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Data collectionBased on the cropland extent, we first introduced a cropland distribution template, the Self-adapting Statistics Allocation Model of Global Cropland (SASAM-GC)16, as shown in Fig. S1. The global cropland extent map used herein was a global cropland synergy map with a 500-m spatial resolution representing approximately the year 2010, developed by the Smart Agriculture Innovation Team of the Key Laboratory of Agricultural Remote Sensing (AGRIRS) of the Chinese Academy of Agricultural Sciences in cooperation with the International Food Policy Research Institute (IFPRI) and the International Institute of Applied Systems Analysis (IIASA). The overall accuracy of the SASAM-GC products is 90.8%, which is higher than those of existing global farmland products such as the Climate Change Initiative Land Cover (CCI-LC), GlobeLand30, and Medium Resolution Imaging Spectrometer (MODIS) products.Vegetation indices are often used to depict crop growth, such as the ratio vegetation index (RVI)17, normalized difference vegetation index (NDVI), and enhanced vegetation index (EVI)18. Among them, the EVI is the most sensitive to high-biomass regions and less susceptible to atmospheric and soil interference19,20. MODIS vegetation index datasets are generated every 8 days or 16 days at spatial resolutions of 250 m, 500 m, and 1000 m. The 250-m spatial resolution is considered the best resolution for detecting crops21,22. Here, we used EVI time series with a spatial resolution of 250 m reported every 16 days in the MODIS product MOD13Q1 as the primary data to calculate the global cropping intensity; these data can be accessed at https://lpdaac.usgs.gov/products/mod13q1v006/.$${rm{EVI}}=2.5times frac{{{rm{rho }}}_{{rm{NIR}}}-{{rm{rho }}}_{{rm{Red}}}}{{{rm{rho }}}_{{rm{NIR}}}+6times {{rm{rho }}}_{{rm{Red}}}-7.5times {{rm{rho }}}_{{rm{Blue}}}+1}$$
(1)
In formula 1, ρNIR, ρRed, and ρBlue represent the reflectivity of the near-infrared band, the red band, and the blue band, respectively.The Food and Agricultural Organization of the United Nations statistical data (FAOSTAT) provides long-time-series cropland-related statistical data at the country level and can be accessed at http://www.fao.org/faostat/en/#data. FAOSTAT cropland data have been widely used in a variety of studies evaluating food security and hindcasting historical land-use changes23,24. Here, we defined the cropland area as the sum of areas characterizing arable land (land under temporary crops, temporary meadows used for mowing or pasture, market and kitchen gardens, and land that is temporarily fallow; abandoned land resulting from shifting cultivation was excluded) and permanent croplands (land cultivated with long-term crops that do not have to be replanted for several years). Additionally, the harvested area is used; this value refers to the area from which a crop is gathered. If the crop under consideration is harvested more than once during a year due to successive cropping (i.e., the same crop is sown or planted more than once in the same field during the same year), the area is counted as many times as the crop is harvested. Therefore, the sown area is recorded only for the crop reported. The statistical cropping intensity is defined as the harvested area divided by the cropland area and is used to validate the consistency between the cropping intensity obtained herein and that reported in the FAOSTAT product at the country level.Brief algorithmIn this study, a sixth-order polynomial function was used to reconstruct EVI time series for brief and rapid calculations at the global scale25. As the global cropping intensity ranges from 0 to 3 and the sixth-order polynomial function can have 3 maxima at most, we chose the sixth-order polynomial function (formula 2) in this study:$${rm{EVI}}({rm{t}})={{rm{a}}}_{0}+{{rm{a}}}_{1}{rm{t}}+{{rm{a}}}_{2}{{rm{t}}}^{2}+cdots +{{rm{a}}}_{6}{{rm{t}}}^{6}$$
(2)
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The arthritic degeneration visualized in the pathological Smilodon specimens could have arisen from one of three etiologies: traumatic, infective or degenerative arthritis. Findings on the specimens make infective or traumatic arthritis less likely. In the case of infective arthritis, the presupposition is that the animal developed typically before an insult that led to infection and subsequent obliteration of the hip joint. This assumption also holds true for the case of traumatic arthritis following an injury or fracture. However, the anatomical distortions of the right femoral head, in conjunction with the obliteration of the right acetabulum, suggest chronic changes that led to degeneration over time (Figs. 3, 4). The degeneration of the femoral head would not be expected if the degenerative change in the hip joint were due to infection or trauma, as the development of the pelvis and femur presumably