintenance, the rest to increasing and maintaining the level of maturity EH and to reproduction once maturity is reached. The rate of change in the structural volume at constant food density is equal to

image(m1)

where t is time, [Em] is maximum reserve density (which, at constant food, reaches steady state at f[Em]), [M] is the somatic maintenance costs per unit volume and [EG] is the total energetic cost of structure (tissue energy content plus overheads for synthesis) per unit structural volume (van der Meer 2006b; Kooijman 2010). For a constant food density, this equation is equivalent to the von Bertalanffy growth curve, although based on very different principles (Kooijman 2010). The rate of change of the reserve density (which must be multiplied by structural volume, converted to mass and added to the structure to obtain a wet weight) is equal to

image(m2)

Once maturity is reached under the standard DEB model, a fixed fraction of assimilates is continually transferred from the reserve to the reproduction buffer (after accounting for maturity maintenance) and then ‘packaged’ as eggs and dispensed as soon as an appropriate threshold amount for a clutch is reached. The energy allocated to the reproduction buffer per unit time is inline image = (1 − κ)inline image − inline image, where inline image = (inline imageAm[E]V2/3)/[Em] − [E](dV/dt) is the reserve mobilization rate and inline imagej = inline imagejEH is the maturity maintenance rate, with inline imagej the maturity maintenance rate coefficient. However, S. undulatus is a seasonal breeder, with a refractory period from mid-summer until mid-winter (Marion 1970). Following Pecquerie, Petitgas & Kooijman (2009), I implemented seasonal reproduction whereby reserve in the reproduction buffer continues to accumulate at all times but is only drawn upon to produce eggs between 1st January and 1st August. The rate that the reserve buffer is transferred to the egg buffer (J h−1) during the reproductive season is inline imageB = (κR/λ)((1 − κ)([Em] ({inline imageAm}/[Em])V2/3) + [inline imageM]V)/(1 + 1/([EG]/κ[Em]/)) − J), where (1 − κR) is the overhead cost of reproduction and λ is a constant that relates to the maximum fraction of the year during which the animal would reproduce if fed ad libitum (Pecquerie, Petitgas & Kooijman 2009), assumed here to be 0·58 (i.e. 7/12 months).

Dynamic Energy Budget Parameter Estimation for Sceloporus undulatus

Dynamic Energy Budget parameters are not directly observable because they relate to the abstracted state variables of structure, reserve and maturity. To estimate DEB parameters (Table 1) for S. undulatus, I applied a new approach called the ‘covariation method’ (Lika et al., in press, a). The covariation method aims to estimate all of the DEB parameters simultaneously from empirical observations of physiological processes. I implemented this estimation procedure in Matlab (The MathWorks, Natick, MA, USA) using the freely available package ‘DEBtool’ (http://www.bio.vu.nl/thb/deb/deblab/debtool/).

Table 1.    (a) Observed and predicted values from the Dynamic Energy Budget (DEB) parameter estimation procedure for Sceloporus undulatus, and (b) DEB resulting core parameter estimates and standard deviations (SD), with rates corrected to 20 °C. The shape coefficient is for snout-vent length (SVL)
(a) Observed and predicted data
DataObs.Pred.UnitsData source
a b, age at birth 62 29·39days (28 °C) Andrews, Mathies & Warner (2000); Parker, Andrews & Mathies (2004)
a p, age at puberty152·5153·8days (28·9 °C) Ferguson & Talent (1993)
l b, length at birth 25·0 25·1mm Tinkle (1972)
l p, length at puberty 58·0 57·3mm Tinkle (1972)
l , maximum length 80·0 79·9mm Tinkle (1972)
W b, mass at birth  0·56  0·56g, wet Tinkle & Ballinger (1972)
W p, mass at puberty  6·9  6·7g, wet Tinkle & Ballinger (1972)
W , maximum mass 18·0 18·4g, wet Tinkle & Ballinger (1972)
R , max reproduction rate 40·1 39·6# year−1 (24 °C)Lordsburg population (Tinkle & Dunham 1986)
(b) Core parameter estimates
ParameterEstimateSDUnits
z, zoom factor (relative volumetric length)1·920·228cm
δM, shape correction factor0·24010·05914
v, energy conductance0·02710·01098cm day−1
κ, allocation fraction to soma0·6050·1904
[M], somatic maintenance10028·61J cm−3 day−1
inline image J, maturity maintenance rate coefficient0·0017510·004541day−1
[EG], cost of structure7725485J cm−3
inline image, maturity at birth14161080J
inline image, maturity at puberty3·649e+0042·434e+004J

In DEB theory, a distinction is made between ‘core’ parameters and ‘auxiliary’ parameters. The core DEB parameters are intimately linked to the underlying assumptions of DEB theory and relate directly to processes controlling state variable dynamics. Auxiliary parameters combine with the core DEB parameters and state variables to define mapping functions from the abstract quantities such as structural volume to real world observations such as wet mass. In the covariation method, empirical observations are obtained for a given species (entered in the ‘mydata.m’ DEBtool script), mapping functions are specified using auxiliary theory (contained in the ‘predict.m’ DEBtool subroutine) that relates the given set of empirical data to the DEB core parameters and state variables, and the set of core and auxiliary parameters that best reflects the empirical data is obtained inversely through a regression proceedure.

