Philippa Kaur

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In June 2018, 180 cars fanned out across Denmark and parts of Germany on a grand insect hunt. Armed with white, funnel-shaped nets mounted on their car roofs, enthusiastic citizen naturalists roamed through cities, farmlands, grasslands, wetlands and forests. The drivers sent the haul from their ‘InsectMobiles’ to scientists at the National History Museum of Denmark in Copenhagen and the German Centre for Integrative Biodiversity Research in Leipzig.The researchers dried and weighed the collections to determine the total mass of flying insects in each landscape. They expected some bad news. The previous year, scientists in Germany had found that the flying-insect biomass in their nature reserves had plunged by 76% over 27 years1. Similar studies had led to news headlines that screamed of an ongoing “insectageddon” and “insect apocalypse”. British columnist George Monbiot wrote in The Guardian: “Insectageddon: farming is more catastrophic than climate breakdown”.But when the researchers tallied the InsectMobile results2, they didn’t see evidence of declines everywhere. Insect biomass totals were higher than expected in agricultural fields, and indeed in all places except cities in their study, which is yet to be peer reviewed2. Aletta Bonn, an entomologist at the Leipzig centre and a co-author of the study, says this could be because the fertilizers that farmers use are leading to lush plant life, which is reverberating through the ecosystem. That said, she cautions, not every insect species in the study area might be doing well; some could be thriving, others not so much.“We do need to understand better what kind of insects are affected and to which degree,” Bonn says. “I think the generalization that all agriculture is bad — I wouldn’t say so.”The findings resonate with what biologist Mark Vellend and his colleagues have seen in their studies of trees at the edge of boreal forests in eastern Canada. They’ve found that spruce, eastern white cedar, eastern hemlock and American beech have been struggling to maintain their roothold since European and American settlers began clearing land more than a century ago. But poplar, paper birch, maple and balsam fir are thriving3. Vellend, who teaches at the University of Sherbrooke in Quebec, Canada, poses a question to his students every year: if they were to count the plant species in the province, would the number have gone up or down since Europeans arrived?Most students so far have got it wrong. “Many of them are surprised to learn that there’s 25% more [species] than there were 500 years ago, before people of European origin laid a foot here,” Vellend says.
Humans are driving one million species to extinction
Something odd is going on in biodiversity studies. Scientists have long warned that animal and plant species are disappearing at an alarming rate. In 2019, an international group of hundreds of researchers produced the most comprehensive report on biodiversity ever assembled, and they concluded that some one million animals and plant species are facing extinction. On top of that, humans have cleared landscapes and chopped down nearly one-third of the world’s forests since the Industrial Revolution — all of which bodes poorly for protecting species.So, scientists naturally assumed that they would find precipitous declines in biodiversity nearly everywhere they looked. But they haven’t. And a consensus is emerging that, even though species are disappearing globally at alarming rates, scientists cannot always detect the declines at the local level. Some species, populations and ecosystems are indeed crashing, but others are ebbing more slowly, holding steady or even thriving. This is not necessarily good news. In most places, new species are moving in when older ones leave or blink out, changing the character of the communities. And that has important implications, because biodiversity at the small scale has outsize importance; it provides food, fresh water, fuel, pollination and many ecosystem services that humans and other organisms depend on.“Ecosystems don’t work at the global scale,” says Maria Dornelas, an ecologist at the University of St Andrews, UK. “I’m interested in what is happening to biodiversity at the local scale, because that’s the scale that we experience.”Scientists say it’s clear that there’s a biodiversity crisis, but there are many questions about the details. Which species will lose? Will new communities be healthy and desirable? Will the rapidly changing ecosystems be able to deal with climate change? And where should conservation actions be targeted?To find answers, scientists need better data from field sites around the world, collected at regular intervals over long periods of time. Such data don’t exist for much of the world, but scientists are trying to fill the gaps in Europe. They are planning a comprehensive network, called EuropaBON, that will combine research plots, citizen scientists, satellite sensors, models and other methods to generate a continuous stream of biodiversity data for the continent. The effort will inform European policymakers, who are pushing for a strong and verifiable global biodiversity agreement when nations next meet to renew the United Nations’ Convention on Biological Diversity (CBD) — an international pact to halt and reverse biodiversity loss.How to measure biodiversityBiological diversity is a shape-shifting term that has been used in many ways. The CBD takes a broad approach, defining it as “the variability among living organisms from all sources”. This includes, it says, “diversity within species, between species and of ecosystems”.“Everybody could sign up to such a definition,” says Chris Thomas, an ecologist at the University of York, UK. “It means that different people can pick on different aspects that are all included within that all-encompassing definition, and find almost whatever trend they want.”Scientists measure biodiversity through many metrics, but the most common is species richness: a simple count of the number of species in the area. They also check the relative abundance of different organisms — a metric called species evenness — and track the identity of species to learn the ‘community composition’. Further complicating matters, scientists sometimes tally biomass instead of species richness, especially when it comes to insects.Using such measures, the clearest signal that the world is losing biodiversity comes from the bookkeeper of species, the International Union for Conservation of Nature. It has found that 26% of all mammals, 14% of birds and 41% of amphibians are currently threatened globally. Insufficient data are available for other groups, such as most plants and fungi. Extinction rates in the past few centuries are much higher than they had been before humans started to transform the planet; some estimates suggest current rates are 1,000 times the background level. One calculation estimates that, if high rates continue, then within 14,000 years, we could enter the sixth mass extinction — an event similar to the one that wiped out about three-quarters of the planet’s species, including dinosaurs, 65 million years ago4. For the most critically endangered species, the death knell could come within decades.

