Ecology

The avian plumage color gamut is much more diverse than previously estimated2. We demonstrate that hummingbird barbule structural colors contribute substantially to the total color diversity of living birds, occurring in areas of the avian color space that were sparsely occupied in Stoddard and Prum2, which most notably included saturated blues, greens, and true purples (blue + red). Such regions of the avian color space were suggested to be unoccupied because these colors are challenging to create, rather than because they might function poorly for communication2. Our results support this hypothesis because hummingbird coloration densely occupies these regions of the avian color gamut (Fig. 2d), using plumage patches that generally play particularly important roles in hummingbird communication, such as throat and crown plumage patches (Supplementary Fig. 5)16,17. The greater color diversity uncovered by our study suggests that barbule structural coloration is the most versatile class of all plumage coloration mechanisms and poses the least constraints on the evolvability of plumage color diversity. Barbule structural colors evolve through changes in the size, shape, spacing, and refractive index of barbule melanosome nanostructures, but little is known about how changes in these parameters themselves evolve18.The UV/V + green region of avian color space remains mostly unoccupied (Fig. 2c, d). It is challenging to create colors with separate reflectance peaks within the wavelength sensitivities of non-adjacent color cones because the peaks must be highly saturated to avoid stimulating neighboring cones2. However, this idea does not explain why there are far more true purple (blue + red) than UV/V + green plumage colors. Notably, birds particularly fail to fill the more UV/V regions (those closer to the UV/V vertex) of UV/V + green color space, which might indicate that it is difficult to create spectra with uv/v wavelength peaks higher than those in the m wavelengths.The differences between our methods and those of Stoddard and Prum2 likely contribute in part to the larger gamut size when comparing species data but not overall data. While the number of species included in our study was comparable to that of Stoddard and Prum2 (114 vs 111 species, respectively), we measured almost twice as many plumage patches as they did (+1600 vs. 965 patches). To prevent erroneous distortion to iridescent colors we did not average the three measurements per patch. Both studies measured six standard patches for all species and additional patches if necessary to capture other plumage color variation. The larger number of plumage patches we measured reflects how color diverse hummingbird plumages are. Our methods preserved the natural variation in hue due to iridescence and avoided the distorted flattening caused by averaging highly saturated peaks with slightly different peak hues. Although our methods are biased toward increasing variation, they are necessary to accurately capture the phenomenon of iridescent hummingbird coloration.There are multiple reasons why the hummingbird color gamut is so diverse. The size of the hummingbird color gamut, like the achieved color gamut of any clade, constitutes a combination of the history of selection on color function, the clade’s evolved capacities for color production, the age of the clade, and the number of species. Hummingbirds excel at all these criteria. The 336 species of extant hummingbirds have radiated rapidly over the last 22 million years19. Hummingbird plumage color diversity has evolved through a long history of persistent sexual and social selection on plumage coloration. Hummingbirds have polygynous breeding systems characterized by female only parental care, female mate choice, and often elaborate male courtship displays. Intersexual selection in hummingbirds has contributed to elaborate radiation in brilliant plumage coloration as well as vocalizations and non-vocal feather sounds14,16,20. Hummingbird plumage color evolution rates have even been shown to positively correlate with hummingbird speciation rates14. Furthermore, in some species, brilliant monomorphic plumage ornaments apparently function in aggressive, intra- and interspecific defense of floral resources21 and appear to be associated with socioecological features related to resource competition19. Our finding that crown and throat patches, which flash brilliantly when the head of the bird is oriented toward the observer, are more diverse in coloration than other plumage regions highlights the role of plumage coloration in direct inter-individual communication and social interactions.The mechanistic properties of hummingbird barbule structural color further explain the exceptional diversity of hummingbird plumage coloration. Hummingbird barbule structural coloration is among the most complex plumage coloration mechanisms, comprised of stacks of hollow, air-filled melanosomes, surrounded by a thin superficial, solid keratin cortex as well as sometimes superficial, miniature melanin platelets which lie just beneath this cortex9,10,11,12,13. Complex nanostructures allow for independent tuning of multiple components, and, hence, greater achievable color diversity12,18,22. Barbule structural color permits the production of any peak-reflected wavelength by varying the thickness of melanosome arrays, which can produce a diversity of single-peak spectra-hues, such as the unusual diversity of greens, blues, and blue + greens seen in hummingbirds (Fig. 2b). Hummingbird melanosomes are among the most unusual in birds in being both disc-shaped and air-filled9,10,11,12,13,23. The air in the center of hummingbird melanosomes approaches the maximum possible biological difference in refractive index (air = 1.0, melanin = ~1.7), which results in the efficient production of brilliant colors with the fewest layers of melanosomes, such that resulting spectra are narrow and near saturation13,24. Such spectra can thereby create colors that extend further in color space (Fig. 2a–c).Barbule structural color also allows for the production of plumage spectra with multiple saturated peaks, creating saturated color combinations that are not as commonly produced via other plumage coloration mechanisms. However, researchers have yet to identify exactly how hummingbird multipeak spectra are produced12,13, emphasizing the need for further analyses of the optics of hummingbird feathers. Many hummingbird melanosome arrays are non-ideal– i.e., the products of the thicknesses and refractive indices of the melanin and air cavity layers are not equal25. Non-ideal thin films can create more highly saturated, pure tone colors of the primary peak while also introducing additional, harmonic spectral peaks at shorter wavelengths25, which allows for complex reflectance spectra with multiple bright peaks within the avian visible spectrum. Also, melanosome arrays with a large average layer thickness ( >~300 nm) can create colors with fundamental interference peaks in the infrared and multiple, harmonic peaks in the avian visible range (300–700 nm). The presence of minute, superficial melanin platelets below the cortex in hummingbird barbules is also correlated with secondary, lower wavelength reflectance peaks, but the precise optical mechanism remains to be established12. These different nanostructural elements all contribute to distinctive multipeak reflectance spectra that can stimulate non-adjacent color cone combinations, which Stoddard and Prum2 identified as particularly difficult to accomplish: UV/V-purple (uv/v + s + l wavelengths; Schistes geoffroyi cheek, Fig. 4g); true purple (s + l wavelengths; Atthis ellioti gorget, Fig. 4h); UV/V-green (uv/v + m; Schistes geoffroyi crown, Fig. 4a); and UV/V-red (uv/v + l; Heliangelus viola, Fig. 4b). With multipeak spectra the potential for creating new and different colors is greatly expanded, allowing for a more versatile evolution of novel colors.Unexpectedly, the hummingbird plumage color gamut is larger in volume when modeled with the VS-type (34.2%) than with the UVS-type (29.6%) visual system. This apparently unique result contrasts notably with both Stoddard and Prum’s2 and our revised estimate of the color gamut of all birds combined– VS gamut = 40.5%; UVS gamut = 47.3%. Multiple previous analyses have shown that the UVS cone-type visual system does a more efficient job of discriminating the colors of natural objects because of the broader separation between the peak spectral sensitivities of the uv and s (blue) cone types2,26,27. Because the UVS-type visual system produces an even greater increase in color volume for a diverse plant color data set over the VS-type visual system, Stoddard and Prum2 rejected the hypothesis that the UVS-type visual system had specifically evolved to expand the diversity of avian color stimuli.However, our observations that the hummingbird plumage gamut is substantially greater in volume with the VS-visual system than with the more efficient UVS-visual system strongly suggests another hypothesis: Hummingbird plumage may have specifically evolved to be more diverse within the hummingbird VS-type color visual system via selection for highly saturated plumage colors. Given diversity in hue, the way to achieve greater color gamut volume, i.e., greater plumage color diversity, is through highly chromatic color vectors that extend toward the limits of the color space. The two visual systems map variation in wavelength to different maximum potential chroma—i.e., wavelengths with color vectors that extend toward the edges, faces, and vertices of the tetrahedron6. Color vectors that extend towards the vertices, i.e., plumage that best corresponds to a singular cone type’s peak sensitivity, have the highest maximum potential chroma because vertices are the regions furthest away from the tetrahedron’s center. Thus, hummingbird plumages may have specifically evolved to have maximum chroma within their own VS-visual system via peaks that correspond most closely to the peak sensitivities of the VS- rather than the UVS-visual system. For example, when comparing the UVS and VS plumage color gamuts for hummingbirds, it is notable that hummingbird coloration extends much further into the UV/V regions of color space for the VS-visual system (Supplementary Fig. 2). While in the VS system these color points map toward the v vertex, in the UVS-visual system they map towards the uv-s edge and the uv-s-l face. Such color vectors that contribute to expanded color volume of the VS gamut could have evolved by sexual or social selection for highly saturated plumage colors that are near in hue to the specific sensitivity peaks of hummingbird receptor cone types. Such selection could note preferences within some hummingbird species for hues with maximally possible chroma, not merely for maximal chroma of a given hue.Hummingbirds have tetrachromatic color vision with substantial sensitivity in the near ultraviolet28,29. Recently, Stoddard et al.30 used a series of elegant experiments with hummingbird feeders and LED lights to demonstrate for the first time that hummingbirds can distinguish non-spectral colors distributed throughout the tetrachromatic color space. However, the presence of this remarkably proficient four-color vision in hummingbirds poses an interesting evolutionary conundrum. Recent phylogenetic analyses have established that hummingbirds and swifts are phylogenetically embedded within the nocturnal caprimulgiforms31,32. The most parsimonious hypothesis is that the immediate ancestors of swifts and hummingbirds were extensively nocturnal for approximately 8 million years before they re-evolved diurnal ecology and behavior31. Given that an evolutionary history of nocturnality can lead to the degradation or loss of opsin genes33,34, it should be a high priority to establish what effect that ancestral nocturnality may have had on the molecular physiology and anatomy of the hummingbird color visual system.Our attempt to document the color diversity of an avian family has revealed that current estimates of the total avian color gamut are likely inaccurately low. Similar studies sampling from other color-diverse families, such as sunbirds (Nectariniidae), parrots (Psittacidae), tanagers (Thraupidae), birds of paradise (Paradiseidae), manakins (Pipridae), and starlings (Sturnidae), most of which have already been studied for their plumage coloration35,36,37,38,39, would help us obtain a better estimate of the true avian color gamut. More