Because core DEB parameters frequently appear concurrently in the mapping functions for different types of data, substantial constraints on their possible values are imposed as the number of types of observations used for parameter estimation increases (e.g. the primary core parameter {Am} appears in the compound core parameters Lm and [Em] which themselves appear in the mapping functions for the von Bertalanffy growth rate, dry mass and body length). This situation allows the parameter values to be estimated via regression based on a set of single data points (single numbers) for a range of different physiological observations, which are hence referred to as ‘zero-variate’ data (in contrast to the more typical situation in regression of using a list of one or more pairs of numbers, e.g., time vs. length, which would be referred to as ‘uni-variate’ data). The general idea behind the covariation method is to let all available information compete to produce the best fitting parameter set and, to this end, it is necessary to estimate all parameters from all data sets simultaneously.

In addition, the estimation proceedure may be guided by prior knowledge of parameter values, not unlike the concept of a ‘prior’ in Bayesian parameter estimation methods. Conceptually, this prior knowledge is treated as data and hence it is referred to it as ‘pseudo-data’. The pseudo-data (rates corrected to 20 °C) for the present study were: energy conductance v = {inline imageAm}/[Em] = 0·02 cm day−1, allocation fraction to soma κ = 0·8, growth efficiency κG = 0·8 (relates to [EG] and is the energy fixed in structure as fraction of the energy required for structure), maturity maintenance rate kJ = 0·002 day−1 and somatic maintenance [inline imageM] = 54 J cm−3 day−1, based on Lika et al. (in press, a).

Finally, when estimating DEB parameters with the covariation method on the basis of diverse data sets, one frequently faces the issue that certain observations have been made with greater confidence or accuracy than others. It is therefore useful to be able to assign relative weights to the different data points on the basis of this prior knowledge.

The covariation method applies the Nelder-Mead simplex method for estimating parameters, using either a maximum likelihood (ML) or weighted least squares (WLS) criterion for model fit. The WLS performs better than the maximum likelihood method when only ‘zero-variate’ data is being used for parameter estimation (Lika et al. , in press, b) as is the case in this study. I used data based largely on the Utah population of S. undulatus, for which growth data in controlled-environments are available (Ferguson & Talent 1993). The observed data and their sources are summarized in Table 1a. The associated Matlab scripts I used to estimate the parameters for S. undulatus can be found at http://www.bio.vu.nl/thb/deb/deblab/add_my_pet/Species.xls.

For energy/mass conversions, the chemical potential of structure μV = 500 000 J mol−1 and the chemical potential of reserve μE = 585 000 J mol−1 were used, adjusting the default DEB values (Kooijman 2010) based on (Vitt 1978).

For the observations in Table 1a, age at birth includes both pre- and post-ovipositional development. Weights at birth, puberty and maximum size were based on the observed sizes (snout-vent length, SVL) and the relationship between SVL and mass in Tinkle & Ballinger (1972) (Fig. S1, Supporting information). I assumed a body water content of 70% and an egg water content of 50% (Vitt 1978). The value for maximum observed reproductive rate used in the parameter estimation proceedure must relate to the value used for the mass of the hatchlings (0·56 g wet mass in this case) and the different water content of eggs and hatchlings. I was unable to find suitable data from the literature on reproduction rate under controlled conditions (food and temperature). An initial guess at this value was based on the maximum observed annual reproduction for the 11 populations in the comparative life-history study of 9·5 g wet mass (Lordsburg, New Mexico, Tinkle & Dunham 1986). Assuming that the eggs (50% water) have an average dry-mass energy content of 27·82 kJ g−1 (Vitt 1978), a 9·5 g clutch contains 132·2 kJ of energy. Assuming that the hatchlings (70% water) have an average dry-mass energy content of 23·58 kJ g−1 (Vitt 1978), a 0·56 g hatchling would contain 3·96 kJ. Thus, a reproductive output of 9·5 g wet mass per year is equivalent to 132·2/3·96 = 33·4 offspring per year. This value resulted in underestimates of observed field reproductive output by approximately 20%, and was adjusted accordingly, decreasing κ slightly from the original estimate.