would have been complete before the insult or injury occurred during the adult cat’s life.The condition of the right acetabulum and right femoral head demonstrates anatomy consistent with developmental distortion. Typically, the head of the femur develops in conjunction with the acetabulum of the pelvis16. The spherical femoral head fits into the concentric-shaped acetabulum to form a ball-and-socket joint that enables a four-legged mammal to ambulate, lie down, sit down, stand up, and generally function normally16. In developmental hip dysplasia, however, the acetabulum does not develop appropriately, and the articulation between the femoral head and acetabulum is lost. An elliptical (as opposed to concentric-shaped) acetabulum causes progressive subluxation (dislocation) of the femoral head17, which can result in coxa plana, or necrosis of the bony nucleus of the femoral head16. This subsequent coxa plana produces flattening and degeneration of the normally spherical femoral head18.Proper anatomical development and ossification of the hip joint rely on continuous and symmetrical pressure of the femoral head on the acetabulum, and dysplasia results from improper positioning of the femoral head within the acetabulum16. Dysplastic hips are characterized by pathological restructuring and accelerated remodeling of the joint in response to abnormal forces and tensions that create stress. This produces formation of new bone in some areas and resorption of bone in others, ultimately causing degenerative joint disease16. Dysplastic hips have varying degrees of deformity and malformation, but typically the acetabula are hypoplastic and deficient in various planes and dimensions (Supplementary Fig. S10).Further inspection of LACMHC 131 demonstrates anatomical changes consistent with chronic degeneration throughout the right hip joint and pelvis. The obliteration of the right innominate likely occurred over many years and progressively resulted in significant bony destruction and remodeling. These findings of a flattened femoral head in LACMHC 6963 in conjunction with a shallow acetabulum in LACMHC 131 are consistent with changes observed with mechanical instability of the hip joint and bony destruction secondary to dysplasia. Repeated subluxation events due to the dysplastic hip likely accelerated the destruction of the cartilage and joint, altering the biomechanical stresses through the joint. This increased stress along with cartilage loss likely led to a progressively hypertrophic and aberrant bone response with subchondral sclerosis and osteophyte formation in the acetabulum and pelvis. The external, anatomical deformities in these specimens are consistent with changes that have occurred over the animal’s lifespan and subsequently resulted in the gross morphology observed, with destruction of the hip joint on both the acetabular and femoral side.This type of pathology starts to impact movement at the time of first walking, although minimal pain tends to ensue at this time because of the animal’s flexibility at its early age19,20. As the joint cartilage wears out, however, bone begins to rub on bone. The resulting forces make the bone stiffer, producing osteophytes or bone spurs as well as sclerosis that manifests on CT imaging as increased bone density (Figs. 3 and 4; Supplementary Video S2; Supplementary Data S1–S2). At this point, loading the limb would cause pain, and range of motion would be limited. Therefore, the animal examined in this study would have spent as little time as possible on its right hindlimb, needing to compensate for the handicap by increasing the load on its left hindlimb. This compensation would explain the exostoses on the left ilium anterodorsal to the non-pathological acetabulum (Fig. 1; Supplementary File S1; Supplementary Video S1), indicating abnormal pulling of the quadriceps femoris muscles originating in this area.Hip dysplasia is a heritable, polygenic condition that affects a range of mammal species16, including humans17. Canine hip dysplasia (CHD) is one of the most prevalent orthopedic diseases in domestic dogs (Canis lupus familiaris)21 and is very well studied, in part because it is similar to developmental dysplasia of the human hip22. Feline hip dysplasia (FHD) has received less clinical attention than CHD, possibly because its functional impairment is less overt or because domestic cats (Felis catus) are able to compensate for the resulting lameness better than dogs20,23. The overall results of physiological changes from dysplasia are mechanical imbalance and instability in the hip joint causing displacement due to opposing forces from the acetabulum and femoral head, and osteophytes in the acetabulum to compensate for cartilage loss16.Embryologically, articular joints differentiate from skeletal mesenchyme in situ with the support of surrounding tissues that sustain mechanical and physiological forces that tend to pull