In their original description of V. rhodanica, Fischer & Riou16 determined that the previously undescribed genus was a Jurassic relative of V. infernalis. This assignment was based on the configuration of the arm crown and armature, fin type, presence of luminous organs, lateral eyes, and the absence of an ink sac. Assuming this assignment is correct, then V. rhodanica is a member of the suborder Vampyromorphina, which includes the family Vampyroteuthidae22,29.Reappraisal of the anatomy shows that V. rhodanica and V. infernalis both have 8 arms and uniserial suckers flanked by cirri. They both possess V. infernalis-like sucker attachments34,36, which are broader at the base and taper up to a radially symmetrical sucker.Both species have distinctly modified arms though the morphology differs in each. V. infernalis, has retractable filaments in the position of arm pair II27,33,34, though there is no evidence of these appendages in V. rhodanica. Instead, the species has elongate dorsal arms (arm pair I) with a unique configuration of suckers and cirri on the distal section.The suckers and cirri of V. rhodanica are more numerous than those of V. infernalis27,37. They are also more closely positioned. Proportionally, the suckers of both species have a consistent ratio to mantle length37, though the diameter of the cirri and infundibulum are greater in V. rhodanica. The V. infernalis-like attachment1,3,34 is present in both species, though in V. rhodanica, the distal part of the neck protrudes into the acetabular cavity. Of note, the sucker stalks on the dorsal arms of V. rhodanica are more elongate than those on the other arms (Figs. 2b,c, and 3a,b). This variation in suckers and their attachments suggests a specialized function between the dorsal and sessile appendages. On the longer dorsal arms, the larger sucker diameter, and more elongate stalks (Figs. 2b and 4) indicate the potential for increased mobility over their extant relatives, and possibly facilitated additional manipulation and prey capture capability.Figure 4Hypothesised reconstruction of V. rhodanica based on the data from this study (A. Lethiers, CR2P). The scale is based on measurements from the holotype (MNHN.B.74247) and the arm crown is completed using dimensions from MNHN.B.74244.Full size imageIn addition to the arm crown specialization, V. rhodanica has a more streamlined shape than V. infernalis, which is caused by a proportionally narrower head. Their muscular body is narrower and more elongate than the gelatinous V. infernalis16,27,37 suggesting a higher energy locomotory style. This is consistent with increased predation relative to the modern form. Observations in this study support many assertions of Fischer & Riou16 about the characters in V. rhodanica, though the presence of luminous organs cannot be confirmed. Rather than luminous organs much larger than those present in the deep-sea, extant V. infernalis, it is possible that these structures represent displaced cartilage prior to fossilization (Supplementary Fig. 6).Two other genera from the La Voulte-sur-Rhône locality, Gramadella and Proteroctopus are, like V. rhodanica, considered to be Incertae sedis Vampyromorpha22. All three share morphological similarities that include an elongated mantle fused with the head, and a longer dorsal arm pair with armature on the distal ends1,16,22,38. Neither the second nor fourth arm pair have been modified. Each has one pair of fins. In Gramadella, the fins are lateral and skirt-like16,38. In V. rhodanica and Proteroctopus these fins are located posteriorly1,16.V. rhodanica shows the greatest length variation between the dorsal and sessile arms (Fig. 4), though proportionally, Gramadella, and Proteroctopus have longer dorsal arms1,31. Fischer & Riou31 and Kruta et al.1 described biserial suckers in their descriptions of Gramadella, and Proteroctopus, respectively. In Proteroctopus, these suckers have a proportionally smaller diameter than the uniserial row in V. rhodanica, and do not exhibit the same tapered pattern.None of these specimens shows evidence of an ink sac, though it is present in contemporaneous genera from the same assemblage (Mastigophora, Rhomboteuthis and Romaniteuthis)8,16. That this character occurs only in some taxa from the same assemblage suggests variation in ecology, possibly associated with the steep, bathymetric relief in the La Voulte-sur-Rhône paleoenvironment11. The mosaic of characters found within the coleoid taxa at La Voulte-sur-Rhône suggests that Mesozoic vampyromorphs co-occurred in different ecological niches during the mid-Jurassic.Today, extant V. infernalis is uniquely adapted to a low-energy, deep-sea mode of life27,28,29,39, though the timing of character acquisition and progression of this ecology is unclear24. It is hypothesised that the vampyromorph Necroteuthis Kretzoi 1942 was already exploiting this niche by the Oligocene29, and that the initial shift to offshore environments was possibly driven by onshore competition24,29. The data obtained here suggests that V. rhodanica, the purportedly oldest-known genus of the Vampyromorphina group, was an active predator following a pelagic mode of life.Indeed, several anatomical details, mainly found in the brachial crown, seem to support this hypothesis. Though we cannot directly compare functionality of the arm crown elements with other Jurassic taxa, we can infer function based on observation in modern forms. In Octopoda, the sister group to Vampyromorpha, suckers are attached to the arm by a cylindrical layer of muscle, encircling oblique musculature40,41, that connects the arm musculature and the lateral margin of the acetabulum34,40,41,42. This facilitates a variety of functions including locomotion, manipulation, and prey retention43. The sucker attaches by flattening the infundibulum against the surface and then the encircling epithelium creates a watertight seal36,40,41,42,43,44,45. Contraction of the radial acetabular muscles provides the pressure differential required to create the suction force43,44,46.The stalked sucker attachments2,34 of decabrachians (Fig. 3d, and Supplementary Fig. 4) are muscular35 and connect the musculature of the arm with the base of the sucker, forming part of the acetabulum33,34. Tension on the sucker stretches this muscular attachment, which pulls locally on the acetabular base. This facilitates a greater pressure differential inside the sucker, allowing the teeth on the sucker ring to maintain the hold47.Extant V. infernalis lack decabrachian-like stalks2,18 and the neck of the attachment joins to the base of the acetabulum (Fig. 3c, and Supplementary Fig. 4), rather than being inserted into it18. The infundibulum is not distinct, and the suckers do not provide strong suction27. Instead, suckers function by secreting mucus to coat detritus—marine snow captured by retractable filaments—which is then moved to the mouth by cirri7,27.A mosaic of these characters is present in V. rhodanica (Fig. 3a,b), therefore, suggesting their potential for increased attachment and hold on prey over extant V. infernalis. These include a larger infundibular diameter, a neck attachment integrated with the acetabular muscles, and the elongated stalks of the dorsal suckers.Additionally, the paired, filamentous cirri observed in extant cirrates48 are present in V. rhodanica (Fig. 4, and Supplementary Fig. 2). In extant forms they are understood to have a sensory function and are used in the detection and capture of prey48. In V. infernalis, they serve to transport the food proximally along the arms to the mouth27. The greater diameters of cirri, and placement along the entire arm in V. rhodanica (Fig. 4), suggests an increased sensory function in these fossil forms.The shape of the arms also contributes to the suction potential49 in coleoids. Functional analysis in Octopoda highlights a positive correlation between distal tapering of the arms and their flexibility. A tapered, flexible arm facilitates more precise adhesion than a cylindrical-shaped one and requires a greater force for sucker detachment49. Suckers detach sequentially, rather than the more simultaneous release observed in models of arms with less taper variation. The tapered diameter of the suckers, like those seen on the sessile arms of V. rhodanica, potentially facilitated this kind of sequential detachment49 allowing them more adherence force and flexibility. Though V. rhodanica has just two suckers on the distal tips of their dorsal arms, the most distal is marginally smaller in diameter than the proximal one. On the dorsal arms, this tapering is observed in conjunction with a well-developed axial nerve cord (Fig. 2b). In extant forms, the nerve cord facilitates complex motor functions42. The combination of these characters in V. rhodanica suggests their arms had increased potential to be actively used in prey capture50 over extant V. infernalis.Though arm crown characters offer insight on the ecology of V. rhodanica, in fossil coleoid phylogenies only a few characters are based on the suckers1, 3. Two studies that have attempted to create a phylogeny using morphological characters that include both fossil and extant taxa return V. rhodanica and V. infernalis as sister taxa1,3. These matrices are, by necessity, heavily influenced by the gladius51 and more than 50% of the characters are based on this feature1,3. Indeed, the authors1 note that the lack of gladius data for some fossil forms, including V. rhodanica, creates an inherent bias in the phylogenetic matrix. Fischer & Riou16 suggested that V. rhodanica and V. infernalis are related on the basis of the observable morphological characters in the family Vampyroteuthidae, though without morphological information on the gladius, a recent systematic synthesis of fossil Octobrachia22 positioned V. rhodanica as Vampyromorpha Incertae sedis.X-ray CT analysis in this study did not allow a reconstruction of the gladius. Nevertheless, it does provide new data on soft tissues, and permits comparisons between extant and fossil taxa. Specifically, we can add distinct states to 4 of the 132 characters in the existing phylogenetic matrix from Sutton et al.3 that was modified and used in Kruta et al.1. These four characters (#89–#92) represent the suckers, and sucker attachments. Detailed examination revealed that the sessile and dorsal arms have the Vampyroteuthis-like attachment. In the dorsal arms, this is more elongated, though it cannot be considered pedunculate like those seen in modern decabrachians. Indeed, the attachment type (plug and base34) is the same, only the length varies. As previously discussed, this variation may have functional implications.When updated with these new data, the matrix from this study returns the same topology seen in Kruta et al.1 that supports the positioning of V. rhodanica and V. infernalis as sister taxa. Further, it strengthens their relationship as they both share a sucker attachment that is not clearly attached to the arm muscles, a state that was previously considered autapomorphic in V. infernalis. However, it is important to note that no additional characters were added for the gladius, which is the cornerstone of coleoid systematics52. Indeed, just 29 of the 132 matrix characters can so far be coded for V. rhodanica, with only 9 of these relating to the 74 states of the gladius.Assuming the phylogenetic work so far is correct, then both species belong to the family Vampyromorphina, and are joined by the Oligocene fossil Necroteuthis hungarica29. While the lack of gladius characters precludes a full phylogenetic understanding of this group, preservation and observation of the soft tissues allow us to infer information regarding palaeobiology.The data obtained in this study demonstrates that the characters observed in V. infernalis, including the sucker attachments and lack of ink sac, were present in Jurassic Vampyromorpha. Comparative anatomy of V. rhodanica and extant V. infernalis revealed that the fossil taxon displayed more morphological variation and were more diversified than previously understood. The assemblage of characters observed in V. rhodanica are consistent with a pelagic predatory lifestyle and corroborate the likelihood of a distinctly different ecological niche. These findings support the hypothesis that a shift towards a deep-sea environment occurred prior to the Oligocene5,29. More