For simulations, egg wet mass was assumed to be 0·36 g, based on the Utah population (Tinkle 1972). In keeping with the assumption of DEB theory that the specific energy content of eggs is the same as the specific energy content of reserve (estimated in this study to be 7343 J g−1 wet weight, 70% water), I assumed that the energy content of one egg (0·36 g wet weight, 50% water) was 4406 J. For simulations of life-history variation, I ran the model either with the Utah body size and clutch size values, or with the locally observed values for these traits. In DEB theory, maximum size Lm = (κ{inline imageAm}/[inline imageM]); it is thus not a core parameter itself but is rather the outcome of the parameters that control the ratio between the incoming energy for growth and maintenance, and the amount of energy consumed by maintenance. Variation in maximum size can be considered the outcome of proportional changes in the ‘extensive’ physical design parameters through the dimensionless ‘zoom factor’z, where Lm2 = zLm1 = (κz{inline imageAm}/[inline imageM]). An adjustment of z reflects a shift in the maximum assimilation rate {Am} as well as in the parameters controlling the size at hatching inline image and maturity inline image (Kooijman 2010), with all other DEB parameters remaining constant. The z adjustment thus imposes a covariation of these three design parameters from a physicochemical point of view with subsequent implications for the life history and provides a null expectation for the implications of size shifts unless selection acts independently on the design parameters. Mean snout-vent length at maturity (SVLmat) for the 11 populations of S. undulatus summarized by Tinkle & Dunham (1986) is indeed tightly related to mean adult SVLmax (linear regression: SVLmat = 0·951SVLmax − 6·742, R2 = 0·874, P < 0·001, Fig. S2, Supporting information). I thus determined, for each population, the value of z that the reference size (Utah population) must be multiplied by to produce the locally observed SVLmat and the corresponding SVLmax that this implied (Table S2, Supporting information). The resulting relationship closely matched the observed one (Fig. S2, Supporting information).

Thermoregulatory Model and Integration With the Dynamic Energy Budget Model

The Niche Mapper system calculates hourly steady-state body temperatures (Tb) from actual or interpolated weather station records given the properties of the animal and its microhabitat (Kearney, Shine & Porter 2009). Following previous studies (Adolph & Porter 1993, 1996; Buckley 2008), solar absorptivity was set to 0·9, and I assumed that the lizards foraged between Tb of 32 and 37 °C, maintaining a Tb of 33 °C whenever possible during active or inactive periods by seeking shade/changing depth within the soil profile. Parameters for the microclimate model follow Kearney & Porter (2004) except that climatic data were drawn from a global data set of monthly mean daily maximum and minimum air temperature and monthly mean daily relative humidity, wind speed and cloud cover (1961–1990; 10’ resolution; http://www.cru.uea.ac.uk/cru/data). The lizards were assumed to experience the predicted wind speed and air temperature calculated for 0·5 cm above the ground when active on the surface.

A FORTRAN script implementing the DEB model was integrated with the Niche Mapper system and was called every hour to estimate structural volume, reserve density and reserve allocated to reproduction, given the body temperature estimate [see Fig. S3 (Supporting information) for an example of daily outputs across average days of each month].

The Arrhenius temperature correction (similar to Q10 correction) was applied to all rates for a common Arrhenius temperature TA of 9600 which provides a good approximation for the observed thermal responses of resting metabolism, assimilation and development rates (Fig. S4, Supporting information). The Sharp adjustment (see Kooijman 2010, p. 21) was also applied to provide an appropriate thermal response outside the range to which the Arrhenius relationship applied (Table 1b, Fig. S4, Supporting information). This adjustment causes rates to reduce outside upper and lower temperature boundaries (parameters TL and TH) at particular rates (TAL and TAH), reflecting enzyme inactivation and producing a classic-shaped thermal performance curve.

Feeding was only allowed when the animal was active on the surface and was assumed to occur ad libitum (i.e. the functional response f = 1). The energy (mass) of food in the stomach Ms started at zero (at hatching) was assumed to fill at a rate dMs/dt = {inline imageAm}V2/3 (f − Ms/MsmV) (Kooijman 2000), where Msm, the maximum energy content of the stomach per lizard mass, was assumed to be 186 J g−1 assuming an adult lizard has a stomach volume of approximately 2·5% of total body volume and that insect prey contain 23·85 J mg−1 dry mass (Buckley 2008). Assimilation was assumed to occur when the stomach was >1% full, otherwise f was set to zero and the reserve density declined according to eqn 2.

The wet mass of the animal as calculated by the DEB model each hour was used in the biophysical calculations of body temperature, dynamically capturing the effect of size on body temperature.