on the joints16. In dogs, hip joints are normal at birth, as teratological factors and mechanical stresses that could displace the femoral head are rare at this time16. Epiphyseal ossification normally begins by 12 days of age; in dogs that eventually develop CHD, anatomical changes of the femoral head and pelvic socket begin before week three24. In dysplastic hips, the teres ligament, which is crucial for holding the femoral head in place, is too short; this produces luxation, or dislocation, of the top of the femoral head, beginning at around seven weeks16. This luxation increases throughout development, degrading the articular cartilage that surrounds the femoral head, delaying ossification of the femur and acetabulum16, and shortening the affected limb, as the femoral head becomes positioned higher in the acetabulum.In clinical reports of hip dysplasia in domestic cats, osteoarthritis (i.e., degenerative joint disease, DJD) of the hip secondary to FHD is well known19. For example, osteoarthritis was recorded in 43 of 45 (95.6%) of cats with FHD25. As well, in 5 of 13 (38.5%) cases of hip osteoarthritis with a radiographically or historically identifiable cause, hip dysplasia was pinpointed as the cause, with the remaining cases resulting from trauma or equivocal between trauma and dysplasia23. A recent study of FHD in Maine Coons—a large-bodied domestic cat breed in which hip dysplasia is known to be common—calculated a prevalence of 37.4%, finding severity to increase with age and body mass26. The same study further highlighted a genetic correlation between FHD and large body size within the Maine Coon26, inviting inquiry into how FHD impacts other breeds and non-domestic felid species across a range of body sizes.Reports of FHD in non-domestic large cats are rarer than in domestic cats. Captive snow leopards have exhibited hip dysplasia; across 14 zoos, seven cases were classified as moderate to severe, and at least two individual snow leopards needed total hip replacement before being able to breed27,28. Accounts of hip functional impairment in other captive large cats have tended to report osteoarthritis, which can be associated with FHD though may also stem from trauma and increased age29,30,31.For wild-caught large cats, the only comprehensive study of which we are aware is a survey of 386 individuals (283 wild-caught) across three felid genera mounted as exhibit skeletons in multiple North American natural history museums30. Though not focusing on hip dysplasia, the study tracked degenerative joint disease, which may be associated with dysplasia23,25. The sample recorded DJD in 9.7% of 31 tigers, 2.3% of 88 African lions, and 5.1% of 59 mountain lions (Puma concolor), and none in five other species of big cat. These frequencies are low compared to domestic cats, perhaps owing to differences in body size, diet, and lifestyle between large wild cats and domestic cats, as well as selective breeding constraining genetic variation in domestic animals. Furthermore, selection against hip dysplasia would be expected in the wild because hip dysplasia would compromise hunting19. Though this study identified instances of non-inflammatory osteoarthritis in the shoulder, elbow, and stifle joint, it found none in the hip. However, 4% of all joints afflicted by spondyloarthropathy—a form of inflammatory arthritis—included the hip30.What is the significance of Smilodon, an extinct Pleistocene predator, having the same congenital defect as living domestic cats and dogs? Previous workers have inferred social behavior from paleopathologies in fossil carnivorans ranging from the extinct Eurasian steppe lion7 to Pleistocene wolf-like canines from Italy8 and China9, interpreting signs of healing as evidence of survival after injury12. Given the severity of many injuries, authors have argued, the animal would have starved to death had it not operated within a social structure. The present hip dysplasia having manifested from a young age—hindering this animal’s ability to hunt prey and defend a home range over the course of its life—supports this assertion, although other inferences are possible.Sociality, the degree to which individuals live with conspecifics in groups32, is difficult to infer in Smilodon given that it has no living analogues or closely related taxa. Estimated to have weighed between 160 and 350 kg (3,14, this study), Smilodon was at least the size of the Amur tiger (Panthera tigris altaica), the largest living cat; some estimates reach 369 to 469 kg, placing Smilodon in the range of the largest extant ursids15,33. No living felid has Smilodon’s elongate, knife-like canines or stocky, powerful build. As well, Smilodon (of the extinct felid lineage Machairodontinae) is only distantly related to extant large felids (Felinae), introducing further uncertainty. Based on its robust morphology (e.g.,34,35) and on evidence from stable isotopes (e.g.,4), it