Life cycle cost modelAn analysis of life cycle costs is an economic analysis of the assessment of the total cost of acquisition, ownership and liquidation of a product. It is applicable during the entire life cycle of the product or a life cycle stage or combination of different stages21 and22.There are five period phases of the vehicle life cycle:Generally, the total costs for the above listed phases are acquisition costs, ownership costs and liquidation costs21 and22. For the LCC model, I recommend to divide the life cycle costs into four categories:$$LCC={C}_{P}+{C}_{M}+{C}_{O}+{C}_{D},$$
(1)
$${LCC}_{s}=frac{LCC}{t},$$
(2)

where LCC—the life cycle cost of vehicles, LCCs—the specific life cycle cost of vehicles, CP—the vehicle purchase cost, CM—the maintenance cost, CO—operating state of vehicle cost, CD—the vehicle disposal cost, t—the time of vehicle operation.The model for evaluating the economic viability of products is based on the general LCC model which is based on acquisition and ownership costs$$LCC={C}_{P}+{C}_{OW},$$
(3)

where CP—purchase cost, COW—ownership costs.Acquisition cost (CP) is represented by the purchase price at the time of acquisition of the assessed passenger vehicle.Ownership cost (COW) is significant during the life cycle of a motor vehicle and varies according to the type of the vehicle. This cost includes the costs of maintenance and operation time can be defined as follows10$${C}_{Ow}={C}_{M}+{C}_{O},$$
(4)

where CM—cost of maintenance, CO—operation cost.The cost of ownership a vehicle (COW) can be defined as follows$${C}_{OW}={C}_{O}+{C}_{MC}+{C}_{MP},$$
(5)

where CO—operation cost, CMC—corrective maintenance cost, CMP—preventive maintenance cost.The cost of ownership (COW) may include the operating and maintenance costs which consist of the corrective maintenance cost (CMC) and the cost of preventive maintenance (CMP) of a motor vehicle.Calculation of operating costsOperating cost CO is determined by the price and amount consumed of conventional or alternative types of fuel. It cover the cost of fuel CF, operating fluids, oils and lubricants COL that are supplied during vehicle operation (not during service inspection), tyres CT, accumulator batteries CAB, vehicle insurance fee and road tax or other mandatory fees CIRT, cost of the motorway tax sticker CMT, mandatory vehicle inspection and emission measurement in special vehicles CETC. The costs are calculated according to$${C}_{O}={C}_{F}+{C}_{OL}+{C}_{T}+{C}_{AB}+{C}_{IRT}+{C}_{MT}+{C}_{ETC}.$$
(6)
Fuel costs (CF) are affected by the average consumption of a given type of propulsion vehicle. Then the comparative fuel costs (CF) can be expressed by the equation$${C}_{F}=frac{{bar{c}}_{aF}}{100}{p}_{F}{t}_{l},$$
(7)

where CF—total fuel costs (EUR), (bar{c})aF—average fuel consumption (l/100 km), pF—fuel price (EUR/l), tl—service life of a passenger vehicle (km).Costs for operating fluids, oils and lubricants (COL) are any costs for operating fluids, oils and lubricants that are replenished during operation and not during service maintenance; it can be expressed by the equation$${C}_{OL}=frac{{bar{c}}_{aOL}}{100}{p}_{OL}{t}_{l},$$
(8)

where (bar{c})aOL—average consumption of oil and lubricant (l/100 km), pOL—price of oil and lubricant (EUR/l).The cost of tyres (CT) can be expressed by the equation$${C}_{T}=frac{{t}_{l}}{{bar{d}}_{aT}}{n}_{T}{p}_{T},$$
(9)

where (bar{d})aT—average life of a passenger vehicle tyre (km), nt—number of tyres on the passenger vehicle (pc), pT—price of one piece of tyre (EUR).Accumulator battery costs (CAB) —can be expressed by the equation$${C}_{AB}=frac{{t}_{l}}{{bar{d}}_{aAB}}{n}_{AB}{p}_{AB},$$
(10)

where (bar{d}_{aB})—average life of one accumulator battery (km), nAB—number of accumulator batteries in the passenger vehicle (pc), pAB—price of an accumulator battery (EUR).Costs arising from laws (CIRT) are the costs of motor vehicle insurance (compulsory liability, accident insurance, or other). Some of them can be omitted in case of the same costs due to the simplification of the model. Otherwise, they can be expressed by the equation$${C}_{IRT}=left({C}_{SI}+{C}_{AI}+{C}_{RT}+{C}_{R}right){t}_{la},$$
(11)
where CS1—price of mandatory annual insurance of a passenger vehicle (EUR), CA1—price of the annual accident insurance of a passenger vehicle (EUR), CRT—price of annual road tax (EUR), CR—price of statutory fee (EUR), tla—operating time of the passenger vehicle until decommissioning (years).The cost of obtaining a motorway sticker (CMT) may be omitted if the same type of passenger vehicle is compared. Otherwise, the cost of a motorway sticker (CMT) can be expressed by the equation$${C}_{MT}={c}_{MT}{t}_{la},$$
(12)

where cMT—price of annual motorway sticker for the passenger vehicle (EUR).The costs of the mandatory vehicle inspection and emission measurement (CETC) include the costs incurred for the measurement of emissions of the drive engine unit (CE) and for the technical inspection of the passenger vehicle (CTC). For the proposed model, the costs of the mandatory technical inspections and emission measurements can be expressed by the equation$${C}_{ETC}=left({C}_{E}+{C}_{TC}right)frac{{y}_{n}}{{t}_{la}},$$
(13)

where CE—costs related to the measurement of passenger vehicle emissions (EUR), CTC—costs of mandatory technical inspection (EUR), yn—number of years of legal validity of emission measurement and technical condition for the given type of the passenger vehicle (years).Calculation of maintenance costThe total costs for vehicle maintenance CM consist of the cost of preventive maintenance CMP and the cost of corrective maintenance CMC10,11$${C}_{M}={C}_{MC}+{C}_{MP}.$$
(14)
Vehicle maintenance costs include the cost of material and the cost of labour$${C}_{M}={(C}_{MCM}+{C}_{MCL}+{C}_{MCF})+left({C}_{MPM}+{C}_{MPL}+{C}_{MPF}right),$$
(15)

where CM—cumulative maintenance costs, CMC—corrective maintenance costs, CMP—preventive maintenance costs, CMCM—costs of material used for corrective maintenance, CMCL—costs of labour force for corrective maintenance, CMCF—costs of workshop equipment used for corrective maintenance, CMPM—costs of material used for preventive maintenance, CMPL—costs of labour force for preventive maintenance, CMPF—costs of workshop equipment used for preventive maintenance.