For simulations of the life-history study sites, hatching was assumed to occur in July–September (as described in the original studies, Table S2, Supporting information) and the hourly simulations were run for 5 years, repeating the loop through average monthly climatic conditions for the appropriate number of days per month. For continent-wide simulations for predicting geographic range, hatching was uniformly assumed to occur at the beginning of August. The simulation was run for 4 years at each location, tallying the total number of eggs and the total activity time for each year. A life table was then constructed, with annual estimates of age-specific fecundities mx (number of female offspring of age class x, assuming a 50 : 50 sex ratio) and survivorships lx (the survivorship to age class x). Following Adolph & Porter (1993), survivorship was estimated according to the empirically derived relationship between annual activity hours a (calculated in this study) and observed adult survivorship across the 11 life-history study populations (linear regression, F1,8 = 5·96, P = 0·041, R2 = 0·427, −0·00020818a + 0·63854795). From this, I calculated the net reproductive rate inline image, the average generation time inline image and the intrinsic rate of increase inline image.

Results

Dynamic Energy Budget Parameter Estimation and Validation

The DEB parameter estimates in Table 1b provided a close fit to the data used in the estimation procedure, with the exception of age at birth which was underestimated. The resulting DEB model successfully captured a wide range of metabolic phenomena not included in the fitting procedure. Most fundamentally, the model predicted the oxygen consumption rate (derived from first principles based on a generalized stoichiometry, Fig. S5a, Supporting information) and the assimilation rate (Fig. S5b, Supporting information). It also closely predicted the trajectory of growth observed in laboratory-reared Utah individuals (Fig. S6a, Supporting information). Reduction in the ‘zoom’ factor z to 0·92 provided a close fit to the growth trajectory of the Oklahoma population (reared under the same conditions as Utah individuals) (Fig. S6a, Supporting information) and predicted the qualitative (and to some extent quantitative) differences between these lineages with respect to size at maturity, post-partum weight and egg size (Table S3, Supporting information). The adjustment of z also qualitatively predicted the observed differences in the scaling of body size with body mass (Fig. S6b, Supporting information), reflecting DEB predictions of the scaling of reserve pools. Reserve was estimated to make up approximately 60% of the wet body mass. A simulation of starvation at 33 °C predicted a wet mass loss of 28% over 10 days, which compares well with the range of 20–30% observed by Dunlap (1995) at the same temperature and over the same time for the related Sceloporus occidentalis.

Field Body Temperatures, Activity Budgets and Growth

The biophysical model accurately predicted potential field body temperature when based on coarse climatic data (Fig. S7, Supporting information). The estimates in the present study of annual activity time were tightly related to those calculated by Adolph & Porter (1993) (linear regression, R2 = 0·909) but were around 700 h (30%) shorter (paired t-test, t8 = 16·84, P < 0·001) because of the inclusion of cloud cover (clear skies were assumed in Adolph & Porter 1993); re-running the simulations without cloud produced statistically indistinguishable results to Adolph & Porter (1993) (paired t-test, t8 = 1·79, P = 0·112, linear regression, R2 = 0·794).

The combined biophysical/DEB model produced results qualitatively and quantitatively consistent with observed field growth rates when driven by long-term monthly climatic averages (Fig. 1a–d). The overall relationship between observed and predicted age-specific body sizes (SVLobs and SVLpred) explained a similar proportion of the variance and was similarly close to a 1 : 1 relationship, whether using the Utah parameter set (linear regression, R2 = 0·856, P < 0·001, SVLobs = 0·97SVLpred + 3·06, Fig. 1a) or adjusting to the local body size via the parameter z (linear regression, R2 = 0·897, P < 0·001, SVLobs = 0·96SVLpred + 4·15, Fig. 4b).

Figure 1.

 Observed and predicted field growth of Sceloporus undulatus derived from a Dynamic Energy Budget model integrated with a biophysical model of heat exchange and thermoregulatory behaviour. The first two panels show contrasting growth trajectories for three sites assuming the Utah size (a) and the local size (b), with points representing observed data and lines representing the DEB predictions. The second two panels show observed and predicted values for SVL at particular ages for all five sites for which there is growth data, again based on either the Utah size (c) or the local size (d). In (c) and (d), the dotted lines represent 1 : 1 and the solid lines represent a linear regression. Observations come from Tinkle & Ballinger (1972) (Colorado, South Carolina, Ohio and Texas) and Tinkle & Dunham (1986) (Arizona). Predictions are made based on long-term monthly average climate, rather than the actual weather of the observations.

Geographic Life-History Variation: Age at Maturity and Reproductive Output

Operational definitions of ‘maturity’ in the life-history studies of S. undulatus are varied but almost always involve some level of egg development having occurred (to be detectable ‘in the hand’) (Tinkle 1972; Vinegar 1975) (Tinkle & Ballinger 1972; Ballinger, Droge & Jones 1981). The closest fit between observed and predicted age at maturity occurred when a threshold of 1/5th through the development of a clutch was used, with the exception Nebraska and Kansas (Fig. 2a,b). These latter two populations were also outliers in Adolph & Porter’s (1996) analysis. Regression analyses including these outlier populations indicated that SVL on its own was the best predictor of age at maturity (Table 2a). However, when the outliers were excluded, the integrated DEB/biophysical model predictions significantly explained large fractions of the variation in month at maturity relative to SVL or activity hours, with the predictions based on the Utah lineage fitting best (Table 2a). Figure S8 (Supporting information) shows the trajectories of body mass (with sudden drops representing oviposition events) as well as midday body temperature and reserve densities for the 11 sites in order of activity time and clearly shows the trend toward earlier maturation with increasing activity windows. Figure 4a shows the spatial variation in predicted age at maturity when the Utah-based model was run across all of the USA as well as Central America.