likely stalked and ambushed prey; therefore, it may have been comparable to the African lion (Panthera leo), which has a similar hunting strategy and is the only truly social extant felid36. Yet sociality varies across felid species, including within a genus; for example, other extant pantherines like tigers (P. tigris) show incipient sociality37, while jaguars (P. onca) are solitary except for females with cubs. Social strategies also can vary within species, e.g., between sexes. For instance, African lion females are philopatric and social throughout their lives, while adult males are often nomadic and solitary until joining a gregarious pride, which itself usually lasts for only a few years38. This social variation complicates behavioral inferences based on ancestral reconstructions.Advocates of the solitary-cat hypothesis39,40 have cited Smilodon’s small relative brain size determined using endocranial casts as support for solitary behavior, because sociality exerts high cognitive demands. However, in 39 species across nine carnivoran families, larger relative brain size was found to correlate with problem-solving capabilities rather than social behavior41. Rather than analyses of overall encephalization across carnivoran families, studies of relative regional brain volume within families and species have been more informative regarding sociality42,43. In both African lions and cougars (Puma concolor, a solitary species), total relative endocranial volume was not sexually dimorphic; however, relative anterior cerebrum volume was significantly greater in female African lions than males, a difference absent in cougars38.Though regional endocranial studies have yet to be performed on Smilodon, the gregarious-cat hypothesis has drawn support from multiple lines of evidence. One is the abundance of Smilodon relative to prey at RLB10,11,34, although detractors have pointed out that some extant large cats aggregate at carcasses despite otherwise being solitary40. A full range of ages is present among RLB Smilodon; in contrast, animals interpreted to be solitary, such as the American lion Panthera atrox, are represented largely by adult individuals44. As well, the proportions of social and solitary species at RLB parallel those drawn to audio recordings of herbivore distress calls in the African savanna, suggesting that RLB Smilodon sample sizes are more consistent with it having been social rather than solitary45,46. The lack of size sexual dimorphism in Smilodon is more typical of modern solitary cats47 but could also be reflective of monogamy within a gregarious species, like modern wolves. Most relevant to the current study, the existence of healed injuries in Smilodon also has been interpreted as evidence for social behavior, with the assumption that surviving long after serious injury would be difficult if not impossible without cooperative sociality12. We now revisit this interpretation considering the novel diagnosis of hip dysplasia in this study.Smilodon’s large body size necessitated preying on megaherbivores for adequate sustenance3. To do so, like most large cats today, it would have used its hindlimbs for propulsion and acceleration48,49, a pounce behavior enabled by its morphology. Smilodon’s ratio of total forelimb to hindlimb length is greater while its ratio of tibia to femur length ranks lower than in living felids34. The shorter hindlimbs lacking the distal limb elongation in cursorial animals suggest that Smilodon was an ambush predator surpassing the ability of felids today50. Hunting large prey is dangerous51. After the initial hindlimb-powered leap, Smilodon would have grappled with its struggling prey, as evidenced by traumatic injuries in the rotator cuff and radiating from the ventral midline dorsolaterally to where the ribs articulate with the spine5. As it subdued prey with robust forelimbs35,48 under enough torque to injure the lumbar vertebrae5, Smilodon would have needed to leverage itself against the ground using its hindlimbs. Therefore, the pelvis and femur would have been critical to multiple phases of its hunting strategy.A dysplastic individual would have encountered much difficulty hunting in this manner. Yet, as evidenced by the complete fusion of its pelvic and femoral epiphyses (Figs. 1, 2) as well as its large body size (Figs. 6, 7), the individual in this study had reached adult age. (Studies of the detailed timing of epiphyseal fusion in large wild cats are lacking, but distal femoral epiphyses fuse at around the same time as or soon after proximal femoral epiphyses in domestic cats and dogs52,53. Given this, the broken distal femur likely had a fused epiphysis, as on its intact proximal end.) Limbs in African lions completely fuse between 4.5 and 5.5 years54,55,56, so it is reasonable to assume that adulthood in Smilodon likely started at around four years old. This estimate is reinforced by bone histological work