Preventive maintenance costs (CMP) are costs that include all costs associated with preventive maintenance performed to reduce degradation and mitigate the likelihood of failure. At present, preventive maintenance is performed at predetermined time intervals (according to the manufacturer’s preventive maintenance program) or when a specified number of kilometres are not covered before the next service maintenance, depending on the time. In practice, for passenger cars, it is usually 1 or 2 years, depending on the use of engine oil. This mainly includes the cost of:

material consumed during preventive maintenance,

work spent on preventive maintenance,

workshop equipment, training of preventive maintenance specialists.$${C}_{MP}=frac{{t}_{l}}{MTB{M}_{p}}left({C}_{MPM}+{(bar{c}}_{p}{bar{t}}_{pm})right),$$
(17)

where MTBMp—mean operating time between preventive maintenances (km), CMPM—costs of material used for preventive maintenance (EUR), (bar{c})p—average hourly cost of labour and workshop equipment used for maintenance (EUR/hour), ̅tpm—mean time of labour-intensity per one preventive maintenance (hour).

Design of a model for the analysis of selected life cycle costs of a passenger motor vehicleThe model for performing an analysis of life cycle costs for the purchase of a new motor vehicle is based on the basic Eq. (3), (18). We will not count the costs of improvement (CE) and the costs of the decommissioning phase (CD) for the mentioned model due to the calculations of costs that are unnecessary for the analysis. Then the model can be expressed as follows$$LCC={C}_{P}+{C}_{O}+{C}_{M}.$$
(18)
Then, the following Eqs. (6), (7), (8), (9), (10), (11), (12), (13), (16) and (17) are substituted into the given equation, and the selected costs can be calculated for individual vehicles. The resulting model for calculating the LCC costs has the following form$$LCC={C}_{p}+frac{{bar{c}}_{aF}}{100}{p}_{F}{t}_{l}+frac{{bar{c}}_{aOL}}{100}{p}_{OL}{t}_{l}+frac{{t}_{l}}{{bar{d}}_{aT}}{n}_{T}{p}_{T}+frac{{t}_{l}}{{bar{d}}_{aAB}}{n}_{AB}{p}_{AB}+{C}_{SI}{t}_{la}+{c}_{MT}{t}_{la}+left({C}_{E}+{C}_{TC}right)frac{{y}_{n}}{{t}_{la}}+frac{{t}_{l}}{MTBF}left({bar{c}}_{m}+{(bar{c}}_{p}{bar{t}}_{pc})right)+frac{{t}_{l}}{MTB{M}_{p}}left({C}_{OMPM}+{bar{(c}}_{p}{bar{t}}_{pm})right).$$
(19)
It is presented in a Fig. 6.Figure 6Structure of model input parameters for LCC model calculation.Full size imageIn this way, the cumulative costs for each passenger motor vehicle are calculated. Since the passenger motor vehicles may have a different service life tl which is expressed in kilometres, it is recommended to convert this equation to specific costs which are related to one kilometre of use. The selected LCCS life cycle specific costs can be expressed by the following equation$${LCC}_{S}=frac{LCC}{{t}_{l}}.$$
(20)
LCC model input values and items affecting ownership costs for alternative drivesThe process of the calculation of selected life cycle costs for the propulsion of passenger vehicles and the structure of individual cost items is shown in Fig. 6. These are the input parameters to the LCC model.The total life cycle costs are divided into two main cost groups, which are the ownership and acquisition costs for a given drive type. Fuel costs are determined by the price and the quantity of conventional or alternative fuel consumed. For the calculation of the selected LCCs, the authors of the paper assume that the availability of conventional and alternative fuels is not limited in any way. It is assumed that the availability of fuels is ideal, which is not entirely true in practice. This is dependent on the support for each alternative fuel in each state.In practice, therefore, multiple costs may arise due to the distance to the refuelling station to provide alternative fuels such as E85, CNG, LPG and hydrogen. In addition, there is a distance to the charging station for electric drives.Another item that affects the cost of operation for hybrid passenger vehicles is the percentage of alternative fuel driving, which can have a significant impact on life cycle costs. Values for this item are given as a percentage, which is then converted into the number of kilometres driven on alternative and conventional fuel.One of the important parameters for calculating the life cycle operating costs for the hybrid-electric and electric drive is the setting of a threshold value for the capacity of the electric vehicle battery (EV battery) when the replacement is performed. For the model calculation, a limit value of 70% of the electric vehicle battery capacity at 20 °C was set.Return on investmentReturn on investment (ROI) is a performance measure used to evaluate the efficiency or profitability of an investment or compare the efficiency of a number of different investments. ROI tries to directly measure the amount of return on a particular investment, relative to the investment’s cost. To calculate ROI, the benefit (or return) of an investment is divided by the cost of the investment. The result is expressed as a percentage or a ratio12,23.For our calculation of the return on investment ROI on alternative and conventional passenger car propulsion the following formula is used, which is expressed as a percentage$$ROI=frac{{LCC}_{A}-{LCC}_{C}}{{LCC}_{C}}100,$$
(21)

where LCCA—selected live cycle costs of the alternative passenger car propulsion (EUR), LCCC—selected live cycle costs of the conventional passenger car propulsion (EUR).The return on investment of an alternative vehicle ROIAV purchase expresses after how many kilometres the increased cost of purchasing an alternative fuel vehicle compared to a conventional one is recovered. If the value is negative, the payback will not occur for various reasons. The following equation is used to calculate ROIAV$${ROI}_{AV}=frac{{C}_{{P}_{AV}}-{C}_{{P}_{CV}}}{frac{{C}_{O{W}_{CV}}-{C}_{O{W}_{AV}}}{{t}_{l}}}$$
(22)

where ({C}_{{P}_{AV}})—purchase cost on alternative vehicle (EUR), ({C}_{{P}_{CV}})—purchase cost on conventional vehicle (EUR), ({C}_{O{W}_{CV}})—ownership cost on conventional vehicle (EUR), ({C}_{O{W}_{AV}})—ownership cost on alternative vehicle (EUR), tl—service life of the passenger vehicle (km).Ownership costs on conventional vehicle are expressed by the following equation$${C}_{{OW}_{CV}}={left(frac{{bar{c}}_{aF}}{100}{p}_{F}{t}_{l}+frac{{bar{c}}_{aOL}}{100}{p}_{OL}{t}_{l}+frac{{t}_{l}}{{bar{d}}_{aT}}{n}_{T}{p}_{T}+frac{{t}_{l}}{{bar{d}}_{aAB}}{n}_{AB}{p}_{AB}+{C}_{SI}{t}_{la}+{c}_{MT}{t}_{la}+left({C}_{E}+{C}_{TC}right)frac{{y}_{n}}{{t}_{la}}+frac{{t}_{l}}{MTBF}left({bar{c}}_{m}+{(bar{c}}_{p}{bar{t}}_{pc})right)+frac{{t}_{l}}{MTB{M}_{p}}left({C}_{OMPM}+({bar{c}}_{p}{bar{t}}_{pm})right)right)}_{CV}.$$
(23)
Ownership costs on alternative vehicle are expressed by the following equation$${C}_{{OW}_{AV}}={left(frac{{bar{c}}_{aF}}{100}{p}_{F}{t}_{l}+frac{{bar{c}}_{aOL}}{100}{p}_{OL}{t}_{l}+frac{{t}_{l}}{{bar{d}}_{aT}}{n}_{T}{p}_{T}+frac{{t}_{l}}{{bar{d}}_{aAB}}{n}_{AB}{p}_{AB}+{C}_{SI}{t}_{la}+{c}_{MT}{t}_{la}+left({C}_{E}+{C}_{TC}right)frac{{y}_{n}}{{t}_{la}}+frac{{t}_{l}}{MTBF}left({bar{c}}_{m}+{(bar{c}}_{p}{bar{t}}_{pc})right)+frac{{t}_{l}}{MTB{M}_{p}}left({C}_{OMPM}+({bar{c}}_{p}{bar{t}}_{pm})right)right)}_{AV}.$$
(24)
The rate of return on investment for the purchase of an alternative vehicle depending on the kilometres travelled to is expressed by the following equation$${ROI}_{AV({t}_{o})}={(C}_{{P}_{AV}}-{C}_{{P}_{CV}})-({C}_{O{W}_{CV}left({t}_{o}right)}-{C}_{O{W}_{AV}left({t}_{o}right)}) quad text{when} ;to = (0-tl)$$
(25)