Figure 2.

 Observed (crosses) and predicted (circles) age at maturity (a, b) and annual reproductive output (c, d) plotted against calculated potential activity time. Predictions are based on either the Utah body size and clutch size (a, c) or the locally observed body and clutch sizes (b, d). Dotted lines are regressions for observed values, and solid lines are regressions for predicted values. All observed data come from Tinkle & Dunham (1986). Predictions are made based on long-term monthly average climate, rather than the actual weather of the observations.

Table 2.   Results of linear regression of observed (a) annual reproductive output and (b) age at maturity as a function of activity hours, body size (snout-vent length, SVL) and the Dynamic Energy Budget (DEB) model predictions. ‘Utah month mature’ refers to model predictions of the month of maturity assuming the DEB parameters for Utah, while ‘z-adjusted month mature’ holds all DEB parameters the same except the scaling parameter, z, which alters size at maturity and asymptotic size according to the assumptions of DEB theory. AIC is the adjusted value of the Akiaike Information Criterion, with the lowest (best model) value in bold. For multiple regressions, statistical significance of individual predictors is indicated by asterisks. Analyses are reported with and without outlier populations
 d.f. F R 2 P AIC
(a) Age at maturity (months)
 All populations
Activity hours1,90·9160·0920·36472·121
SVL1,96·0570·4020·036 67·527
Activity hours + SVL2,73·3080·4530·0971·796
Utah month mature1,93·0940·2560·11269·937
Utah month mature + SVL*2,76·4840·6180·02167·826
z-adjusted month mature1,92·5420·220·14570·451
 Excluding Kansas and Nebraska
Activity hours1,78·8340·5580·02155·421
SVL1,72·490·2620·15960·028
Activity hours + SVL2,63·9370·5680·08162·423
Utah month mature1,7124·40·947<0·001 36·376
Utah month mature** + SVL2,670·9230·959<0·00141·127
z-adjusted month mature1,728·2830·8020·00148·209
(b) Annual reproduction (kJ)
 All populations
Activity hours1,81·3630·1460·227104·385
SVL1,88·1440·5040·02198·938
Activity hours + SVL*2,78·7430·7140·01299·437
Utah reproduction1,84·4080·3550·069101·569
Utah reproduction + SVL*2,78·6320·7110·01399·528
z-adjusted reproduction1,89·4880·5410·015 98·161
Grant and Porter reproduction1,88·8680·5260·01898·498
 Excluding Arizona
Activity hours1,70·9990·1250·35196·695
SVL1,76·9120·4970·03491·715
Activity hours* + SVL**2,612·360·8050·00790·398
Utah reproduction1,73·8970·3580·08993·913
Utah reproduction* + SVL*2,611·0080·7860·0191·227
z-adjusted reproduction1,727·6250·7980·00183·508
Grant and Porter reproduction1,745·2250·866<0·001 79·809

Regression analyses comparing observed to predicted geographic variation in annual reproductive output are presented in Table 2b. Observed annual reproductive output was poorly explained by annual activity hours alone. Body size alone explained 50% of the variation in reproduction, but multiple regression with body size and annual activity explained over 70% of the variation. Similarly, regression of residual annual reproduction (from the regression on SVL) against annual activity hours showed a strong positive relationship (R2 = 0·62, F1,8 = 13·1, P = 0·007). The DEB-based model provided a poor prediction of reproduction when based on a constant body size (that of Utah), unless SVL was included as a covariate. However, the size-adjusted DEB model explained a similarly large fraction of the variation as did the more detailed empirical energy budget model of Grant & Porter (1992). In all cases, Arizona was a significant outlier from the analyses, as found previously (Grant & Porter 1992). Exclusion of this site substantially increased the variances explained but the qualitative differences among models were largely unchanged. The size-adjusted DEB model fitted best (adjusted AIC) when all populations were considered, and the fit of this model was second to the model of Grant & Porter (1992) when Arizona was excluded from the analyses. Annual clutch frequency was predicted to decline from 13 in Central America to 1 or 0 in northern USA (Fig. 4b).

Geographic Range Limits

Life-table calculations based on the Utah DEB energy budget calculations and empirical survivorship vs. activity time curve produced very similar results to the observed life table for this site reported by Tinkle (1972) (Table S4, Supporting information). When spatial variation in rmax (based on the Utah parameters) was calculated using gridded climatic data, the threshold where rmax > 0 provided a coarse match with the observed geographic range limits but tended to under-predict in the south and north west, and to over-predict in the north-east and far west (Fig. 4c).