quantifying at least four to seven lines of arrested growth (LAGs; one per growth year) in limb bones with fused epiphyses belonging to Smilodon fatalis from the Talara asphaltic deposits in Peru57. Some LAGs in the Talara histological specimens likely have been masked by secondary bone remodeling, which may be more extensive in larger-bodied taxa57, making these specimens possibly older than the number of visible LAGs suggest. Therefore, four years represents a likely minimum age for this individual, although it could have been much older.Ontogenetic growth patterns in teeth and bone further support inferences of sociality. In Smilodon, teeth appear to mature earlier than when sutures and long-bone epiphyses fuse, suggesting delayed weaning, prolonged juvenile dependence, and extended familial care until the adult hunting morphology—saber canines and robust limbs—was complete47. At RLB, most sampled Smilodon specimens show significant pulp cavity closure of the lower canine (14 of 19 specimens over approximately 80% closure), a sign of dental maturation58. This contrasts with RLB pantherine pulp cavities, which are more evenly distributed across the closure percentage range, suggesting that teeth mature earlier in Smilodon than in pantherines. (Other age assessments have ruled out the possibility that Smilodon juveniles were underrepresented relative to pantherines44). At Talara, age determination by dentition yields low estimates of juveniles (zero based on skulls; 8% based on dentaries), but age determination based on limb epiphyseal fusion yields higher estimates (41% juveniles)57. Histology of Talara Smilodon long bones reinforces this mismatch, as an apparent adult femur with fused epiphyses and seven LAGs was found to lack avascular and acellular subperiosteal lamellar bone57, suggesting that it had not yet finished growing. Further, prolonged parental care was interpreted in a recent description, from Pleistocene deposits in Corralito, Ecuador, of two subadult Smilodon fatalis individuals inferred to have been siblings and associated with an adult that was likely their mother59. This scenario of prolonged parental care, like that in the social African lion, would help explain how the individual in this current study survived to adulthood given its debilitating handicap.Novel application of CT visualization to an old question of paleopathology has enabled diagnosis of hip dysplasia, a lifelong condition, in an individual Smilodon fatalis saber-toothed cat. This individual was likely not the only Smilodon afflicted with hip dysplasia: multiple RLB Smilodon pelvic specimens, especially that described by Shermis11, exhibit gross morphology similar to the pathological pelvis examined in this study (Supplementary Figs. S6–S9). The individual examined in this study reached adulthood (at least four to seven years of age) but could never have hunted nor defended territory on its own, given its locomotor impairment that would have been present since infancy. As such, this individual likely survived to adulthood by association with a social group that assisted it with feeding and protection.Further conclusions are limited by the lack of a comprehensive and systematic comparative dataset comprising pathological post-crania from extant species, a persistent limitation of paleopathological studies5. Natural history museums may acquire cranial remains from zoos or similar institutions but often lack storage to accommodate postcranial skeletons, especially for large mammals. As well, while radiographic studies on domestic cats and dogs illustrate the nature of hip dysplasia, these studies tend to examine pathological bones in situ, still embedded in a muscular framework (e.g., Supplementary Fig. S10). This is opposed to the bones-only, flesh-free context of paleopathological specimens. Computed tomography and digital data may be key to building a comparative paleopathology dataset in the future.Within the scope of this study, we cannot rule out the hypothesis that the pathological animal was a scavenger and may have obtained food outside the context of a social structure. It is also possible that, regardless of its disability, its large size and fearsome canines made it a strong interference competitor. However, the pathological specimens examined here are consistent with the predominance of studies supporting a spectrum of social strategies in this extinct predator. In many extant carnivorans, sociality offers the benefits of cooperative hunting and rearing of young (e.g.,60): benefits that likely also applied to Smilodon in the late Pleistocene. As Smilodon coexisted with a rich megafaunal carnivore community including dire wolves (Aenocyon dirus), American lions (Panthera atrox), and short-faced bears (Arctodus simus), cooperative sociality may have aided its success as a predator in a crowded field. More