Lionfish have invaded the Red Sea, one example of species changes seen in many places.Credit: Alexis Rosenfeld/Getty

More bad news comes from the United Nations-backed Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES) — the organization behind the 2019 report warning that about one million species were threatened by extinction. The report also found that the abundance of native species in local terrestrial ecosystems has dropped by an average of around 20% as a result of human activities.Another biodiversity report that draws considerable attention comes from the conservation organization WWF and the Zoological Society of London, among other groups. Every year, they produce the Living Planet Index, which has amassed data for 27,695 populations of 4,806 vertebrate species. Last year, the report stated that population sizes of birds, mammals, fish, amphibians and reptiles declined, on average, by 68% between 1970 and 2016.Some researchers worry that such averaged figures can hide a lot of nuance, because many people might assume incorrectly that the average applies to most species. Dornelas likes to illustrate the danger by pointing out that the ‘average human’ has one breast and one testicle, and doesn’t exist.
Why deforestation and extinctions make pandemics more likely
Last year, Brian Leung, a biologist at McGill University in Montreal, and his colleagues re-analysed the Living Planet Index data from 2018 and found that a handful of populations are declining catastrophically, strongly pulling down the average. If these outliers are dropped from the computation, 98.6% of the populations on the index are holding steady or increasing or declining more slowly5. “We’re not saying there are not problems,” says Leung, who stresses that declines are still bad. “But there should be some caution about using these really broad-based global metrics, even though they are pretty powerful statements. But they can mask a whole lot of variation and be driven by extreme outliers.”When scientists talk about the world entering a sixth mass extinction, what sometimes gets lost is the timescale. Extinction rates for past periods of Earth’s history are calculated per one million years, and at present, researchers are seeing vertebrate species disappear at a rate of about 1% every century, and most of that has happened on islands.It’s clear there is a biodiversity crisis right now, although the pace is uncertain, says Henrique Pereira, a conservation biologist at the German Centre for Integrative Biodiversity Research , and a co-chair of an IPBES expert group. “It doesn’t mean that there is no decline. It means that if there is a decline, it’s much smaller than what maybe we thought.”So is the sixth mass extinction happening? “Well, not yet, if you want my scientific assessment of it. But is it going to be starting? Yes, maybe starting,” says Pereira.Difficult messageIn 2012, Vellend and his colleagues decided to see what’s happening with plant biodiversity by looking at a collection of individual field sites around the world. They compiled more than 16,000 studies in which scientists had monitored plants for at least 5 years, and found that only 8% of the studies noted a strong decline in the total number of species. Most plots showed either no change, smaller declines or even an increase in biodiversity6.The study was rejected by Nature, and one reviewer worried that journalists would garble the results and give the false impression that there were no problems with biodiversity. A Nature spokesperson says the peer-review process is confidential and that editorial decisions are not driven by considerations of potential media coverage. (Nature’s news team is editorially independent of its journal team.)