where to—operation of the passenger vehicle (km). More

Experimental designThe comparative field microcosm experiments were conducted on Laojun Mountain in China (26.6959 N; 99.7759 E) in September–October 2013, and on Balggesvarri Mountain in Norway (69.3809 N; 20.3483 E) in July 2013, designed to be broadly representative of subtropical and subarctic climatic zones, respectively, as first reported in Wang et al.29. In the Laojun Mountain region, mean annual temperatures ranged from 4.2 to 12.9 °C, with July mean temperatures of 17–25 °C. In the Balggesvarri Mountain region, mean annual temperatures ranged from −2.9 to 0.7 °C, with July mean temperatures of 8–16 °C. The experiments were characterised by an aquatic ecosystem with consistent initial DOM composition but different locally colonised microbial communities and newly produced endogenous DOM. While allowing us to minimise the complexity of natural ecosystems, the experiment provided a means for investigating DOM-microbe associations at large spatial scales by controlling the initial DOM supply. Briefly, we selected locations with five different elevations on each mountainside. The elevations were 3822, 3505, 2915, 2580 and 2286 m a.s.l. on Laojun Mountain in China, and 750, 550, 350, 170 and 20 m a.s.l. on Balggesvarri Mountain in Norway. At each elevation, we established 30 aquatic microcosms (1.5 L bottle) composed of 15 g of sterilised lake sediment and 1.2 L of sterilised artificial lake water at one of ten nutrient levels of 0, 0.45, 1.80, 4.05, 7.65, 11.25, 15.75, 21.60, 28.80 and 36.00 mg N L−1 of KNO3 in the overlying water. To compensate for nitrate additions shifting stoichiometric ratios, KH2PO4 was added to the bottles so that the N/P ratio of the initial overlying water was 14.93, which was similar to the annual average ratio in Taihu Lake during 2007 (that is, 14.49). Thus, we use “nutrient enrichment” to indicate a series of targeted nutrient levels of both nitrate and phosphate, the former of which was used to represent nutrient enrichment in the statistical analyses. Each nutrient level was replicated three times. The lake sediments were obtained from the centre of Taihu Lake, China, and were aseptically canned per bottle after autoclaving at 121 °C for 30 min. Nutrient levels for the experiments were selected based on conditions of the eutrophic Taihu Lake, and the highest nitrate concentration was based on the maximum total nitrogen in 2007 (20.79 mg L−1; Fig. S19). We chose the nutrient level of this year because a massive cyanobacteria bloom in Taihu Lake happened in May 2007 and initiated an odorous drinking water crisis in the nearby city of Wuxi.The microcosms were left in the field for one month allowing airborne bacteria to freely colonise the sediments and water. To keep the microbial dispersal events as natural as possible, we did not cover the experimental microcosms in case of rainfall. To avoid or minimize potential influence of extreme nature events, we (i) left the top 20% of each microcosm empty to prevent water from overflowing during heavy rains, and (ii) checked the experimental sites twice during each experimental period, and added sterilized water to obtain a final volume of approximately 1.2 L. The bottom of our microcosm was buried into the local soils by 10% of the bottle height, partly to reduce UV exposure to sediments. More considerations of the experimental design were detailed in the Supplementary Methods. To avoid the effects of daily temperature variation, we measured the water temperature and pH within 2 h before noon at all elevations in the day before the final sample collection. At the end of the experimental period, we aseptically sampled the water and sediments of the 300 bottles (that is, 2 mountains × 5 elevations × 10 nutrient levels × 3 replicates) for the following analyses of physiochemical variables, bacterial community and DOM composition.Physiochemical variables and bacterial communityWe measured environmental variables, namely, the total nitrogen (TN), total phosphorus (TP), dissolved nutrients (that is, NOx−, NO2−, NH4+ and PO43−), total organic carbon (TOC), dissolved organic carbon (DOC) and chlorophyll a (Chl a) in the sediments, and the NO3−, NO2−, NH4+, PO43− and pH in the overlying water (Table S2, Fig. S20), according to Wang et al.29.The sediment bacteria were examined using high-throughput sequencing of 16S rRNA genes. The sequences were processed in QIIME (v1.9)45 and OTUs were defined at 97% sequence similarity. The bacterial sequences were rarefied to 20,000 per sample. Further details on physicochemical and bacterial community analyses are available in Wang et al.29.ESI FT-ICR MS analysis of DOM samplesHighly accurate mass measurements of DOM within the sediment samples were conducted using a 15 Tesla solariX XR system, a ultrahigh-resolution Fourier transform ion cyclotron resonance mass spectrometer (FT-ICR MS, Bruker Daltonics, Billerica, MA) coupled with an electrospray ionization (ESI) interface, as demonstrated previously46 with some modifications. It should be noted that FT-ICR MS does not identify molecules, but only molecular formulae in terms of elemental composition and there can be many molecular structures sharing the same elemental compositions. DOM was solid-phase extracted (SPE) with Agilent VacElut resins before FT-ICR MS measurement47 with minor modifications. Briefly, an aliquot of 0.7 g freeze-dried sediment was sonicated with 30 ml ultrapure water for 2 h, and centrifuged at 5000 × g for 20 min. The extracted water was filtered through the 0.45 μm Millipore filter and further acidified to pH 2 using 1 M HCl. Cartridges were drained, rinsed with ultrapure water and methanol (ULC-MS grade), and conditioned with pH 2 ultrapure water. Calculated volumes of extracts were slowly passed through cartridges based on DOC concentration. Cartridges were rinsed with pH 2 ultrapure water and dried with N2 gas. Samples were finally eluted with methanol into precombusted amber glass vials, dried with N2 gas and stored at −20 °C until DOM analysis. The extracts were continuously injected into the standard ESI source with a flow rate of 2 μl min−1 and an ESI capillary voltage of 3.5 kV in negative ion mode. One hundred single scans with a transient size of 4 mega word (MW) data points, an ion accumulation time of 0.3 s, and within the mass range of m/z 150–1200, were co-added to a spectrum with absorption mode for phase correction, thereby resulting in a resolving power of 750,000 (FWHM at m/z 400). All FT-ICR mass spectra were internally calibrated using organic matter homologous series