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To capture the first-order characteristics of the distribution of eDNA shed by mesopelagic species, we used a mechanistic model with one spatial dimension (the vertical dimension). It treats migrating organisms as a continuous point source of eDNA and simulates temporal evolution of the eDNA concentration vertical profiles in a one-dimensional ocean with climatological conditions. Horizontal variation of the distribution of eDNA shed by mesopelagic species is not considered in this study. The point source in the model represents a number of individuals within a single species. The simulation includes relevant eDNA transport processes using a vertical Price-Weller-Pinkel (PWP) dynamical module42. We use this model to first explore the sensitivity of eDNA concentration profiles to a variety of biological and physical parameters and to assess vertical and temporal variability of the eDNA concentration. Table 1 outlines the parameters, separated by type, and ranges of values for each parameter used in the sensitivity analysis (see “Sensitivity analysis” section below) and in a case study of how the model can be used to determine the percent of individuals that migrate (see “Application to ecological questions” section below). We then discuss how the mechanistic model can be used in conjunction with field sampling to test ecological hypotheses, highlighting how analysis of eDNA concentration profiles can provide information regarding where and when organisms are located in the water column. Finally, we comment on implications of this work and how the model can be used to both design and interpret observations.Table 1 Parameters separated by category.Full size tableModel overviewGoverning equationsTo maintain simplicity and also provide a more realistic scenario where both settling and neutrally buoyant eDNA particles28 are considered, we assume that the eDNA material at any given time in the system consist of eDNA particles of two size classes. Here, large eDNA particles are on the order of 10 s of µm to 1 mm and represent materials such as fecal pellets, chunks of tissue, or gametes that would be subject to both settling and physical breakdown over time. Small eDNA particles are on the order of 0.1 µm to 10 s of µm and would be any eDNA shed from an organism that has a density such that they are near-neutrally buoyant including extracellular eDNA or forms such as sperm, urine, blood, or single cells. This size class of small particles is so small that we can neglect their settling. In practice, these two size fractions of eDNA would be captured using the common method of filtering water through a < 1 µm pore size filter43. Both small and large particles in the model are subject to the same eDNA decay rate constant and large particles break down over time into small particles.The equation describing the change of eDNA concentration in the form of large particles is:$$begin{aligned} frac{partial {C_{LP}}}{partial {t}} = - w_vfrac{partial {C_{LP}}}{partial {z}} + frac{partial {}}{partial {z}}left( kappa _zfrac{partial {C_{LP}}}{partial {z}}right) - kC_{LP} - delta C_{LP} + w_sfrac{partial {C_{LP}}}{partial {z}} + hat{S_{LP}} end{aligned}$$ (1) where (C_{LP}) is the concentration of large eDNA particles, (w_v) is the vertical velocity [L/T] with a positive value pointing upward, (kappa _z) is the vertical diffusivity [L(^{2})/T], k is the first order decay rate constant [1/T], (delta) is the breakdown rate of particles [1/T], (w_s) is the settling rate of eDNA [L/T], and (hat{S_{LP}}) is the shedding rate of new large eDNA particles.Similarly, the equation describing the temporal evolution of the concentration of small particles of eDNA, (C_{SP}), is:$$begin{aligned} frac{partial {C_{SP}}}{partial {t}} = - w_vfrac{partial {C_{SP}}}{partial {z}} + frac{partial {}}{partial {z}}left( kappa _zfrac{partial {C_{SP}}}{partial {z}}right) - kC_{SP} + delta C_{LP} + hat{S_{SP}} end{aligned}$$ (2) where the parameters are the same as Eq. (1). Note that there is no settling term in Eq. (2) as small eDNA particles are assumed to be neutrally buoyant and that the (delta) term in Eq. (2) has the opposite sign as that in Eq. (1), reflecting eDNA volume conservation during the breaking-down of the large particles into small particles.Physical modelThe 1-D vertical model is implemented in MATLAB. The physical module is based on a previously published 1-D model of physical processes in the mixed layer42 and is modified to include the relevant organism and eDNA parameters. The model spans from the surface down to 1500 m deep with a vertical resolution of 0.5 m and has a time step of 10 s. Each simulation runs for a total of 90 days, representing one of the four seasons: summer (July, August, September - JAS), fall (October, November, December - OND), winter (January, February, March - JFM), and spring (April, May, June - AMJ). The model has a zero-gradient open boundary condition at the bottom. The