A life-sized Heyuannia huangi body model was constructed using polystyrene foam and wood for the skeletal framework, and cotton, bubble paper, and cloth for soft tissue. Heyuannia huangi was chosen as the model because both it and its egg Macroolithus yaotunensis have been well studied (Lü, 2003; Cheng et al., 2008; Sato et al., 2005; Wiemann et al., 2017, 2018; Yang et al., 2018, 2019a; Bi et al., 2021). In the oviraptorid adult-associated clutches (Osborn et al., 1924; Dong and Currie, 1996; Clark et al., 1999; Fanti et al., 2012; Norell et al., 2018; Bi et al., 2021), the clutch was primarily covered by the trunk region (limbs included) from the first dorsal vertebra to the posterior tip of the ilium (Clark et al., 1999; Norell et al., 2018; Bi et al., 2021). Accordingly, the head, neck, and tail were not reconstructed in the model.

The lengths of long bones (femur and tibia) of Heyuannia huangi were acquired from specimens HYMV1–1 and HYMV1–2 in Lü (2003). Since the width and the height of the Heyuannia huangi holotype were not available due to poor preservation, the width was approximated with an oviraptorid Nemegtomaia barsboldi skeleton displayed at the National Museum of Natural Science, Taiwan (NMNS) based on the reason that Nemegtomaia barsboldi is the sister taxon of Heyuannia with a similar body length (Fanti et al., 2012; Funston, 2020). Transverse planes were constructed at an interval of 7.5 cm (2.5% of the total body length of the Nemegtomaia barsboldi skeleton; Seebacher, 2001) starting from the first dorsal vertebra to the posterior tip of the ilium. In each transverse plane, the width (twice the maximum medial-lateral distance of the right rib cage) was measured. Given that the body size (mass) is positively correlated with the length of the long bones (Christiansen, 1999a, 1999b; Campione and Evans, 2012), the measured widths were rescaled using the average of the femoral length ratio between Heyuannia huangi and Nemegtomaia barsboldi and the tibial length ratio between the two species, a value of 0.947, to estimate the width of Heyuannia huangi in each transverse plane.

Since the NMNS Nemegtomaia barsboldi skeleton lacked gastralia, the height (the dorsal-ventral distance from the tip of the spinous process, or the linearly interpolated height between the two neighboring spinous process, to the gastralia) of each transverse plane was estimated with the Heyuannia huangi reconstruction illustrated by Scott Hartman (Hartman, 2003). Height measurements were taken at an interval of 7.1 cm (7.5 cm × 0.947). The dorsal-ventral distance from the tip of the spinous process to the width of the right rib cage of each transverse plane of the Nemegtomaia skeleton was also measured and rescaled (× 0.947) to calculate the intersection point between the width and the height in the Heyuannia model (Figure 1). These transverse planes were then lofted to create the trunk region.

The internal structure in the Heyuannia model was supported by polystyrene foam with a cross-section of 4 cm × 5.4 cm. Wood with a cross-section of 0.9 cm × 0.9 cm was cut into the dimensions of calculated widths and heights. The wood pieces were inserted into the foam every 7.1 cm and intersected perpendicularly at the intersection point of each transverse plane. After gluing the cross-like structure along the foam, cotton and bubble wrap were packed into the remaining space to represent flesh. A piece of blue cloth covered the exterior of the model. The final model measured 63.9 cm in length, 45.6 cm at its widest point, and 33.2 cm at its deepest section (Figure 1).

After the Heyuannia model was completed, a heating pad (THOMSON Electric Blanket SW-W03BS) was attached to the ventral side of the model as the heat source. The body model and the heating pad together constituted the oviraptorid incubator. The portion of the pad that encircled the model represented the belly of the brooding oviraptorid, hereinafter referred to as the “core”. The folded-up portion of the heating pad resembled the limbs of an adult covering the rim of the clutch, hereinafter referred to as the “periphery” (Figure 2). Although feathers had so far been found in basal oviraptorosaurians, many studies suggested that feathers were a shared characteristic in Oviraptorosauria (Hopp and Orsen, 2004; Foth et al., 2014; Ksepka, 2020). A piece of approximately 0.3 cm-thick blanket (synthetic fiber and polar fleece) covered the incubator to mimic the insulative effect of feathers (Figure 2). The incubator was positioned on the clutch based on the postures of IGM100/979, IGM100/1004, and LDNHMF2008, with the anterior end (first dorsal vertebra), the posterior end (posterior tip of the ilium), and the limbs (represented by the folded heating pad) contacting the rim of the clutch (Clark et al., 1999; Norell et al., 2018; Bi et al., 2021).

Modeling clay was molded into an oviraptorid egg shape based on the dimensions of NMNS CYN-2004-DINO-05, which had been previously assigned to Macroolithus yaotunensis (Wiemann et al., 2017). The blunt end of the clay egg was attached to a clay egg cup. About 60 g of casting resin was applied on the surface of the clay egg and a layer of fiberglass was attached to the resin to reinforce the shell. After the resin hardened, about 50 g of resin was evenly spread on the first resin layer. The resin egg was then detached from the clay egg cup and heated at 60 °C to melt the modeling clay from the blunt-end opening, leaving a hollow resin egg with a shell thickness ranging from 1.6 to 1.9 mm, a value consistent with the thickness of the Macroolithus eggshell (Yang et al., 2019b; Weiß, 2020). Lids (blunt ends) were made following the same procedure and attached to the resin eggs.

Inside each hollow resin egg, water was chosen to represent egg white based on their similar heat transfer performances (Sabliov et al., 2002). To measure the temperature of the blunt end, where the adult’s belly would have contacted, and the embryo would have occupied (Turner, 2002; Yang et al., 2019a), a hole was drilled to insert a thermometer [digital thermometer NTC (10K/3435)] 2.5 cm into the water-filled cavity. After the hole was sealed with resin, waterproof paint (Aqua Seal: Mold and Mildew Resistant) was applied to the egg to reduce water loss.