An experiment to trap and identify moth species in the Netherlands.Credit: Edwin Giesbers/Nature Picture Library

Vellend eventually published the study in the Proceedings of the National Academy of Sciences in 20136.His conclusions were soon backed up by Dornelas and her colleague Anne Magurran, an ecologist at the University of St Andrews, who have been compiling a database of biodiversity field studies, called BioTIME, since 2010. The database now has more than 12 million records for about 50,000 species from 600,000 locations around the world.In a study of 100 field sites worldwide, Dornelas and her colleagues had expected to find declines in species richness and abundance, but the data showed otherwise. Many sites were declining in biodiversity, but an equal proportion were improving. And about 20% showed no change over time. Overall, there wasn’t a clear trend7.At first, the researchers didn’t believe the results, so they reanalysed the data several ways and finally published the findings in 2014.“It was this tremendous shock. What’s going on?” says Pereira, who wasn’t involved in the study.Dornelas says reactions were mixed. Some people worried that the results could be misconstrued to suggest that everything’s fine with biodiversity. Others went even further. “Some people questioned our integrity, which is something that I take offence at, because being an ethical scientist is at the core of what I do,” she says. “But other people reached out to us and said, ‘Oh, interesting, that sort of matches my experience.’”Since then, many studies looking at biodiversity in the oceans, rivers, among insects — almost any grouping or biome one can think of — have found that there is no clear trend of decline. But that doesn’t mean the ecosystems are remaining static. Dornelas and her colleagues have continued to mine the BioTime database and have found that the mix of species in local communities is changing rapidly almost everywhere on Earth8 (see ‘Life on the go’). As some inhabitants disappear, colonizers move in and add to species richness, so the ‘average ecosystem’ shows no change or even an increase in the number of species, she says, with her usual cautions about averages9.

Source: Ref. 8

“Species are at the moment playing musical chairs,” says Dornelas.This can be seen most clearly on isolated islands, where 95% of the world’s extinctions have happened. Take New Zealand, where there were no mammalian predators before humans first settled there, less than 800 years ago. Since then, nearly half of New Zealand’s endemic birds have gone extinct.But despite the extinctions, biodiversity, measured by species richness, has improved over time in New Zealand, Vellend says. Continental birds have replaced the lost endemics. Plant biodiversity is doing well; fewer than 10 native species have gone extinct, and there are now 4,000 plant species on the islands, up from 2,000 before human settlers. And there are more than two dozen new land mammals.The lesson is that species richness or abundance figures might not tell the whole story, says Dornelas. Rather, scientists need to know the identity of all the species in a community, and track their relative abundances. This will allow them to learn which species are declining and which could be targeted for conservation.The story is similar on the continents, except with fewer complete extinctions. In Denmark over the past 140 years, 50 plant species have declined in abundance and range, but 236 have expanded their habitats. The large majority are holding steady10. Scientists looking at Europe’s birds since 1980 have found that 175 species are declining while 203 are increasing11. Rare birds are doing better than more common species, such as the house sparrow (Passer domesticus). A study of vertebrates in North America and Europe by Leung and his colleagues found that, whereas amphibians are declining across the board, other taxa have winners and losers in roughly equal measure12.Even corals seems to show the same pattern: between 1981 and 2013, 26 genera in the Caribbean and Indo-Pacific became more abundant, while 31 declined13.With studies piling up, it’s become increasingly acceptable for scientists to say that biodiversity isn’t declining everywhere and for all taxa, says Dan Greenberg, an ecologist at University of California, San Diego. “The tide is turning,” he says, “but the field is grappling with how to translate that to a public audience, or what does that mean in terms of social consequences.”That doesn’t mean there’s no biodiversity crisis, stresses Helmut Hillebrand, an ecologist at the University of Oldenburg in Germany. Some scientists worry that unusually high turnover, together with signals of instability in some populations, could itself portend ecological collapse. Humans are carrying species into new environs, leading to colonization. Whereas climate change is spurring warm-loving species to expand into new zones, cold-adapted species are losing out. Plus, generalist species that are fast-growing and less particular about where they live are thriving in human-modified landscapes.Specialists that need highly specific environments or that disperse poorly get easily isolated, which increases their extinction risk, says Greenberg. Case in point: amphibians. “If something changes in that environment, you can’t really hop over to another site very easily,” he says.