model is forced on the surface by climatological, 8-hourly meteorological conditions from the NCEP/NCAR Reanalysis at a location in the Northeast Atlantic Ocean (37.047(^circ) N, -71.25(^circ) W). Inputs include air temperature, short wave radiation, long wave radiation, precipitation, air pressure, air humidity, and winds. The wind stress and heat fluxes are calculated using the bulk formulae44.The initial vertical distribution of the temperature and salinity in the model are seasonal climatological profiles obtained from the NCEI World Ocean Atlas 2018 at a site in the Northwest Atlantic slope sea (39.125(^circ) N, −70.875(^circ) W; north of the Gulf Stream). The model also simulates the influences of vertical mixing and advection on the temperature, salinity and eDNA particles. The mixing influence is incorporated by prescribing synthetic vertical profiles of vertical diffusivity that combines observed seasonal mean mixed layer depth45 and simulated seasonal mean vertical diffusivity profile from an operational model46, both at a site in the Northwest Atlantic slope sea (see Supplemental Text S1 and Supplemental Fig. S1). Vertical advection profiles are set to increase linearly from 0 ms(^{-1}) at the surface to the maximum value of ((10^{-4}) ms(^{-1})) at 200 m and then decrease linearly back to 0 ms(^{-1}) at 400 m (Supplemental Fig. S2). This prescribed advection profile does not change seasonally and represents enhanced vertical motions associated with sub-mesoscale processes in the upper water column (e.g.,47). Note that the result of this study is not sensitive to the prescribed vertical profiles of diffusivity or vertical velocity (see below).The physical parameters that will be adjusted in this study include: the initial temperature and salinity profiles, the mixed layer depth, the vertical diffusivity profile, and the vertical velocity profile (see Ocean parameters in Table 1).Organism movementThis study focuses on eDNA shed by vertically migrating organisms living in the mesopelagic ocean. For simplicity, we assume the organisms reside at the constant depth of 500 m during the day and the constant depth of 50 m during the night. These depths fall within the range of where migrating organisms are found at these times41,48. As the objective of this study is to develop a qualitative understanding of the vertical distribution of the eDNA concentration, the exact depth a particular mesopelagic species resides at in the daytime is not crucial. Organisms begin migrating to the surface two hours before sunset, and the upward migration ends one hour after sunset. The time of sunset and sunrise are determined seasonally using NOAA’s ESL solar calculator for the year 2019 at 42.35(^circ) N, −71.05(^circ) W. Both upward and downward migrations are assumed to be in constant speed, and the migration times do not change within each season (Fig. 2).Figure 2Representative organism migration curves for each season. Migration times change with sunrise and sunset for each season. This illustration assumes that 50% of individuals migrate. The shading shows the percentage of individuals with black indicating 100%, i.e., all organisms, and grey indicating 50%. These numbers vary among the simulations.Full size imageThe migration parameters that can be adjusted in the model include: daytime residing depth, nighttime residing depth, the start and end times of migrations (and thus duration), the percent of individuals that migrate, and the layer thickness (i.e., “width” of school). (See Organism parameters in Table 1). In this study, we focus on the start and end times of migrations (and thus duration) and the percent of individuals that migrate.eDNA parametersWe assume organisms are continuously shedding eDNA, and the shedding rate is fixed in time in each simulation. The eDNA then remains in the water column, where it is subject to transport and decay as described by Eqs. (1) and (2). Although several studies have characterized eDNA shedding rates of different marine organisms29,49,50, shedding rates are highly variable across studies and have high error associated with them, likely due to high temporal variability29. Here, we use a constant shedding rate of 1 mass unit per time step (10 seconds). Note that this study focuses on the relative vertical distribution of the eDNA concentration and its temporal variability, rather than its absolute value. The particle size distribution of eDNA (and the breakdown rate of large to small particles) and the settling rate of eDNA are largely unknown and expected to be time-varying28. Here we use a breakdown rate of fecal pellets to apply to large eDNA particles breaking down into small eDNA particles51. The most well characterized parameter of eDNA is decay rate, which several studies have characterized as a function of water temperature29,49. The upper and lower limits of the decay rate have also been well