The eggs’ lengths (the distance between the tips of the blunt end and the acute end) ranged from 19.3 cm to 21.2 cm, and the widest widths ranged from 7.3 cm to 8.3 cm. The eggs weighed between 740 g and 867 g, falling within the estimated weight range of Macroolithus eggs (Tanaka et al., 2018a). The overall dimensions of the eggs were similar to those described in Bi et al. (2021). These eggs were repeatedly used throughout the experiments. Due to water loss, some eggs were repaired by injecting water into the eggs with needles until the eggs reached their original weights.

Two types of clutches were reconstructed using the abovementioned resin Macroolithus eggs, an original clutch and a generalized clutch. The original clutch replicated the egg arrangement of the double-ring clutch NMNS004529-F003855 (see Yang, 2012). The eggs in this clutch had linearituberculate ornamentation, which only appeared in Macroolithus sp. among all Elongatoolithidae eggs (Zhao, 1975; Yang et al., 2019b; Weiß, 2020). Based on the ornamentation, the lengths, and the widest width of the eggs, the eggs were assigned to Macroolithus. After measuring the eggs’ squashed inclination angles relative to the ground, the reconstructed angles were calculated based on the formula in Yang et al. (2019b). To avoid inaccurate estimation caused by deformation during fossilization, only eggs with intact blunt ends were included, yielding an average inclination angle of 29.6°, a value less than that reported in Yang et al. (2019b). There were 19 eggs in the specimen, but either because of deformation or burial by sediment, only 13 eggs could be reconstructed. The outer-ring eggs were labeled from 1 to 9. The inner-ring eggs were labeled from 10 to 13 and were located below egg 1, 2, and 3 (see Supplementary Figure S1). The eggs were positioned according to the observed arrangement in the clutch. For the eggs excluded in the reconstruction, the clutch rim was built to the height of the neighboring outer-ring eggs. The blunt ends pointed towards the clutch center at the reconstructed angles, and approximately five centimeters from the tip of the blunt end was exposed in the air (Wiemann et al., 2017). A one-centimeter-thick layer of soil laid between the two rings of the clutch. The soil thickness covering the outermost ring was unknown but assumed to be one centimeter thick (Yang et al., 2019b). While the original clutch was more realistic than the generalized clutch, the small number of inner-ring eggs makes a comparison of the incubation effects on the inner- and outer-ring eggs difficult. To better compare the temperatures of the inner-ring eggs and the superimposing outer-ring eggs, a generalized clutch with parameters like that of the original clutch was constructed.

The generalized clutch had a configuration following several published complete clutches (Wiemann et al., 2017; Tanaka et al., 2018a; Yang et al., 2019b). In these fossils, the egg-free center was oval; the central angle between the egg pairs varied; the outer ring did not stack precisely on top of the inner ring; and each egg had a different inclination angle (Wiemann et al., 2017; Tanaka et al., 2018a; Yang et al., 2019b). Nevertheless, in the generalized clutch, the center was assumed to be a circle with a diameter of 24 cm; the outer ring stacked directly on top of the inner ring; and the eggs uniformly had 30° inclination relative to the ground. A one-centimeter-thick soil layer was laid between the two rings. As a complete clutch, it would have been a double-ring clutch with each ring consisting of 5 pairs, with a central angle of 20° within the egg pairs and 50° between the egg pairs. To simplify the experiment, only three out of the five pairs were reconstructed for the experiments. There were twelve eggs in total in the generalized clutch, with six in the inner ring and six in the outer ring. The eggs in the outer ring were named from 1-o to 6-o (“o” denotes outer). The inner-ring eggs were named from 1-i to 6-i (“i” denotes inner) (Figure 2A). Each stacked pair of inner- and outer-ring eggs was collectively referred to as a set (e.g., set 1 consists of egg 1-o and egg 1-i). There were two eggs per set, six sets in total. Egg 1-o, 2-o, and 6-o were contacted by the core of the incubator (hereinafter referred to as the “core eggs”). Other eggs were contacted by the periphery of the incubator (hereinafter referred to as the “peripheral eggs”) (Figure 2).

As oviraptorid Elongatoolithidae eggs had been discovered in fine-grained (in China) and coarse-grained (in Mongolia) soil, soil composition of the clutch substrate might be less important to oviraptorids than to other buried nesters, which had specific preferences for soil for incubation (Tanaka et al., 2018b). Even so, we collected soil from the experimental site (24°04’15.4”N 120°45’18.1”E) as it texturally resembled the sandy paleosol in the Nanxiong Group and Nemegt Formation (Fanti et al., 2012; Ma et al., 2018; Bi et al., 2021). When dried, it had a bulk density (1460 kg/m³) similar to that of sandy loam (Robin et al., 1997). Since the composition, moisture, porosity, and bulk density of the soil can influence its thermal conductivity (Yang et al., 2005), experimenting with a similar type of soil better approximated heat transfer of the adult-associated clutch.

Three experiments were performed to study the thermodynamic effects of clutch attendance: the absence of an adult (Experiment I), clutch attendance in a colder environment that resembled the temperature of the Nemegt Formation (Experiment II), and clutch attendance in a hotter environment that resembled the temperature of Nanxiong Group (Experiment III). All experiments were performed outdoors on a hill in Wufeng District, Taichung City, Taiwan (24°04’15.4”N 120°45’18.1”E). Thermometers (digital thermometer NTC (10K/3435)) were calibrated in air, water, and local soil before the experiment and all measured within 0.3 °C of the mean. Temperatures were recorded manually every twenty minutes and compiled in Excel. Soil moisture (volumetric water content) of the clutch was measured at two locations at the start and the end of each experiment using two Capacitive Moisture Sensors v1.2 powered by an Arduino Uno board. These sensors were calibrated beforehand using local soil with different moisture levels. The correlation coefficient between the measured and observed moisture level was 0.94.