separated by 14 Da (-CH2 groups). The mass measurement accuracy was typically within 1 ppm for singly charged ions across a broad m/z range (150–1200 m/z).Data Analysis software (BrukerDaltonik v4.2) was used to convert raw spectra to a list of m/z values using FT-MS peak picker with a signal-to-noise ratio (S/N) threshold set to 7 and absolute intensity threshold to the default value of 100. Putative chemical formulae were assigned using the software Formularity (v1.0)48 following the Compound Identification Algorithm49. In total, 19,538 molecular formulas were putatively assigned for all samples (n = 300) based on the following criteria: S/N  > 7, and mass measurement error  0.80, P ≤ 0.001; Fig. S9). Similar conclusions were also obtained with either OTUs or genera when relating the pairwise distances of molecular traits with SparCC correlation coefficient ρ values among DOM molecules in Fig. 4c. To reduce type I errors in the correlation calculations created by low-occurrence genera or molecules, the majority rule was applied; that is, we retained genera or molecules that were observed in more than half of the total samples (≥75 samples) in China or Norway. The filtered table, including 1340 and 1246 DOM molecules, and 75 and 49 bacterial genera in China and Norway, respectively, was then used for pairwise correlation calculation of DOM and bacteria using SparCC with default parameters35.Finally, bipartite network analysis at a molecular level was performed to quantify the specialization of DOM-bacteria networks (Box 1). The specialization considers interaction abundance and is standardised to account for heterogeneity in the interaction strength and species richness, which describes the levels of “vulnerability” of DOM molecules and “generality” of bacterial taxa27. The threshold correlation for inclusion in bipartite networks was |ρ| = 0.30 to exclude weak interactions and we retained the adjacent matrix with only the interactions between DOM and bacteria. We then constructed two types of interaction networks (i.e., negative and positive networks) based on negative and positive correlation coefficients (SparCC ρ ≤ −0.30 and ρ ≥ 0.30, respectively). According to resource-consumer relationships, negative networks likely indicate the degradation of larger molecules into smaller structures, while positive networks may suggest the production of new molecules via degradation or biosynthetic processes. The SparCC ρ values were multiplied by 10,000 and rounded to integers, and the absolute values were taken for negative networks to enable the calculations of specialization indices. A separate negative and positive sub-network was obtained for each microcosm by selecting the DOM molecules and bacterial taxa in each sample based on its bacterial and DOM compositions. For the network level analysis, we calculated H2′, a measure of specialization27, for each network:$${H}_{2}=-mathop{sum }limits_{i{{mbox{=}}}1}^{i}mathop{sum }limits_{j{{mbox{=}}}1}^{j}{{mbox{(}}}{{{mbox{p}}}}_{{ij}}{{{{{{rm{ln}}}}}}}{{{mbox{p}}}}_{{ij}}{{mbox{)}}}$$
(2)
$${H}_{2}{prime} =frac{{H}_{2{max }}{-}{H}_{2}}{{H}_{2{max }}{-}{H}_{2{min }}}$$
(3)
where ({{{mbox{p}}}}_{{ij}}{{mbox{=}}}{{{mbox{a}}}}_{{ij}}{{mbox{/}}}m), represents the proportion of interactions in a i × j matrix. ({{{mbox{a}}}}_{{ij}}) is the number of interactions between DOM molecule i and bacterial genus j, which is also referred as “link weight”. m is the total number of interactions between all DOM molecules and bacterial genera. H2′ is the standardised H2 against the minimum (H2min) and maximum (H2max) possible for the same distribution of interaction totals. For the molecular level analysis, we calculated the specialization index Kullback–Leibler distance (d′) for DOM molecules (di′) and bacterial genera (dj′), which describes the levels of “vulnerability” of DOM molecules and “generality” of bacterial genera, respectively:$${d}_{i}=mathop{sum }limits_{j=1}^{j}left(frac{{{{mbox{a}}}}_{{ij}}}{{{{mbox{A}}}}_{i}}{{{mbox{ln}}}}frac{{{{mbox{a}}}}_{{ij}}m}{{{{mbox{A}}}}_{i}{{{mbox{A}}}}_{j}}right)$$
(4)
$${d}_{i}{prime} =frac{{d}_{i}-{d}_{{min }}}{{d}_{{max }}-{d}_{{min }}}$$
(5)
where ({A}_{i}) = (mathop{sum }limits_{j{{mbox{=}}}1}^{j}{{{mbox{a}}}}_{{ij}}) and ({A}_{j}) = (mathop{sum }limits_{i{{mbox{=}}}1}^{i}{{{mbox{a}}}}_{{ij}}), are the total number of interactions of DOM molecule i and bacterial genus j, respectively. di′ is the standardised di against the minimum (dmin) and maximum (dmax) possible for the same distribution of interaction totals. The equations of dj′ are analogous to di′, replacing j by i. Weighted means of d′ for DOM were calculated for each network as the sum of the product of d′ for each individual molecule i (di′) and relative intensity Ii divided by the sum of all intensities d′  = Ʃ(di′ × Ii)/Ʃ(Ii). Weighted means of d′ for bacteria were calculated as the sum of the d′ of each individual bacterial genus j (dj′) and relative abundance of bacterial genus Ij divided by the sum of all abundance. All calculations were performed using the R package FD V1.0.12. The observed H2′ and d′ values ranged from 0 (complete generalization) to 1 (complete specialization)28 (Fig. S21). Specifically, elevated H2′ or d′ values indicate a high degree of specialization, while lower values suggest increased generalization, that is, higher vulnerability of DOM and/or higher generality of microbes. To directly compare the network indices across the elevations or nutrient enrichment levels, we used a null modelling approach. We standardised the three observed specialization indices (Sobserved; that is, H2′, d′ of DOM, and d′ of bacteria) by calculating their z-scores63 using the equation:$${z}_{S}=({S}_{{{{{{rm{observed}}}}}}}-overline{{{S}}_{{{{{{rm{null}}}}}}}})/({sigma }_{S_{{{{{rm{null}}}}}}})$$
(6)
where (overline{{{S}}_{{{{{{rm{null}}}}}}}}) and ({sigma }_{S_{{{{{rm{null}}}}}}}) were, respectively, the mean and standard deviation of the null distribution of S (Snull). One hundred randomised null networks were generated for each bipartite network to derive Snull using the swap.web algorithm, which keeps species richness and the number of interactions per species constant along with network connectance. This null model analysis indicates that interactions between DOM and bacteria were non-random as the observed network specialization indices were generally significantly lower than expected by chance (P  0.05), which tests whether the model structure differs from the observed data, high comparative fit index (CFI  > 0.95) and low standardised root mean squared residual (SRMR  More