The battle for the soul of biodiversity
Turnover could lead to distant communities that increasingly resemble each other — a process called homogenization that has been documented in particular regions and taxa. In 2015, César Capinha, a biogeographer at the University of Lisbon, and his colleagues found that snail populations in temperate regions as far flung as Virginia, New Zealand and South Africa had species in common, thanks to human travel and trade14. Similarly, in the plant study in Denmark, scientists found that plant communities are increasingly looking like each other and are dominated by generalists. Scientists worry that such landscapes might not be resilient to environmental change.Dornelas urges caution in interpreting the changes seen so far. There hasn’t yet been a robust global study of homogenization to know the extent to which this is happening. And there is also increased habitat fragmentation, which can counter this process. “We don’t often talk about both of those at the same time,” Dornelas says. “I’ve now learned not to assume I know what’s going on until I’ve seen what the data show.”Scientists have also observed cases in which a colonizer mixes with a resident to rapidly form a new hybrid species, especially in plants, says Thomas. But it’s unclear how long these hybrids will persist, and most other groups usually take one million years or so to form new species. Many of the beasts of today could go extinct before that process can catch up, says Dolph Schluter, an evolutionary biologist at the University of British Columbia in Vancouver, Canada. “We are going to lose a lot of the ancients. And no amount of evolution in the short term is going to replace those,” Schluter says.Keeping tabs on lifeGlobal studies of biodiversity have important biases owing to data gaps. Most of the records of species come from Europe and North America; there are very few data from the tropics, where rainforests house half of all species in just 7% of the Earth’s surface. And even in the most richly monitored places on Earth, such as Europe, the data are patchy. “We are trying to read the book, but we have only a few letters,” says Pereira.Pereira and his colleagues are designing a top-down monitoring network in Europe called EuropaBON that can add in more letters, and maybe even sentences. The project has received 3 million (US$3.5 million) from the European Commission, and was launched last December. Pereira and Jessica Junker, the scientific coordinator of EuropaBON and a conservationist at Martin Luther University Halle-Wittenberg in Germany, have assembled a 350-strong community of national conservation authorities, non-governmental organizations, scientists and government officials. Among the first goals is to create a map that identifies data gaps as well as a list of metrics to be tracked, Pereira says. At the end of the initial three-year stage, EuropaBON aims to set up a coordinating centre for the monitoring network.It’d have to be affordable to be replicable and maintained over time. Lack of funds has hampered a global version of this project, called GEO BON, on which EuropaBON is based, says Dornelas. To contain costs, EuropaBON intends to use existing long-term monitoring sites. Where there are data gaps, the scientists would launch new tracking efforts using technology such as sensors, weather radar and drones, or citizen volunteers, who already do 80% of the biodiversity monitoring in Europe.EuropaBON would also use satellite data of land cover, vegetation growth and other indicators of local biodiversity. The data streams would be combined with modelling to generate seamless biodiversity data over time and across Europe. The plan is that data from the project will help the European Commission to decide what research to fund on the continent’s biodiversity, says Pereira. In a stakeholder meeting in May for EuropaBON, Humberto Delgado Rosa, the director for natural capital at the European Commission, said that the European Union wants to make “huge leaps internationally in biodiversity, as it has done with climate in Paris”. EuropaBON should help Europe to meet its international commitments to report on its biodiversity, Rosa said.“This new global biodiversity framework needs quantification, measurability,” he said. “In a nutshell, we need knowledge.”Dornelas, who is also part of EuropaBON, says she would like to expand this initiative across the world. Canada is exploring a national version, called CanBON. But for now, monitoring remains sparse in the poorer parts of the world, where most of the planet’s biodiversity remains.“Europe is one of the best monitored parts of the planet, and where we’re really, really missing data is from other parts of the world,” she says. “But I guess we got to start somewhere.”