After Experiments II and III, the exposed surfaces of the eggs were colored by acrylic paint. The incubator, with a piece of white cloth fastened on its ventral side, was then placed on top of the clutch in the same posture as during the experiments. The colored spot on the cloth was later analyzed with ImageJ (https://imagej.nih.gov/ij/) to represent the contact areas between the incubator and the eggs.

Arithmetic means of incubation temperatures were calculated for comparison of temperature distribution between inner- and outer-ring eggs. The egg temperatures of the final four hours of Experiment II were averaged arithmetically. For Experiment III that had fluctuating egg temperature, the readings over a 24-hour period were arithmetically averaged based on the linear degree-hours model (Georges et al., 1994).

Furthermore, for egg temperatures that reached equilibrium, the adult incubation efficiency was calculated using Equation 1, which was modified from Hogan and Varricchio (2021), to assess the incubator’s efficiency in elevating the egg’s temperature above the ambient temperature.

The original equation in Hogan and Varricchio (2021) used the difference between the maximum incubator temperature and the lowest ambient temperature in the denominator. The modified efficiency equation better accounts for the average contribution of the incubator’s temperature to the egg’s temperature, while preventing an arbitrarily low ground temperature value from influencing the calculation of the difference, as intended in Hogan and Varricchio (2021). The substitution of extremum values with average values in the equation is especially important in experimental settings with fluctuating ambient temperatures.

Among the experiments, only the eggs in Experiment II reached equilibria. While some of those eggs only reached equilibria in the final four hours, estimating the incubation efficiency solely from the final four hours would provide an overestimation since the contribution of environmental heat to egg temperatures is discounted by the declining ambient temperature after sunset. To avoid overestimation of the incubation efficiency, temperature data from the final seven hours of the experiment were used to calculate the incubation efficiency.

Two types of heat transfer numerical simulations were designed—the posture simulations and the non-posture simulations—and performed under the environmental conditions of Experiment II and Experiment III with COMSOL Multiphysics 6.1.

The posture simulations were designed to closely match the settings of the two experiments. Eggs were assigned different contact areas based on measured values. Contact temperatures from the incubator closest to each egg, either derived from measurements at the core or the periphery, were applied to the corresponding contact areas to represent the incubation temperature. These simulations were used to verify the temperature distribution trends observed in Experiment II and Experiment III.

In the non-posture simulations, extreme scenarios were created based on the experimental data. The maximum recorded contact area and contact temperature were applied to all outer-ring eggs. The homogeneous setting minimized the influence of the adult’s posture over the eggs to help examine the effects of heat sources other than the incubator, such as the sun, on the temperature distributions within the clutch.

The simulated clutch replicated the dimensions of the generalized clutch and was built in the COMSOL Multiphysics model builder. The below-ground soil was modeled as a cylinder with one meter in both the diameter and the depth.

A two-dimensional oviraptorid egg shape was rotated along its longest axis to form a three-dimensional oviraptorid egg. It was offset to form two bodies, generating an outer layer (the shell) and the inner layer (the egg white). The outer layer had the same dimension as NMNS CYN-2004-DINO-05 described in Wiemann et al. (2017). The inner layer was 1.8 mm less along each boundary compared to the outer layer, creating a space between the two layers that represented the eggshell with a thickness of 1.8 mm, a value derived from the eggshell of Macroolithus yaotunensis (Yang et al., 2019b; Weiß, 2020). The shell was assigned chicken eggshell material properties. The inner layer was assigned chicken albumen material properties.

The clutch rim had a right-triangular lateral view. The inner wall was tilted 60° relative to the ground while the outer wall was tilted 30° relative to the ground (Figure 4). All eggs were arranged as they were in the generalized clutch, with the only difference that the central angle between the eggs within a pair was 30° instead of 20° as in the experiment because of the limiting drawing capacity of the software.

In the posture simulations, the blunt ends of the eggs were sliced to match each egg’s respective contact area. When simulating Experiment III, an area of 15 cm² of the clutch was sliced next to egg 6-o to represent the position where the incubator contacted the soil. In the non-posture simulations, however, every egg was sliced uniformly to have the maximum recorded contact area. Free tetrahedra were used for meshing.

In the model, the incubator was not drawn. Instead, the incubator’s contact temperatures (T_c) were applied to the slices that matched the measured contact area of the eggs (Figure 4). The values of T_c varied based on the position of the egg (core or periphery). For posture simulations with the environmental settings of Experiment II, T_c were the average temperatures of either the core or the periphery over the last seven hours (16:02-23:02) of the experiment. T_c of 36.4°C was assigned to egg 1-o, 2-o, and 6-o, whereas a T_c of 32.3°C was assigned to egg 3-o, 4-o, and 5-o. For simulations with the environmental settings of Experiment III, T_c was assigned as the linearly interpolated functions of the recorded incubator core or periphery temperatures because the incubator temperature had diel fluctuation.

In all simulations, forced convection (h_f) (e.g., blown by the wind) was applied to the exterior of the clutch. The surrounding temperature for parameterizing h_f was based on the ambient temperature data from the meteorological station G2F820 (Ta) (Figure 4A). Convective heat flux of natural convection in the air (h_n) was applied to the surface of the eggs and of the soil inside the clutch (Figure 4A). The surrounding temperature for h_n was based on the measured air gap temperature (T_gap) at different positions inside the clutch. The convective heat transfer coefficient (h) of the heat flux was calculated based on a simplified geometry.