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Model 1: Evolution of multiple mating and mate monopolisation under ancestral monogamyIn all models, I assume a large population with a 1:1 sex ratio. I begin with what is possibly the simplest model set-up for deriving Bateman functions in a scenario that is completely symmetrical aside from gamete number. Assume a monogamous, externally fertilising population where parents pair up and release their gametes into a nest. That is, every individual in the initial population participates in exactly one fertilisation event (the equivalent of a mating). Now consider a mutant individual that can attract multiple mates of the opposite type to release gametes into its nest, with no competition from other individuals of its own type. This simple set-up avoids asymmetries arising from internal fertilisation, and the complication of direct gamete competition for the multiply mating mutant individual (which is examined in Models 2–3), placing focus directly on the core of the problem: the asymmetry arising in fertilisation from imbalanced gamete numbers. All gametes are released in one burst by all individuals, but the focal individual may achieve ‘multiple matings’ simply by monopolising multiple mates at its nest. The reproductive success of the focal individual is then equivalent to the number of fertilisations that take place in that nest. Our aim is to understand how the reproductive success of an individual deviating from the monogamous population strategy and instead mating with (hat{m}) individuals of the opposite type is altered. A strong positive relationship between (hat{m}) and reproductive success then indicates a steep Bateman gradient. If Bateman’s assertion is correct, the resulting gradient should be steeper for the type that produces the larger number of gametes. Note that there is a game-theoretical25 flavour to this setting, where the focus is on the fitness of a rare mutant in a population with a fixed resident strategy.The two types are labelled with x and y, which could correspond to the two sexes, depending on what gamete numbers are assigned to them. The number of gametes produced by a single individual is labelled nx and ny, and the total number of gametes in a nest (or more generally, a fertilisation arena which could be internal or external) is labelled with Nx and Ny. To compute the number of fertilisations in a nest with a total of Nx and Ny gametes, I use a fertilisation function first derived by Togashi et al.24 purely from biophysical principles, treating the two gamete types symmetrically, with no pre-existing assumptions about differences between females and males or their gametes (for a broader context and comparison to other functions, see Table 1 and function F7 in19). Any sex-specific differences arise only retrospectively after different gamete numbers are assigned to x and y of which either one could be male or female. The fertilisation function is (fleft({N}_{x},{N}_{y}right)={N}_{x}{N}_{y}frac{{e}^{a{N}_{x}}-{e}^{a{N}_{y}}}{{{N}_{x}e}^{a{N}_{x}}-{N}_{y}{e}^{a{N}_{y}}}), where a is a parameter controlling fertilisation efficiency (for the special case Nx = Ny the function is defined as (fleft({N}_{x},{N}_{y}right)=frac{a{N}_{x}^{2}}{1+a{N}_{x}})19,24, which is also the limit of f when Ny → Nx).In a monogamous resident pair, we have simply Nx = nx and Ny = ny. But if a mutant individual of type x is able to attract (hat{m}) fertilisation partners of type y, then for that individual ({N}_{y}=hat{m}{n}_{y}), and the corresponding Bateman function is$${b}_{x}left(hat{m}right)=fleft({N}_{x},{N}_{y}right)=fleft({n}_{x},hat{m}{n}_{y}right)$$
(1)
where the fertilisation function f is as described above. Because of symmetry, the corresponding function for y is found simply by swapping x and y. This function can reproduce the characteristic Bateman gradient asymmetry as gamete numbers diverge (progressing from isogamy to anisogamy in Fig. 1), showing how Bateman’s assertion follows from biophysical effects that arise from unequal numbers of fusing particles: the fertilisation function f is derived solely from such biophysical effects, not from any sex-specific assumptions. Equation (1) makes no reference to sexes, and they only become specified when values are assigned to nx and ny. For example, if nx = 10 and ny = 10,000, the female Bateman function is ({b}_{x}left(hat{m}right)) and the male Bateman function ({b}_{y}left(hat{m}right)), where for the latter all xs in Eq. (1) are replaced with ys and vice versa. The labels x and y are truly just labels. While there are inevitably assumptions built into the equations, crucially we can be certain there are no sex-specific assumptions. Yet the typical shapes reminiscent of Bateman gradients arise from the model when different values are specified for nx and ny (Fig. 1).Fig. 1: The Bateman function of Eq. (1).This figure shows how the basic Bateman gradient asymmetry arises from simple biophysics and mathematics of fertilisation. The population is monogamous aside from a mutant individual, whose number of fertilisation partners (‘matings’) varies on the horizontal axes within panels. a–d show the effect of variation in sex-specific gamete numbers under efficient fertilisation, while e–h show the effect of variation in sex-specific gamete numbers under inefficient fertilisation. Parameter values used are shown in the figure. Females (gamete number nx) are indicated by blue crosses and connecting lines, while males (gamete number ny) are indicated by black dots and connecting lines. Under isogamy, females and males are undefined, and the two colours overlap. The typical sex-specific shapes of Bateman gradients arise from a single equation (which itself is not sex-specific) when a difference in gamete numbers is assigned to nx and ny, confirming Bateman’s intuition that the primary cause of the difference in selection is that females produce fewer gametes than males. The entire range of gamete number ratios presented in the figure is observed in nature, from equal gamete size in many unicellular organisms39 to vertebrates, where sperm count per ejaculate can commonly exceed 109 (see ref. 40 and Supplementary Information therein).Full size imageGamete limitation changes the results quantitatively so that under conditions of poor fertilisation efficiency a larger imbalance in gamete numbers is needed for Bateman gradients to diverge to a similar extent. However, even under inefficient fertilisation, the Bateman gradients do not reverse.Model 2: An external fertiliser model with population-level polygamy and gamete competitionModel 1 presented the simplest possible scenario, where all individuals except a rare mutant mate only once, and gamete competition (sperm competition26, but without assigning either gamete type to be sperm) was thus excluded for the focal mutant individual. Now I generalise from this to a situation that remains entirely symmetrical, but where the resident number of matings can take on any value, and then derive the Bateman function for a rare mutant that deviates from this population-level value. This set-up allows for gamete competition for the focal mutant individual, a crucial addition because of the empirical and theoretical importance of sperm competition26, as well as earlier theory suggesting that polyandry decreases the sex difference in Bateman gradients2.The biological set-up is such that there is a large population and a large number of patches (fertilisation arenas) where multiple individuals of both sexes can release their gametes for fertilisation. After all individuals have released their gametes, those in each patch mix freely and fertilisations take place randomly. Set up in this way, the model is again identical from the perspective of both sexes, and gamete number can be isolated as the sole possible causal factor in any subsequent differences that may arise, extending from the initially monogamous and gamete competition-free scenario of Model 1. All individuals of both sexes are assumed to initially have the same strategy: to divide their nx or ny gametes equally between m patches, and distribute themselves in such a way that gametes from m individuals of each type release gametes into each patch (the number of individuals of each sex per patch need not necessarily be strictly equal to m, but this is the simplest assumption to account for the fact that gamete competition tends to increase with multiple ‘matings’). Now, if a rare x mutant divides its gametes evenly into (hat{m}) randomly selected patches, its gamete number per patch and consequently competitiveness in each patch is altered. Therefore, gametes of a mutant of type x will gain, on average, a fraction ({c}_{x}=left({n}_{x}/hat{m}right)/{N}_{x}) of the fertilisations in that patch, where ({N}_{x}={n}_{x}/hat{m}+(m-1){n}_{x}/m). To compute the number of realised fertilisations in a patch, I use the same fertilisation function as in Model 1, where the mutant number of gametes in a patch is Nx as above and the number of gametes of the opposite type is ({N}_{y}=mfrac{{n}_{y}}{m}={n}_{y}). All the components are now in place to write down the Bateman function corresponding to this scenario, for a mutant of type x:$${b}_{x}left(hat{m},mright)=hat{m}{c}_{x}fleft({N}_{x},{N}_{y}right)$$
(2)
where cx, Nx and Ny are as defined above, and the fertilisation function f is as in Model 1. For completeness, define bx(0, m) = 0, which is necessarily true, but useful to define separately because division by 0 renders Eq. (2) formally undefined when (hat{m}=0).As in Model 1, Eq. (2) makes no reference to sexes, and they only become specified when values are assigned to nx and ny (Fig. 2).Fig. 2: The Bateman function of Eq. (2) for an externally fertilising population with potential for population-wide polygamy and gamete competition.Results are shown for two values of resident matings (m = 1 and m = 2). a–h show the effect of variation in sex-specific gamete numbers and in fertilisation efficiency with m = 1, while i–p show the same with m = 2. Parameter values used are shown in the figure. The value m = 2 is used here because it is comparable to the mean number of matings in Bateman’s1 work (see Fig. 3 for corresponding results with internal fertilisation, but note that the aim of the models is not to quantitatively reproduce Bateman’s results). Females (gamete number nx) are indicated by blue crosses and connecting lines, while males (gamete number ny) are indicated by black dots and connecting lines. Under isogamy, females and males are undefined, and the two colours overlap. Further variation in m is examined in Fig. 4.Full size imageModel 3: An internal fertiliser modelModels 1–2 were set up with the central aim of full symmetry and exclusion of any sex-specific assumptions. Internal fertilisation breaks this symmetry by introducing a sex-specific assumption other than gamete number. Bateman gradients are, however, most commonly applied to situations with internal fertilisation where females are gamete recipients and males are gamete donors27. I therefore construct a model accounting for internal fertilisation. Where Eqs. (1) and (2) allowed no sex differences aside from gamete number, here I additionally consider the fact that females receive gametes while males donate them.As in model 2, there is a very large population, and I assume that in the resident population, all females and males mate exactly m times. It is then considered how a rare mutant individual’s (of either sex) fitness depends on its number of matings (hat{m}).I use the same fertilisation function as in Models 1-2. Consider first the female perspective (labelled with x). A female produces nx gametes and retains them internally. Each female mates with m males, who also mate with m females, dividing their gametes evenly over these matings. Therefore a mutant female receives (hat{m}frac{{n}_{y}}{m}) male gametes, and her reproductive success is$${b}_{x}left(hat{m},mright)=fleft({n}_{x},hat{m}frac{{n}_{y}}{m}right)$$
(3)
A mutant male, on the other hand, mates with (hat{m}) females, each of which mates with m−1 additional males. Therefore, the mutant male’s mating partners will receive a total of ({{N}_{y}=n}_{y}/hat{m}+(m-1){n}_{y}/{m}) male gametes. Thus, the mutant male gains a fraction ({c}_{y}=left({n}_{y}/hat{m}right)/{N}_{y}) of the fertilisations with each female, while the total reproductive success per female is f(nx,Ny). The mutant male’s reproductive success is therefore$${b}_{y}left(hat{m},mright)=hat{m}{c}_{y}fleft({n}_{x},{N}_{y}right)$$
(4)
To avoid division by 0, we can again define by (0, m) = 0, analogous to Model 2. In contrast to Models 1–2, there are now separate equations for each sex because of the additional sex-specific assumption of internal fertilisation, but no further sex-specific assumptions are used in their derivation. Visually the Bateman functions (Fig. 3) are nevertheless very similar to Model 2, and again reproduce the sex-specific shapes first proposed by Bateman1 when fertilisation is efficient. However, an interesting exception arises when relatively weak asymmetry in gamete numbers is combined with inefficient fertilisation and gamete limitation. When these conditions are combined with internal fertilisation, Bateman gradients can theoretically be reversed.Fig. 3: The Bateman functions of Eqs. (3) and (4) for internal fertilisation.Where Figs. 1 and 2 show that the sex-specific shapes of Bateman functions are ultimately caused by differences in gamete number, Fig. 3 shows that internal fertilisation does not invalidate this outcome when fertilisation is efficient. As in Fig. 2, results are shown for two values of resident matings (1 and 2), and the value m = 2 is used because it is comparable to the mean number of matings in Bateman’s1 work. a–h show the effect of variation in sex-specific gamete numbers and in fertilisation efficiency with m = 1, while i–p show the same with m = 2. Parameter values used are shown in the figure. Inefficient fertilisation combined with relatively low asymmetry in gamete numbers and the added asymmetry of internal fertilisation can in principle reverse the Bateman gradients (second and fourth row). Females (gamete number nx) are indicated by blue crosses and connecting lines, while males (gamete number ny) are indicated by black dots and connecting lines.Full size image More