established29,31. Finally, there are currently no estimates of eDNA settling rates. We use values of marine snow settling rates52 to apply to large eDNA particles that are subject to settling in the model.The eDNA-related parameters that are adjusted in this mechanistic model include: settling rate of large particles; ratio of large particles to small particles that are shed by an organism; eDNA breakdown rate (large particles to small particles); the eDNA decay rate. (See eDNA parameters in Table 1). Note that shedding rates in the model do not change over the course of each simulation.Sensitivity analysisA series of 90-day simulations were carried out with altered values of the aforementioned parameters to examine the sensitivity of the eDNA vertical distribution to the parameters. Table 1 provides a list of the parameters that were adjusted in the sensitivity analysis. Note that the diffusivity profile (both the shape of the profile and the mixed layer depth), the temperature profile (which impacts the eDNA decay rate), and the daytime length (which impacted migration times) all change with the season. Other sensitivity parameters include vertical advection, settling rate of large particles, the ratio of large to small eDNA particles shed by organisms, and the percent of individuals that migrate. A total of 972 simulations were conducted with the following sets of parameters: 4 mixing profiles (one for each season), 3 vertical advection profiles (upwelling, downwelling, none), 3 settling rates of large eDNA particles, 3 decay rate constant scenarios (two constant values representing high and low values in the literature31, one temperature-dependent rate29 for each season, Supplemental Fig. S3), 3 scenarios for the percent of individuals that migrate, and 3 ratios of large to small eDNA particles shed by the organisms.In order to assess the impact of the parameters on the eDNA profiles, several metrics were defined. First, the water column was divided into three sections: surface layer (0–100 m), mid-depth (100–450 m), and deep water (450–550 m), based on our representative daytime and nighttime depths (500 m and 50 m) used for the simulations. For each simulation, the mean and maximum eDNA concentration in each depth bin was recorded. Also, the cumulative eDNA concentration in each depth bin was normalized to calculate the proportion of eDNA in each depth bin at any given time during the simulation. For each simulation, the mean and standard deviation of the proportion of eDNA in each depth bin were calculated over the course of the 90 day simulation.A first-order comparison of the vertical length scales of eDNA transport by different processes were conducted. Here, we use a time scale of eDNA decay, T(_{90}), i.e., the time it takes for 90% of the released eDNA to decay, to determine the vertical length scale of each process. In particular, we would like to estimate, the vertical distances eDNA is transported by advection, mixing, and settling before 90% of the released eDNA has decayed, L(_{mix}), L(_{advect}) and L(_{settling}), respectively. These quantities are defined mathematically as,$$begin{aligned} T_{90} = frac{-ln(0.1)}{k_{avg}} end{aligned}$$ (3) $$begin{aligned} L_{mix} = sqrt{kappa _v T_{90}} end{aligned}$$ (4) $$begin{aligned} L_{advect} = w_{vm} T_{90} end{aligned}$$ (5) $$begin{aligned} L_{settle} = w_s T_{90} end{aligned}$$ (6) where (k_{avg}) is the average decay rate constant [T(^{-1})] for the whole water column over the 90 day simulation, (kappa _v) is the maximum vertical diffusivity coefficient [L(^{2})T(^{-1})] , (w_{vm}) is the maximum vertical velocity [LT(^{-1})] , and (w_s) is the settling rate [LT(^{-1})]. Note that because k is temperature dependent, the value of the decay rate constant will change both vertically (e.g., deeper water will be colder and have a lower decay rate constant) and temporally (e.g., at a given depth, the water temperature will be warmer in the middle of the day and thus will have a higher decay rate constant).Application to ecological questionsAfter obtaining a first-order understanding of the eDNA concentration profile and the impact of the parameters on the eDNA distribution, we ran another set of model simulations to examine if vertical and temporal eDNA concentration variability can shed light on an ecological question regarding what percentage of individuals within the same species migrate on a daily basis. To do this, for each season, all parameters were held constant except for the percent of individuals that migrate, P(_{m}). In each season, the value of P(_{m}) varies from 0 to 100% with an increment of 10%. A total of 44 such simulations (4 seasons x 11 percent migrate scenarios) were carried out. We then compared mean and maximum eDNA concentrations in the surface layer (0–100 m) to those in the deep water (450 to 550 m), and examined the relationship between the surface-to-deep ratios and P(_{m}). More