Surface-to-ambient radiation was applied to account for solar radiation and the radiation between the ground and the sky. Since the clutch was covered by the incubator and the feathers during the experiment, the sun (G_ext) only radiated to the ground and up to 12 cm above the ground on the exterior clutch wall, and the same area radiated to the sky (G_amb) (Figure 4A). Radiation within the clutch was not considered since heat conduction was the dominant mechanism of heat transfer due to a low temperature difference between the incubator and the eggs. Additionally, calculating radiation among irregular shapes constructed by the incubator and the inner side of the clutch would require a substantial computational power that is beyond our capacity and defeat a simulation’s purpose of simplifying a real-world context. Thermal insulation was applied on the side of the soil cylinder. The bottom of the soil cylinder was assigned the soil temperature measured one meter underground by the meteorological station (Figure 4A).

The thermophysical properties implemented in the simulations were acquired from measurements, literature review, and the material library of COMSOL Multiphysics. When calculating thermal conductivity of the soil, it was assumed that the local soil had a quartz (SiO2) content of 0.9, the same as loamy sand (Tarnawski et al., 2011). The thermal properties of the air were calculated using the relationship described in McQuillan et al. (1984). The thermal properties of the oviraptorid eggs were assumed to be the same as chicken eggs, with the chicken eggshell material property assigned to the eggshell and the chicken egg white material property assigned to the egg white in the model. Egg yolk was not represented in the model. The values and the sources of the properties were shown in Table 1.

Since the ambient temperature had a small range in the experiments, the thermal properties of the air at its average temperature were used to calculate the dimensionless Grashof number (Gr), the Prandtl number (Pr), and the Rayleigh number (Ra) to derive the Nusselt Number (Nu), which was then used to calculate the convective heat transfer coefficient (h) (Incropera et al., 2006). Weather data of the average wind speed over the experimental period were retrieved from the meteorological station G2F820. When calculating forced convection, the wind speed was assumed to be the same as the average wind speed over the experimental period. For posture simulations that required assigning convective heat transfer coefficients (h), the clutch was simplified into basic geometries which had known relations between the dimensionless numbers and Nu because of a lack of research on the convective heat transfer coefficient of objects similar to an oviraptorid clutch. Geometries used for calculating h for laminar flow included: a sphere for the blunt end of the egg, an inclined plate for the interior clutch wall and part of the exterior clutch wall, and a horizontal plate for the center devoid of eggs. The sphere had a diameter equal to the average of the major and the minor axes of the blunt end. The interior clutch wall had a characteristic length of 0.192 m, while the upper part of the clutch wall was 0.075 m. Nu and h were calculated from the correlations of these geometries. The lower exterior clutch wall and the rest of the surface exposed to the environment were assumed to have the same correlation as a flat plate in turbulent flow (Kumar and Mullick, 2010).

The emissivity of the soil (ϵ_soil) was assumed to be 0.975, the same as sandy soil (Mira et al., 2007). Hourly total solar irradiance (MJ/m²) was acquired from the meteorological station G2F820 and converted to the hourly average solar irradiation, Gext (W/m²), for the simulations. The ambient temperature to which the soil emits radiation, the effective sky temperature, T_sky, was derived by first calculating the sky emissivity (ϵ_s) using the dew point temperature and the ambient air temperature (T_a) acquired from the meteorological station (Berdahl and Fromberg, 1982), then using Equation 2 to estimate the effective sky temperature (Gliah et al., 2011).

Surface to ambient radiation was applied on the soil surface as Equation 3:

The soil was assumed to be an opaque medium, where the absorptivity and the emissivity of the soil surface were assumed to be corresponding to a diffuse surface. The ambient was approximated as a blackbody at the effective sky temperature.

For heat transfer in solids, the transient heat transfer is governed by conduction as in Equation 4:

where

In Equations 4 and 5, ρ is density of the material (kg/m³), C_p is the heat capacity under constant pressure (J/(kg·K)), T is the absolute temperature (K), q→ is the heat flux (W/m²), k is the thermal conductivity, Qtotal is the heat source in solid (W/m³).

The average of the temperatures of the eggs in the same ring at the start of the experiments was assigned as the initial egg temperature for eggs in that specific ring. Soil temperature within the raised clutch rim was taken directly from the measurements. For the underground soil, temperature data were obtained from the meteorological station G2F820. Since the above-ground rim raised the soil temperature below it, the soil temperature measured at the depths of 20 cm underground and below were assigned to the soil with depths 20 cm above that of the measured depth in the simulation (e.g., 30 cm becomes 10 cm underground). For soil deeper than 50 cm, the soil in the simulation had the same temperature as that recorded by the meteorological station G2F820. The initial temperature of the air-gap air in the non-posture simulation was a linearly interpolated function with respect to the z-axis that reflects the temperature of the nearby soil.

The model was time-dependent. For the simulations replicating Experiment II and Experiment III, the overall simulated time was 900 and 1,800 minutes, respectively.

At the onset of the experiment, the inner-ring eggs generally had higher initial temperatures than the outer-ring eggs, with a mean difference of 1.2°C (29.0°C vs. 27.8°C). As the experiment progressed, the mean temperature of all eggs rose from 28.2°C to 42.0°C at time 13:26, with egg 1 reaching 46.0°C. Between corresponding outer- and inner-ring eggs—for instance egg 2 and 12, and egg 3 and 13—the outer-ring eggs had lower maximum temperatures than the inner-ring eggs. For egg 1 and 11, however, the outer-ring egg 1 had a higher maximum temperature.

The shaded ambient temperature ranged from 23.2°C to 30.8°C during the experimental period. Overall, the ambient temperature, the eggs, and the soil had similar temperature profiles. They all reached their maximum temperature at around 13:26 and declined thereafter. The egg temperature varied more than either soil temperature or shaded ambient temperature. The temperature difference experienced by an individual egg ranged from 11.3°C in egg 3 to 18.3°C in egg 1.

In this experiment, the incubator contacted none of the inner-ring eggs. Among the outer-ring eggs, egg 1-o was most contacted, with an area of 12.32 cm². The least contacted egg was egg 4-o, with an area of 0.76 cm².