Discussion

Constraints on Energy and Mass Budgets

The characterization of energy and mass budgets is basic to understanding life histories (Dunham & Overall 1994), population dynamics (Nisbet, McCauley & Johnson 2010) and range limits (Kearney, Wintle & Porter 2010). Life-history theory aims to predict how evolution will shape the allocation of assimilated resources to the potentially competing destinations of maintenance, growth, development and reproduction such that fitness is maximized (e.g. Stearns 1992). Although the range of allocation ‘strategies’ that organisms could take is potentially vast (Dunham & Overall 1994), the options may be substantially constrained by both the environment (Grant & Dunham 1990; Grant & Porter 1992; Adolph & Porter 1993, 1996; O’Connor, Sieg & Dunham 2006) and by the extent that the ‘architecture’ of a metabolic system imposes mechanistic couplings among life-history traits (Lika & Kooijman 2003).

The results of the present study show that the general principles of biophysical ecology and the DEB metabolic theory can, when integrated, capture much of the observed geographic variation in the life history of S. undulatus and, ultimately, the limits on its geographic range. A distinguishing feature of this analysis lies in the allocation rules of the DEB model, which prescribes simple yet somewhat unorthodox constraints on allocation. Specifically, the κ-rule means that growth does not compete directly with reproduction but only with somatic maintenance. The ultimate, asymptotic size then emerges from the model once the flux of assimilates assigned to growth and somatic maintenance is fully consumed by maintenance; the trajectory of growth is therefore predicted to be unaffected by the transition to maturity, consistent with observation (Kooijman 2000). The estimate for κ in the present study implies that 40% of the energy/mass intake of S. undulatus is allocated to reproduction (and maturity maintenance) at the point of maturity. Where were these resources going prior to maturity, given the constant value for κ assumed throughout ontogeny? The DEB theory states that they were being allocated to maturation, i.e., to increasing the complexity or information content within the organism (such as general tissue differentiation, preparing reproductive organs, developing the immune system) (Kooijman 2010). At the transition from juvenile to adult, the DEB model predicts a discontinuity in the respiration rate because 1 − κ of the reserve flux is redirected from a dissipative destination (maturation) to being largely fixed as reproductive biomass (Fig. S5a, Supporting information).

The other relatively unusual concept in DEB theory is the partitioning of mass into ‘structure’ and ‘reserve’. This study estimates that the reserve comprises around 60% of the total wet body mass and implies realistic starvation times compared with empirical data (Dunlap 1995), as described in the Results. Some predicted starvation times for an adult lizard from the Utah population are: 25 days at 33 °C, 58 days at 25 °C and 117 days at 20 °C. In the simulations for this study, which assumed food was available ad libitum when lizards were able to forage (as constrained by the biophysical model), the longest periods of time when animals were prevented from feeding were during the winter dormancy periods. Because of the cool temperatures predicted in the retreat sites at these times, reserve densities were never predicted to drop to seriously low levels (e.g. Fig. S8, Supporting information). However, if activity budgets were to be limited by high temperatures (e.g. in warm environments where available shade was minimal) and food abundance was low, death by starvation would be predicted in the order of months. This would be further exacerbated for juveniles, with lower absolute amounts of reserve (see Kooijman 1986).

Implications for Life-History Variation in Sceloporus undulatus

The results of the present study contribute further to our understanding of the causes of life-history variation in the classic example of S. undulatus. Adolph & Porter (1993, 1996) showed that thermal constraints on activity could cause geographic variation in life-history patterns qualitatively similar to those expected under adaptive arguments derived from life-history theory. Using S. undulatus as a case study, they predicted how phenomena such as age at maturity and reproductive output should vary with the local potential activity time as inferred from a biophysical model. The qualitative predictions they made are highly consistent with the quantitative predictions of the present study, which are based on growth and maturation rates as constrained by the thermal environment (Figs 1 and S8, Supporting information). First, age at maturity showed threshold shifts of approximately 1-year intervals as potential activity time increased (Figs 2a,b and 4a). Second, the number of clutches produced per year increased as potential activity time increased (Figs 2c,d and 4b). The results of the present study are also highly consistent with the quantitative predictions of Grant & Porter (1992) for annual reproductive output (Fig. 3).

Figure 3.

 Observed annual reproductive vs. predicted values from the present study (filled circles) and from Grant & Porter (1992) (open circles). Predictions are based on either the Utah body size and clutch size (a) or the locally observed body and clutch sizes (b). The dotted lines show the 1 : 1 relationships and the solid lines are linear regressions excluding the outlier population of Arizona. All observed data come from Tinkle & Dunham (1986). Predictions from the present study are made based on long-term monthly average climate, rather than the actual weather of the observations while those of Grant and Porter are based on meteorological observations from nearby weather stations for the period of the observations.