Nature 596, 22-25 (2021)
doi: https://doi.org/10.1038/d41586-021-02088-3

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Paleotemperature estimatorsPhanerozoic paleotemperature data are derived from oxygen isotope (δ18O), clumped isotope (Δ47), and organic geochemistry (TEX86) data. Many published studies have used different conversion formulae to get from measured values to temperatures for each paleothermometry method, which makes it difficult to compare results from various studies. Because of this, we used a uniform formula to re-calculate paleotemperatures. Different estimators generally show consistent trends for a given time interval, e.g. Cretaceous planktonic foraminifera δ18O and TEX8640, and Jurassic belemnite δ18O41.Most Paleozoic and Mesozoic paleotemperature data come from oxygen isotope paleothermometry. Cenozoic paleotemperature data derive mainly from TEX86 and oxygen isotope of planktonic foraminifera. At present, there are major debates over the first-order trend of Phanerozoic surface temperatures calculated from oxygen isotopes12,42,43,44. For instance, it is uncertain whether δ18Osw evolved towards more enriched values due to exchange at mid-ocean ridges, potentially impacting interpretations of seawater temperatures in the early Paleozoic, for instance43. However, this is not an issue in our study because we focus only on changes in temperature rather than absolute values.The data we used for each time bin was selected from a large paleotemperature dataset12 and published literature by adopting the following criteria: 1) data with well-constrained ages; 2) data measured at high time resolution; 3) data from tropical/subtropical regions. For multiple sets of data in the same time bin, we use the average values. Where possible, in order to test the reliability of climate events, we use temperature data from two different proxies or two different regions. For example, δ18O of planktonic foraminifera and TEX86 were used for the late Eocene (Pg4) and Maastrichtian (K8) intervals; and conodont δ18O from South China and Armenia were used for the Lopingian (P5) interval. The results show that data from two different proxies or different regions have a similar magnitude and rate of temperature change (Supplementary Fig. 4).Oxygen isotopesOxygen isotope values from carbonate fossils (i.e., planktonic foraminifera, brachiopod, oyster, and belemnite) are converted to SSTs using the BAYFOX Bayesian model (https://github.com/jesstierney/bayfoxm)45. For phosphate fossil conodont, we used Monte Carlo simulations to propagate parameter uncertainties to temperature estimation by using the equation of Pucéat et al.46:$${SST}=118.7(pm4.9)-4.22(pm0.20)times left({delta }^{18}{O}_{{con}}-{delta }^{18}{O}_{{SW}}right)times 1000$$
(1)
where δ18Ocon is the oxygen isotope composition of conodonts, and δ18OSW is the oxygen isotope composition of seawater.Seawater oxygen isotope composition is affected by salinity and changes in continental ice volume43,47. As such, δ18O data from localities with abnormal salinities (e.g., evaporite facies, upwelling regions) were excluded (see paleotemperature estimators above). Changes in continental ice volume were also considered in our calculation of paleotemperatures. To do this, the oxygen isotope value of seawater was set to −1‰ (VSMOW) for ice-free time intervals48 and +1‰ (VSMOW) during glacial maximum intervals, e.g., the Pennsylvanian Glacial Maximum and the Pleistocene Last Glacial Maximum49. Most oxygen isotope data are from tropical and subtropical regions (Supplementary Fig. 5), therefore, no latitudinal seawater corrections were made. In addition, seawater pH has a further important influence on the δ18O of foraminifera50. During global warming events that are related to CO2 release, seawater pH can decrease, which would lead to underestimated SSTs50,51. Unfortunately, pH estimates for the Phanerozoic time bins are few, and their values are highly uncertain52. Therefore, we did not pH-correct δ18O estimates. Oxygen isotope values that were likely affected by diagenetic alteration were also removed from the database. Diagenetic screening criteria included δ18O values that are unrealistically negative or positive, and δ18O values from carbonate fossil shells with [Mn]  > 250 ppm and [Sr] 0.4 were not used in this study. Other screening criteria of TEX86 were also employed. Notably, data were removed from the database if Methane Index (MI) >0.5 (ref. 58), delta-Ring Index (ΔRI) > 0.3 (ref. 59), %GDGT-0 >67% (ref. 60), and/or fCren′:Cren′ + Cren >0.25 (ref. 40).Age modelGeochronologic constraints can provide absolute age at the initiation and termination of warming/cooling events. In our database, only one warming event, at the Permian-Triassic boundary, meets this criterion. Other dates were obtained using a comprehensive approach including isotope geochronology, astrochronology, and biostratigraphy with reference to the Geologic Time Scale 201261. Age data were applied in the following order of priority: isotope geochronologic age, astrochronologic age, and biostratigraphic age. If isotope geochronologic ages were available, we gave priority to these absolute dates. In the absence of absolute age data, astrochronologic ages were preferred. If neither of these numerical data was available, the climatic events were timed based on the age of biozones in the Geologic Time Scale 201261. Relative ages within individual biozones were constrained based on the stratigraphic position. Age uncertainty was calculated for all events by using Monte Carlo simulation based on the assumption of uniform distribution of ages.The magnitude and rate of temperature changeThe maximum magnitudes (ΔT) of temperature change within each of the 45 defined time intervals were calculated as:$$Delta T={T}_{1}-{T}_{0}$$