The inner-ring eggs had a higher initial average temperature than that of the outer-ring eggs, 19.1°C and 17.5°C respectively. As the experiment progressed, the core outer-ring eggs, egg 1-o, 2-o, and 6-o, reached equilibrium at 17:02 and stayed within ± 0.2°C (Figure 5). Comparing the average temperature over the final four hours (19:22-23:02) of the experiment of these three eggs, egg 1-o had the highest value of 30.3°C, followed by 28.0°C for egg 6-o, and 25.7°C for egg 2-o. For the peripheral outer-ring eggs, egg 3-o, 4-o, and 5-o, their temperatures declined gradually after sunset at 17:42 and reached lower equilibrium temperatures than the core outer-ring eggs. Their average temperatures over the final four hours of the experiment were 25.6°C of egg 3-o, 25.1°C of egg 5-o, and 24.4°C of egg 4-o. The inner-ring eggs reached equilibrium about three hours later than the corresponding outer-ring eggs in the same set. Contrary to their outer-ring counterparts, the temperatures of eggs 3-i, 4-i, and 5-i did not decline. The average temperatures over the final four hours of the inner-ring eggs were 28.5°C for egg 1-i, 26.3°C for egg 2-i, 24.7°C for egg 3-i, 23.7°C for egg 4-i, 25.1°C for egg 5-i, and 25.8°C for egg 6-i.

The soil and the eggs had similar temperature profiles. The temperatures of the inner-ring eggs were approximately the same as those of the surrounding soil, while the outer-ring eggs had temperatures higher than those of the surrounding soil.

At the end of the experiment, the maximum temperature difference within the outer ring was 6°C between egg 1-o and egg 4-o (30.2°C vs. 24.2°C). The maximum temperature difference within the inner ring was 3.8°C between egg 1-i and egg 4-i (28.5°C vs. 23.7°C). All egg temperatures were well above the ambient air temperature.

Zooming in on the comparison between sets, it was observed that in the core sets, the outer-ring eggs had higher average temperatures over the final four hours than the inner-ring eggs. For example, in set 1, egg 1-o was 1.8°C higher than egg 1-i. In the peripheral sets, the differences were much smaller. For instance, in set 4, egg 4-o was 0.7°C higher than egg 4-i (Figure 6A1). In set 5, the average temperature was the same, and the temperature of egg 5-o fell below that of egg 5-i by 0.6°C by the end of the experiment. Moreover, the core sets generally had higher equilibrium temperatures than the peripheral sets. The results showed two general trends: (a) the closer a set was to the core, the larger the temperature difference between the inner- and outer-ring eggs was; (b) the closer an egg was to the core, the higher its equilibrium temperature was (Figure 6A1).

The highest equilibrium temperature over the final seven hours of the experiment was observed in egg 1-o (30.2°C), whereas the lowest was in egg 4-i (23.4°C). The average ambient air temperature over the final seven hours was 18.9°C; the average incubator temperature over the same period was 36.4°C. Calculating from Equation 1, the estimated oviraptorid adult incubation efficiency was 26%~65%.

As in Experiment II, the incubator contacted none of the inner-ring eggs. Among the outer-ring eggs, egg 4-o was most contacted, with a contact area of 10.2 cm². Egg 3-o was least contacted, with a contact area of 5.3 cm². Egg 1-o and 2-o were contacted by the core, egg 3-o, 4-o, and 5-o were contacted by the periphery, and egg 6-o was not contacted (Please refer to Figure 2A for egg arrangement in the clutch).

The partial clutch heated directly by the sun showed extreme temperature fluctuations. The outer-ring egg reached its maximum temperature of 54.1°C at 14:13. The inner-ring eggs had lower maximum temperatures, which were still higher than the unshaded ambient temperature. At night, the lowest temperature of the outer-ring egg was 26.4°C, which was still higher than the ambient temperature at that time. The outer-ring egg had a higher maximum and a lower minimum temperature than the inner-ring eggs, with a maximum temperature difference of 27.6°C.

Due to pre-heating, the initial temperature of the incubator core was 39.6°C, but this value soon dropped to 36.5°C after 40 minutes and maintained a mean of 37.1°C for the rest of the experiment. For the incubator periphery, the temperature was more unstable. It first rose from its initial value of 34.4°C to 41.5°C at 14:13, and then declined to 32.5°C at 18:53. It maintained around 32.8°C until 06:33, then rose again at dawn, reaching 41.6°C at 12:53 of the second day.

For the generalized clutch, the egg temperatures had diel fluctuations (Figure 7). The maximum and minimum temperatures appeared later in comparison to the ambient temperature, resembling a phase shift. All outer-ring eggs had higher maximum temperatures and lower minimum temperatures than the inner-ring eggs, with a temperature range of 5.2°C for egg 1-o and 3.5°C for egg 1-i over the 24-hour period (12:13, July 25 to 12:13, July 26). Moreover, egg sets closer to the core reached their maximum temperature about an hour earlier than the eggs at the periphery. In addition to the difference in timing, the core eggs had a lower maximum temperatures and higher minimum temperatures than the peripheral eggs (e.g., egg 2-o had 36.2°C for the maximum temperature while egg 4-o had 37.4°C; egg 2-o had 30.6°C for the minimum temperature while egg 4-o had 30.4°C). The average egg temperature over the 24-hour period was higher at the core (33.1°C for egg 1-o and 32.8°C for egg 6-o) than at the periphery (32.5°C for egg 4-o; Figure 6B1). For the average temperature difference between the outer- and inner-ring eggs (outer minus inner) within the same set, the trend that core sets had higher temperature differences than the peripheral sets was also observed. In some sets, such as set 4, the difference was slightly negative.

Soil temperatures also had diel fluctuations. Generally, the higher the soil was above the ground, the higher its maximum and the lower its minimum temperature were. Among the surface soil of the exterior clutch wall, the soil covered by the blanket had much lower temperature than sites that were not, with more than 10°C difference in the maximum temperature. The lowest maximum temperature of the surface soil of the exterior clutch wall exceeded 40°C. Some, such as those not covered by the blanket, reached as high as 57.6°C.