The advance of the present study over these important previous studies is that the predictions have been made quantitatively from first principles (cf. Adolph & Porter 1993, 1996) and on the basis of a more general and formal mechanistic framework for whole life-cycle metabolism (cf. Grant & Porter 1992). This permits some novel interpretations of the fit of the energy budget predictions with the observed life-history patterns. First, climatic influences on reproductive output were only strongly apparent once local variation in body size had been accounted for, both in the previous analyses (Grant & Porter 1992; Adolph & Porter 1993) and in the present study (Figs 2d and 3b, Table 2b). This is because body size dominates total reproductive output. Although body size adjustments by Grant & Porter (1992) and by Adolph & Porter (1993) were made based on the average adult size, adjustments in the present study were made through the ‘zoom factor’z, which affects size at birth, maturity and the asymptotic maximum through influences on the ‘extensive’ parameters inline image, inline image and {Am}, respectively (see Kooijman 2010, p. 300). The DEB theory thus predicts that, all else being equal, selection for increased size should have correlated effects on size at birth and size at maturity. I also assumed local clutch size was equal to that empirically observed (Table S2, Supporting information). However, the fit of observed to predicted total annual reproductive output differs little whether clutch and egg size were held constant at the values for the Utah population (R2 = 0·718) or allowed to vary as described above (R2 = 0·798). Thus, an increase in body size alone increased the absolute reproductive output.

However, even once body size was accounted for, the Arizona population was a consistent outlier in all studies in having a lower than expected reproductive output. Grant & Porter (1992) suggested that this was because reproduction at this site was limited by ingestion rate (i.e. prey density) rather than food processing rate. The DEB-based predictions provide support for this hypothesis in that the predicted asymptotic size (given the size at maturity, see Materials and methods) was lower than observed (Figs 1b and S2, Supporting information), and ultimate size in the DEB model is predicted to decrease under low food availability. The generally close match between observed and predicted field growth and reproduction rates has the remarkable implication that S. undulatus is typically not limited by prey availability in much of its range. This is consistent with the findings of Huey, Pianka & Vitt (2001) that only 13·2% of field-active lizards (and only 1·5% of North American iguanid lizards) are observed to have empty stomachs.

Constraints on Geographic Range

Biophysical models have been applied to infer geographic range constraints by predicting regions where a necessary physiological process fails (Kearney & Porter 2009). For example, Kearney & Porter (2004) predicted regions in Australia where foraging or egg development was not possible in a nocturnal lizard. For S. undulatus, the northern range limit has been proposed to be limited by potential for egg development (Parker & Andrews 2007) but my calculations of potential development in shallow nests suggest otherwise (Figs S10 and S11, Supporting information). The present study also predicts at least some annual activity is possible through the region considered. However, the length of the active season required for successful reproduction cannot be inferred without an energy budget. The energy budget developed in the present study predicts the northern distribution limit to occur around the 1100 activity hours threshold (Fig. S9a, Supporting information). Similarly, Buckley et al. (2010) predicted that S. undulatus could not produce sufficient eggs to offset mortality (22·5 eggs per annum) below an annual activity time of 1315 h. However, on this basis, Buckley et al.’s (2010) model precluded S. undulatus from all but the very southern parts of USA (see Fig. 1 in Buckley et al. 2010). In these areas, the present study estimated potential annual activity of >2000 h (Fig. S9a, Supporting information). This suggests that the poor agreement between the observed and predicted range limit based on the ‘biophysical threshold’ model in Buckley et al. (2010) is because their biophysical model substantially underestimated the potential activity hours.

The integrated biophysical ecology/DEB approach taken here allowed inference of age-specific reproductive output as constrained by the thermal environment (Fig. S9b, Supporting information). When combined with the empirically observed relationship between potential activity time and survivorship (Fig. S9c, Supporting information), it was possible to estimate rmax and, therefore, to map a more energetically specific depiction of the niche to infer range constraints (Fig. 4c). This prediction, being based on empirical patterns of survivorship, is potentially more reflective of the realized niche and is sensitive to local variations in predation pressure. Buckley et al. made the distinction between ‘threshold’ models that delimit ranges based on an environmentally limited physiological process vs. those that explicitly compute population dynamics such as the ‘foraging energetic’ model of Buckley (2008). Although the latter study estimated carrying capacity (via a spatially implicit model of optimal foraging and a constant empirically determined prey density) and hence population density, the predicted range limits were in fact a reflection of the threshold where rmax is non-negative, as in the present study. Thus, the ‘biophysical threshold’ distinction seems inappropriate. It is perhaps more helpful to categorize species distribution models with respect to the extent that processes are represented explicitly (through process models) vs. implicitly (i.e. through statistical associations), and whether explicitly stated process are depicted by empirical functions (e.g. allometric regressions) or formal theories (e.g. DEB theory or heat-transfer physics).