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Geography and climatology of study areaThe area of study, Ghana, is on the coastal edge of tropical West African, bounded in latitude 4.5° N and 11.5° N and longitude 3.5° W and 1.5° E, and characterized by a tropical monsoon climate system23,24. Figure 1 shows map of the study area indicating the selected twenty two (22) sunshine measurement stations distributed across the four main climatological zones and Table 1 summarizes the geographical positions of selected stations.Figure 1Adapted from Asilevi27.Map of the study area showing all twenty two (22) synoptic stations distributed in four main climatological zones countrywide.Full size imageTable 1 Geographical position and elevation for study sites.Full size tableAtmospheric clarity over the area is closely connected to cloud amount distribution and rainfall activities, largely determined by the oscillatory migration of the Inter-Tropical Discontinuity (ITD), accounting for the West African Monsoon (WAM)25,26.Owing to the highly variable spatiotemporal distribution of cloud amount vis-à-vis rainfall activities, resulting in contrasting climatic conditions in different parts of the region, the country is partitioned by the Ghana Meteorological Agency (GMet) into four main agro-ecological zones namely, the Savannah, Transition, Forest and Coastal zones as shown in Fig. 123. As a result, the region experiences an estimated Global solar radiation (GSR) intensity peaks in April–May and then in October–November, with the highest monthly average of 22 MJm−2 day−1 over the savannah climatic zone and the lowest monthly average of 13 MJm−2 day−1 over the forest climatic zone27.Research datasetsGround-based measurement dataDaily sunshine duration measurement datasets (n) spanning 1983–2018 where derived for estimating Global solar radiation (GSR). The measurements were taken by the Campbell-Stokes sunshine recorder, mounted at the 22 stations shown in Fig. 1, under unshaded conditions to ensure optimum sunlight exposure. The device concentrates sunlight onto a thin strip of sunshine card, which causes a burnt line representing the total period in hours during which sunshine intensity exceeds 120.0 Wm−2 according to World Meteorological Organization (WMO) recommendations27. The as-received daily records were quality control checked by ensuring 0 ≤ n ≤ N, where N is the astronomical day length representing the possible maximum duration of sunshine in hours determined by Eq. 1 from the latitude (ϕ) of the site of interest and the solar declination (δ) computed by Eq. 227:$$ {text{N}} = frac{2}{15}cos^{ – 1} left[ { – tan phi tan {updelta }} right] $$
(1)
$$ {updelta } = 23.45sin left[ {360^{{text{o}}} times frac{{284 + {text{J}}}}{365}} right] $$
(2)
where J represents the number for the Julian day of the year (first January is 1 and second January is 2).NASA-POWER Global solar radiation (GSR) reanalysis dataThe satellite-based Global solar radiation (GSR) dataset for specific longitudes and latitudes of all 22 stations, assessed in the study, were retrieved from the National Aeronautics and Space Administration-Prediction of Worldwide Energy Resources (NASA-POWER) reanalysis repository based on the Modern Era Retrospective-Analysis for Research and Applications (MERRA-2) assimilation model products, developed from Surface Radiation Budget, and spanning equal study period (1983–2018). The datasets are accessible on a daily and monthly temporal resolution scales at 0.5° × 0.5° spatial coverage via a user friendly web-based mapping portal: https://power.larc.nasa.gov/data-access-viewer/17. The advantage of the NASA-POWER reanalysis GSR, is the wide spatial coverage, and thus can be used to develop a high spatial resolution of solar radiation across the study area.The POWER Project analyzes, synthesizes and makes available surface radiation related parameters on a global scale, primarily from the World Climate Research Programme (WCRP), Global Energy and Water cycle Experiment (GEWEX), Surface Radiation Budget (SRB) project (Version 2.9), the Clouds and the Earth’s Radiant Energy System (CERES), FLASHFlux (Fast Longwave and Shortwave Radiative Fluxes from CERES and MODIS), and the Global Modeling and Assimilation Office (GMAO)17. Table 2 shows the source satellites and the corresponding temporal coverage used in the development of NASA-POWER GSR products.Table 2 Satellites providing the NASA-POWER GSR datasets20.Full size tableThe monthly average NASA-POWER all-sky shortwave surface radiation reanalysis products are statistically validated, showing reasonable biases of − 6.6–13%, against a global network of surface radiation measurement metadata in an integrated database from the Baseline Surface Radiation Network (BSRN) of the World Radiation Monitoring Center (WRMC)20,22. The datasets are widely used in renewable energy application16,22, agricultural modelling of crop yields28, crop simulation exercises29, and plant disease modelling30.Furthermore, in order to assess the suitability of the NASA-POWER surface solar radiation products for the study area, a synthetic sunshine duration based Global solar radiation (GSR) is developed from the Angstrom-Prescott sunshine duration model by Eq. 3 for comparisons27.$$ {text{GSR}} = left[ {{text{a}} + {text{b}}frac{{text{n}}}{{text{N}}}} right]{text{H}}_{{text{o}}} $$
(3)
were Ho (kWhm−2 day−1) is the daily extraterrestrial solar radiation on an horizontal surface, n is the daily sunshine duration measurements obtained from the Ghana Meteorological Agency (GMet), and N is the maximum possible daily sunshine duration or the day length in hours determined by Eq. 1. Generalized regression constants a = 0.25 and b = 0.5 for the study area were determined by Asilevi27 from experimental radiometric data based on correlation regression analysis between atmospheric clarity index (GSR/Ho) and atmospheric cloudlessness index (n/N), for estimating solar radiation over the study area, and compared with other satellite data retrieved from the National Renewable Energy Laboratory (NREL) and the German Aerospace Centre (DLR)27. Ho was calculated from astronomical parameters by Eq. 4:$$ {text{H}}_{0} = frac{{24{ } cdot { }60}}{pi } cdot {text{G}}_{{{text{sc}}}} cdot {text{d}}_{{text{r}}} left[ {omega_{{text{s}}} sin varphi sin delta + cos varphi cos delta sin omega_{{text{s}}} } right] $$
(4)
where Gsc is the Solar constant in MJm−2 min−1, dr is the relative Earth–Sun distance in meters (m), (omega_{s}) is the sunset hour angle (angular distance between the meridian of the observer and the meridian whose plane contains the sun), (delta) is the angle of declination in degrees (°) and (varphi) is the local latitude. A detailed presentation of the calculation was published in a previous work27.Statistical assessment analysisFor the purpose of assessing the NASA-POWER derived monthly mean GSR (GSRn) datasets in comparison with the estimated Global Solar Radiation (GSRe) datasets used in this paper, the following deviation and correlation methods in Eqs. 5–11, each showing a complimentary result were used: Standard deviation (({upsigma })), residual error (RE), Root mean square error (RMSE), Mean bias error (MBE), Mean percentage error (MPE), Pearson’s correlation coefficient (r), and Willmott index of agreement (d) for n observations31,32,33,34,35. GSRe, GSRn, and RE represent the estimated GSR, NASA-POWER GSR, and the residual error between GSRe and GSRn respectively. A positive RE indicates that sunshine-based estimated GSR is larger than the NASA-POWER reanalysis dataset, while a negative RE indicates that sunshine-based estimated GSR is smaller than the NASA-POWER reanalysis dataset. The arithmetic mean of any dataset is µ.The standard deviation (({upsigma })) was used to check the upper and lower limits of distribution around the mean deviations between GSRe and GSRn in order to ascertain violations between both datasets33. The RMSE is a standard statistical metric to quantify error margins in meteorology and climate research studies, and by definition is always positive, representing zero in the ideal case, plus a smaller value signifying a good marginal deviation31. The MBE is a good indicator for under-or overestimation in observations, with MBE values closest to zero being desirable. The MPE further indicates the percentage deviation between the GSRe and GSRn individual datasets35.$$ {upsigma } = sqrt {frac{1}{{{text{n}} – 1}}mathop sum limits_{{{text{i}} = 1}}^{{text{n}}} left( {{text{GSR}} – {upmu }} right)^{2} } $$
(5)
$$ {text{RE}} = {text{GSR}}_{{text{e}}} – {text{GSR}}_{{text{n}}} $$
(6)
$$ {text{RMSE}} = sqrt {frac{1}{{text{n}}}mathop sum limits_{{{text{i}} = 1}}^{{text{n}}} left( {{text{RE}}} right)^{2} } $$
(7)
$$ {text{MBE}} = frac{1}{{text{n}}}mathop sum limits_{{{text{i}} = 1}}^{{text{n}}} left( {{text{RE}}} right) $$
(8)
$$ {text{MPE}} = frac{1}{{text{n}}}mathop sum limits_{{{text{i}} = 1}}^{{text{n}}} left( {frac{{{text{RE}}}}{{{text{GSR}}_{{text{e}}} }} times 100{text{% }}} right) $$
(9)
$$ {text{r}} = frac{{mathop sum nolimits_{{{text{i}} = 1}}^{{text{n}}} left( {{text{GSR}}_{{text{e}}} – {upsigma }_{{text{e}}} } right)left( {{text{GSR}}_{{text{n}}} – {upsigma }_{{text{n}}} } right)}}{{left( {{text{n}} – 1} right){upsigma }_{{text{e}}} {upsigma }_{{text{n}}} }} $$
(10)
$$ {text{d}} = 1 – left[ {frac{{mathop sum nolimits_{{{text{i}} = 1}}^{{text{n}}} left( {{text{GSR}}_{{text{e}}} – {text{GSR}}_{{text{n}}} } right)^{2} }}{{mathop sum nolimits_{{{text{i}} = 1}}^{{text{n}}} left( {left| {{text{GSR}}_{{text{e}}} – {text{GSR}}_{{{text{nave}}}} left| + right|{text{GSR}}_{{text{n}}} – {text{GSR}}_{{{text{nave}}}} } right|} right)^{2} }}} right] $$
(11)
Further, as with other statistical studies in meteorology36, the Pearson’s correlation coefficient (r) was used to quantify the strength of correlation between GSRe and GSRn. Finally, the Willmott index of agreement (d) commonly used in meteorological literature computed from Eq. 7 is used to assess the degree of GSRe/GSRn agreement34. More