Egg temperatures stabilized in the final few hours of Experiment II but did not do so in Experiment III. The difference can be explained by the sun acting as an additional, powerful heat source in Experiment III.

The solar effect in Experiment II was not as powerful as in Experiment III. Even though the air and the soil temperatures in Experiment II displayed diel fluctuations, they remained below the incubator temperature. Thus, the air and soil acted as heat sinks, whereas the incubator acted as a heat source. Over time, the eggs reached temperatures between the ambient and incubator, consistent with the results reported by Hogan and Varricchio (2021) and Hogan (2024).

In Experiment III, nevertheless, the sun raised the temperatures of the soil along the exterior clutch wall well above that of the incubator during the day. Instead of acting as a heat sink, the soil, alongside the incubator, heated up the eggs and raised their temperatures. After sunset, when the ambient temperature fell below the egg temperature, the air and surrounding soil again functioned as heat sinks, producing the pronounced diel fluctuations observed in Experiment III.

Despite the differences in temperature profiles, both experiments showed consistent temperature distribution patterns: (1) The core outer-ring eggs had higher average temperatures than the peripheral outer-ring eggs. (2) The temperature differences between the outer- and inner-ring eggs were larger in core sets than in peripheral sets. These patterns were also present in the posture simulation of Experiment II, but they were less pronounced in the posture simulation of Experiment III. More detailed initial conditions for the simulation may help reproduce the small temperature differences observed in Experiment III.

It should be emphasized that the above generalizations are based on a clutch with 12 eggs. An oviraptorid clutch could contain 10 pairs (20 eggs) or more, which could have yielded very different results. Furthermore, caution is warranted against overgeneralizations since this study was designed based on specific interpretations of oviraptorid behaviors and physiology for clutches of a specific size class. We thus refrain from applying the interpretations to a general condition and instead limit our discussions to the incubator model and clutches built for this study.

Comparing the posture and the non-posture simulations of Experiment II, it can be observed that the abovementioned temperature distribution patterns are much more pronounced in the posture simulations but to a lesser extent in the non-posture simulations, which by design excluded the effects of different contact areas and temperatures due to the adult’s posture. Since both types of simulations experienced the same time-dependent solar radiation specific to the experimental settings, solar radiation may contribute to the observed patterns much less than the adult’s posture. As the incubator and the sun were the only two heat sources in the system, by deduction, the adult’s posture is most likely the major contributor to the patterns.

The variations in incubation temperature within the reconstructed clutch resulting from the adult’s position may have been caused by unequal contact areas and temperatures, as well as by differing rates of heat loss. As observed in the experiments, the incubators had unequal contact areas with the eggs. In extant bird nests, such a condition may arise when many eggs crowd the nest, causing some eggs to be only partially contacted and therefore receive less heat from the brooding adult (Bortolotti and Wiebe, 1993). Additionally, peripheral eggs may experience higher rates of heat loss. In extant birds, eggs located nearer the periphery are more exposed to the environment and cool more rapidly when the incubating adult is absent (Boulton and Cassey, 2012). Our interpretation of inter-ring temperature variation as being caused by differences in contact area potentially provides an explanation for the inter-ring temperature variation observed in Hogan (2024), despite the paper’s unrealistic radially symmetric design of the incubator. Importantly, however, if our incubator model had been capable of movement, the temperature differences described above might not have been as pronounced.

If the embryos of insects, turtles, and lizards are not incubated at extremely high or low temperatures, their development rates have positive linear relationships with the incubation temperature (Wagner et al., 1984; Georges et al., 1994, 2005). Based on the avian developmental temperature range of 26°C to 40.5°C (Conway and Martin, 2000), it is reasonable to assume that the recorded incubation temperatures of 30.4°C to 37.4°C over the 24-hour period (12:13, July 25^th to 12:13, July 26^th) of Experiment III do not approach the extremes of oviraptorid developmental temperature.

While some paleontologists hypothesize that the adult-clutch association primarily served to protect eggs from predation (Norell et al., 1995; Dong and Currie, 1996; Deeming, 2002; Yang et al., 2019b), shielding eggs from external environmental stressors, such as sunlight, may also have driven the evolution of oviraptorid incubation behavior. Our experiments suggest that solely relying on ambient heat sources for incubation would have increased embryonic mortality due to exposure to lethal incubation temperatures. Unlike Argentinian sauropods, there is no evidence that oviraptorosaurians utilized geothermal heat to incubate their eggs (Grellet-Tinner and Fiorelli, 2010). If ambient heat came only from solar radiation and an oviraptorid clutch was left unattended, the eggs in the partially open nest would have experienced extreme diel temperature fluctuations that reduced hatching success. For avian embryos, exposure to temperatures above 41°C for several hours leads to mortality (Webb, 1987). In the experiments, some eggs reached 46°C in Experiment I and 54.1°C in the partial clutch of Experiment III. In both cases, the egg temperatures remained above 41°C for more than four hours. As oviraptorid embryos exhibit several avian features and may have physiology similar to that of precocial birds (Norell et al., 2001; Grellet-Tinner and Makovicky, 2006; Weishampel et al., 2008), it is likely that oviraptorid embryos cannot withstand the extreme environmental conditions under direct solar exposure. Moreover, birds that expose eggs to sunlight before clutch completion, such as ostrich (Struthio camelus), have eggs with white eggshell, which absorb less heat compared to the colored ones, and a larger volume that confers greater heat capacity (Bertram, 1992). By comparison, oviraptorid eggs are smaller and contain blue-green pigmentation, potentially making them more vulnerable to overheating under solar exposure than ostrich eggs (Downs and Ward, 1997).