Figure 4.

 Spatially explicit predictions of (a) age at maturity, (b) annual clutch frequency and (c) the intrinsic rate of increase, rmax. All predictions are based on the estimated parameters for the Utah population. The solid black line represents the approximate range limit within the USA, while the filled triangles represent the sites for which detailed life-history data are available, summarized in Tinkle & Dunham (1986).

Advantages from Applying a Formal Metabolic Theory: Parameters, Generality, Testability

A number of studies have successfully integrated biophysical principles with energy budgets to study life history, population dynamics and range limits of ectotherms using S. undulatus as a model (Grant & Porter 1992; Adolph & Porter 1993, 1996; Dunham 1993; Angilletta et al. 2004; Buckley 2008). The key distinction between the present study and these important prior efforts is in the application of a more formalized, general and mechanistic theory of metabolism. If the core assumptions and propositions of the DEB theory are indeed correct, then a number of practical and theoretical gains follow from its application.

First, the models derived from DEB theory have substantially fewer parameters relative to the number of processes that are explained. To illustrate this, I have compared the number of processes, parameters and variables involved in the energy budget for S. undulatus as derived here and in two previous, less formalized approaches for the same species (Table S5, Supporting information). Grant & Porter’s (1992) pioneering energy budget analysis of S. undulatus required 22 parameters and one environmental variable (Tb) to capture potential reproduction at a fixed adult body size. Buckley’s (2008) model required 29 parameters and two environmental variables (Tb and prey density) to estimate potential reproduction at a fixed size, optimal foraging radius, and the implications of these for rmax and carrying capacity K. In contrast, the DEB model applied in the present study required only 24 parameters and one environmental variable (Tb) to predict ontogenetic growth, size- and age-specific reproduction, the phenology of growth, maturation and reproduction, the dynamics of the gut and the reserve (and hence starvation and body condition), and rmax. Note also that, while the number of parameters in these models may seem high, most of them are directly observable ‘first-principle’ parameters that can be measured with high precision (e.g. energy or water content, multi-parameter thermal response curves).

A second advantage stems from the generality of the model; the standard DEB model applied in the present study was developed to be applicable to most kinds of animal (Kooijman 2010) and the scaling responses are fixed and based on first-principles arguments, rather than through species-specific allometric curve-fitting. This should enhance the capacity for meaningful comparisons of parameter values estimated among populations, species and higher taxonomic groups (Lika et al., in press, b).

The third advantage lies in the testability of the model, particularly when considering the plausibility of ontogenetic or spatial (local adaptation) changes in parameters. For example, Angilletta (2001a) observed that laboratory-measured assimilation rate was higher (at one of three temperatures considered) in a South Carolina population compared with a New Jersey population. Although DEB theory would predict that the South Carolina population would, as a result, reach a larger size, in fact the converse is true (Angilletta et al. 2004). As there is no evidence for differences in adult somatic maintenance rate between these population, DEB theory implies a higher value for κ in the South Carolina population and hence an increased proportion of energy flowing to maturation and reproduction rather than growth and maintenance. An explicit test of this would therefore be that respiration rates during growth and development are higher for the South Carolina population. Another inconsistency is the substantially earlier observed ages at maturity of the Kansas and Nebraska populations, which may reflect lower energetic thresholds for maturity. An explicit test of this hypothesis would therefore be that the cumulative energy invested in growth and maturation (as reflected in O2 consumption per gram of tissue produced) is lower for these two outlier populations. The important point here is that DEB theory provides clear and testable predictions of how the metabolism of a species should behave as a consequence of such shifts in parameter values (e.g. Zonneveld & Kooijman 1989). Detailed empirical studies aimed at testing these predictions will provide potent tests of the assumptions of both the DEB theory and of life-history theory in general.

Conclusion

The aim of this study was to assess the potential for integrating DEB theory with biophysical ecology to make first-principles estimates of the energy budget, life-history patterns and geographic range limits of a terrestrial ectotherm. This can be thought of as attempting to infer a species’ fundamental niche on the basis of its functional traits and how they connect to environmental gradients (Kearney & Porter 2004; Kearney et al. 2010). The results for S. undulatus indicate that such an approach is indeed feasible and effective, capturing and explaining a wide range of phenomena ranging from basic elemental fluxes to life-history variation and geographic range limits. The overall approach should provide a firm basis for inferring the impact of changing environments on the distribution and abundance of organisms.

Acknowledgements

I thank Reid Tingley for assistance with construction of the climatic database, and Bas Kooijman, Raymond Huey and Warren Porter for advice and comments on the MS. This work was supported by an Australian Research Fellowship from the Australian Research Council (DP110102813).

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