Shielding the eggs from extreme temperature fluctuations may also have benefited embryonic survival. Without the incubator, the egg temperatures dropped substantially at night, sometimes by more than 20°C, as recorded in the partial clutch. The magnitude of fluctuation is far greater than that observed in crocodile nests, where egg temperatures are maintained within 3°C (Magnusson, 1979). For avian embryos, the continuous fluctuation between the optimal developmental temperature and the physiological zero, the temperature above which the embryos begin to develop (Lundy, 1969; Webb, 1987), may impair development (Stoleson and Beissinger, 1995). Although unincubated embryos have a broader thermal tolerance and are better at surviving temperatures below the physiological zero (Booth, 1987; Stoleson and Beissinger, 1995), previous experiments have shown that no ostrich embryo survives after 15 days of outdoor exposure (Bertram and Burger, 1981).

Due to the risk of reducing embryonic survivorship, oviraptorids probably would not have left their eggs exposed to the environment without care for an extended period. Adult clutch attendance likely assisted embryonic development by reducing the extreme temperature variation and preventing eggs from experiencing lethal incubation temperature (Norell et al., 1995), an adaptation also observed in modern birds (Wang and Beissinger, 2011). Such an interpretation complicates Hogan (2024)’s hypothesis on the role incubation plays in subaerial nesting strategies, as the brooding adult may have regulated clutch temperature against large-magnitude fluctuations in addition to conferring thermal advantages beyond environmental heat sources.

To understand whether the thermodynamic advantage conferred by adult clutch attendance fulfills the criteria of TCI, the prerequisites are examined.

The first prerequisite of contacting all eggs is not fulfilled in our experimental model. In the generalized clutch, the outer ring prevents the incubator from contacting the inner ring. Since the eggs are partially buried and highly organized in the clutch, the brooding adult may not be able to manipulate the inner-ring eggs or contact them (Clark et al., 1999; Yang et al., 2019b). This result agrees with the inferences of Deeming (2002) and Yang et al. (2019b). Among the outer-ring eggs that are contacted, the greatest contact area is about 3% of the egg’s total surface area (calculated from Wiemann et al., 2017). In contrast with the 8-10% contact area of chicken eggs (Ar et al., 2012), this proportion is relatively small and may be the cause of less efficient heat transfer and could be a factor in the longer incubation times expected in non-avian theropods (Varricchio et al., 2018).

The second prerequisite of the adult being the major heat source is fulfilled but with an incubation efficiency far lower than that of modern birds. The calculated incubation efficiency in Experiment II reaches at most 65%, well below the 84.3% reported for the common eider (Somateria mollissima) (Hogan and Varricchio, 2021). The low efficiency indicates that an oviraptorid may have partially relied on ambient heat sources to incubate the eggs in a clutch. With more reliance on solar heat than birds, an adult could still maintain eggs temperatures within the developmental ranges of crocodilian and avian embryos. In Experiment II, the equilibrium temperatures of the eggs ranged from 23.7°C to 30.2°C. This range overlaps with the field-observed 23.3°C to 32.8°C temperature range of Alligator mississippiensis eggs (Joanen, 1969). In Experiment III, nighttime minimum egg temperature was 30.4°C and daytime maximum temperature reached 37.4°C, consistent with the 30°C to 38°C incubation temperatures reported for Nanxiong oviraptorids (Amiot et al., 2017; Bi et al., 2021). The average 24-hour temperature, in the low 30s°C, also approximates the mean natural incubation temperature across a wide phylogeny of birds (Webb, 1987).

The third prerequisite of maintaining all eggs within a narrow range of temperature is not fulfilled by our experimental model. As discussed in 4.2.3, the temperature differences within a clutch in Experiment II could result in intra- and inter-ring asynchronous hatching. While there are few intra- or inter-ring temperature differences in Experiment III, the diel fluctuations of eggs are greater than those found in extant birds. Direct comparison of the temperature profile in Experiment III with that of wild ostrich eggs incubated in an environment with greater ambient temperature fluctuation shows that the experimental oviraptorid egg temperatures fluctuated by about 2°C more (Bertram and Burger, 1981). While it remains unknown how such temperature fluctuations might influence embryonic development, the wider range of egg temperatures indicates that the oviraptorid adult simulated in our experiments was less capable of regulating egg temperature to a similar extent to that of birds.

The prerequisites critical for TCI are not fulfilled. By induction, the results of this study do not support the hypothesis of oviraptorids conducting TCI. It is worth noting, nevertheless, that other interpretations of adult postures in the clutch, such as one in which the brooding adult places its legs in the egg-free center (Hogan, 2024), might have resulted in other patterns of contact between the brooding adult and the eggs and, accordingly, other temperature distribution patterns.

Instead of utilizing TCI, oviraptorids and the sun may be co-incubators. While an oviraptorid relies on the sun to raise the clutch temperature, the comparison between the attended and unattended clutch demonstrates that clutch attendance can maintain a substantially more stable microenvironment for embryonic development (Wang and Beissinger, 2011). The shade provided by the adult reduces the risk of embryonic hyperthermia. The heat conferred by the adult raises egg temperatures above the ambient temperature during nighttime. This comparatively less efficient incubation behavior, relative to modern birds, combines adult incubation and ambient heat sources, which could be a more ancestral state of incubation and the precursor of the TCI behavior.

In extant animals, the development of offspring gonads can be determined by the sex chromosome or the accumulation of certain genetic products, a mechanism known as genotypic sex determination (GSD). Another type of sex-determining mechanism depends on the environmental conditions, such as temperature, social dynamics, or stress. It is known as environmental sex determination (ESD) (Weber and Capel, 2021). Among different types of ESD, crocodiles use temperature-dependent sex determination (TSD) (Deeming and Ferguson, 1991; Lang and Andrews, 1994), which is considered the ancestral state of Reptilia based on amino acid analysis (Janes et al., 2014). Birds, in contrast, use GSD. The initiation of avian testis development based on the dosage of the dmrt1 gene on the Z chromosome is an apomorphy (Ioannidis et al., 2021). Previous studies suggest that the dinosaurs used the GSD system based on their low survivorship over the Cretaceous–Paleogene boundary compared to other TSD species (Deeming and Ferguson, 1991; Silber et al., 2011). Grellet-Tinner and Fiorelli (2010) also suggest that the neosauropods use the GSD system based on the constant incubation temperature maintained by the hydrothermal heat source. In contrast, Varricchio (2021) speculates that the temperature differences reported in Bi et al. (2021) indicate the use of the TSD system in oviraptorids. Here, we provide new insights into the sex-determination system of oviraptorids.

Extant TSD species incubate their eggs near the pivotal temperature, which is the temperature that yields 50% of offspring of each sex in a nest (Mrosovsky and Yntema, 1980), to avoid skewing the sex ratio of offspring (Ferrando-Bernal and Lao, 2022). Take crocodiles, for example, different species’ pivotal temperatures range from 31.5°C to 31.8°C (for female-male transition) and 32.5°C to 34°C (for male-female transition) (Lang and Andrews, 1994). A field study documented that the eggs of the saltwater crocodile (Crocodylus porosus) are incubated in a cavity of 32.5 °C, which falls within the abovementioned pivotal temperature range (Magnusson, 1979). The pivotal temperature is relatively constant. It varies slightly both across different populations of loggerhead turtles (Caretta caretta) (less than 1°C; Mrosovsky, 1988) and related species (about 2.5°C across different genera of turtles compared by Mrosovsky, 1988; about 0.3°C among crocodiles compared in Lang and Andrews, 1994). Given that Heyuannia and Nemegtomaia are sister clades (Funston, 2020), it is parsimonious to assume that they share a similar pivotal temperature range if both species were to use TSD. However, in the experiments, the lower limit of Heyuannia’s incubation temperature (30.4°C) does not overlap with the upper limit of Nemegtomaia’s (30.2°C). If the two species were to share a similar pivotal temperature range, most Heyuannia eggs would have been incubated above the presumed pivotal temperature, whereas most Nemegtomaia eggs below it, resulting in predominantly single-sex offspring for both species. In extant turtles and tuatara, deviation from the pivotal temperature tends to decrease hatching success in both male and female-biased populations (Grayson et al., 2014; Hays et al., 2017). The compounding influence of having sexually biased populations and reduced hatching success could greatly increase extirpation risk for the populations (Grayson et al., 2014).

Moreover, oviraptorids appeared to have occupied regions that are less favorable for TSD species. Extant TSD species tend to inhabit areas with significantly warmer ambient temperatures (mean 24.38°C), whereas GSD species with short breeding seasons (one to four months) inhabit areas with significantly colder ambient temperatures (Cornejo-Páramo et al., 2020). Given that the estimated maximum incubation period of the two oviraptorid species is less than four months (calculated from Deeming et al., 2006), the presence of Nemegtomaia barsboldi in an environment with a low warmest-month temperature is consistent with the biogeographic distribution of extant GSD species with short breeding season.

The hypothesis that oviraptorids utilized GSD is not contradicted by molecular clock analyses of avian ZW chromosomes. Although the recombination suppression of the ZW chromosomal segment appeared after the divergence between non-avian dinosaurs and Avialae species around 155 Ma (Nam and Ellegren, 2008; Wang and Lloyd, 2016), proto-sex chromosomes could take several millions of years or longer to accumulate enough mutations to become heterologous and thereby suppress recombination (Kratochvíl et al., 2021). Since the molecular clock cannot ascertain the precise timing of the ZW chromosome differentiation (Organ and Janes, 2008; Ellegren, 2013), oviraptorids could have used the GSD system.

Although uncertainty remains regarding the existence of multiple tiers in caenagnathid clutches, caenagnathids and oviraptorids may have had similar clutch configurations and potentially shared similar brooding behaviors (Tanaka et al., 2018a, contraHuh et al., 2014; Yang et al., 2019b). Our interpretation of oviraptorid reproductive biology, therefore, may apply to the clade Oviraptorosauria. If oviraptorosaurians used GSD, as birds do, the GSD system may have evolved in the common ancestor of oviraptorosaurians and birds, which could be Pennaraptora or even earlier lineages (Figure 8). While Grellet-Tinner and Fiorelli (2010) suggest that sauropods from Auca Mahuevo, Argentina, also use the GSD system, it is unknown if the GSD systems in theropods and sauropods have a shared evolutionary origin.

In contrast to most birds, which induce hatching asynchrony by incubating some eggs prior to the others in a nest, the temperature difference within an oviraptorosaurian clutch may have led to asynchronous embryonic developments, reaffirming Yang et al. (2019a)’s hypothesis of hatching asynchrony as a autapomorphy of oviraptorosaurians (Figure 8). Such an inference is also consistent with the understanding that troodontid eggs hatch synchronously and that hatching asynchrony has evolved independently in the avian lineage several times (Stoleson and Beissinger, 1995; Varricchio et al., 2002).

Regarding the evolution of the TCI behavior, the adult incubation efficiencies of Troodon (75.6%~83.1%) and birds (84.3%) are much higher than that of oviraptorids (26%~65%) (Hogan and Varricchio, 2021). It is likely that the TCI behavior evolved after the divergence of oviraptorosaurians from the lineage leading to troodontids and birds, possibly within Paraves. Nevertheless, the incubation efficiency and the embryonic hatching pattern of dromaeosaurids remain unknown despite the discoveries of an adult-associated egg of dromaeosaurid Deinonychus antirrhopus (Grellet-Tinner and Makovicky, 2006), a single-layered, center-devoid-of-egg dromaeosaurid clutch in North America (Zelenitsky and Therrien, 2008), and a likely small dromaeosaurid clutch in Japan (Tanaka et al., 2020). The TCI behavior could as well have evolved independently in troodontids and birds.