(2)
where ({T}_{0}) and ({T}_{1}) represent the initial and terminal temperature of warming/cooling events, respectively. Supplementary Fig. 1 illustrates how the parameters ({T}_{0}) and ({T}_{1}) are derived from a climatic time series (t).The time scale Δt represents the duration of time during which the temperature increases/decreases. Δt was calculated as:$$Delta t={t}_{1}-{t}_{0}$$
(3)
where ({t}_{0}) and ({t}_{1}) represent the initial and terminal time of warming/cooling events, respectively.The rate (R) of temperature change was computed as:$$R=Delta T/Delta t$$
(4)
The ratio (R) represents the rate of a single warming/cooling event over geological time.ΔT, R and their uncertainties were calculated by using Monte Carlo simulation based on the distribution of data from two groups around t0 and t1.Temperature databaseThe temperature database is composed of the most significant warming/cooling events in 45 time intervals from the Late Ordovician (445 Ma) to early Miocene (16 Ma) (Supplementary Data 162). The time intervals are consistent with the time bins used to compute biodiversity and evolutionary rates63,64, and are defined by one or several neighboring geologic stages with roughly uniform durations (averaging 9.71 Myr). Time bins range between 4.7 and 18.9 Myr. The 45 bins are Katian and Hirnantian (Or5), Llandovery (S1), Wenlock (S2), Ludlow and Pridoli (S3), Lochkovian and Pragian (D1), Emsian (D2), Eifelian and Givetian (D3), Frasnian (D4), Famennian (D5), Toutnaisian (C1), early Visean (C2), late Visean and Serpukhovian (C3), Bashkirian (C4), Moscovian and Kasimovian (C5), Gzhelian (C6), Asselian and Sakmarian (P1), Artinskian (P2), Kungurian and Roadian (P3), Wordian and Capitanian (P4), Wuchiapingian and Changhsingian (P5), Induan and Olenekian (T1), Anisian and Ladinian (T2), Carnian (T3), Norian (T4), Rhaetian (T5), Hettangian and Sinemurian (J1), Pliensbachian (J2), Toarcian and Aalenian (J3), Bajocian-Callovian (J4), Oxfordian (J5), Kimmeridgian and Tithonian (J6), Berriasian and Valanginian (K1), Hauterivian and Barremian (K2), Aptian (K3), Albian (K4), Cenomanian (K5), Turonian-Santonian (K6), Campanian (K7), Maastrichtian (K8), Paleocene (Pg1), Ypresian (Early Eocene, Pg2), Lutetian (early Middle Eocene, Pg3), Bartonian and Priabonian (late Middle-Late Eocene, Pg4), Oligocene (Pg5), and Aquitanian and Burdigalian (Early Miocene, Ng1).Original proxy data, calculated temperature data, geologic age, time span, paleolatitudes, and relevant references are provided in Supplementary Methods and Source Data. Most paleotemperature data are sea surface temperatures (SST) from tropical and subtropical regions (between 40°N and 40°S, Supplementary Fig. 5). Only one collection is from a mid-latitude region with a paleolatitude of 42.71°N (Source Data). Paleolatitudes were reconstructed using PointTracker v7 rotation files published by the PALEOMAP Project65.For most time bins, the major climate change events (warming/cooling) were restricted to that bin. A few events occurred that cross two adjacent bins, e.g., the rapid climate warming around the Permian-Triassic and the Triassic-Jurassic boundaries17,66,67. Both these events had relatively short durations (61–400 kyr) and were initiated towards the end of the first (earliest) time bin. Therefore, these warmings would have primarily impacted organisms in the first bin too. Accordingly, we didn’t divide the warming event into two intervals belonging to the two adjacent bins, but take instead the warming event as belonging to the first bin.Extinction rateThere are a few well-established methods to calculate extinction rate in geological history, e.g., boundary-crosser, gap-filler (GF), and three-timer (3 T) rates63,68,69. Both the GF and 3 T rates are calculated from occurrence data and have higher accuracy than the boundary-crosser method64. The boundary-crosser method is more susceptible to major biases such as the Signor-Lipps effect and sampling bias because it uses full age ranges instead of investigating sampling patterns across limited temporal windows64. The 3 T method can be noisy in the case of high turnover rates and poor sampling. The GF method is more precise when sampling is very poor.Gap-filler and three-timer extinction rates of marine animals were calculated using data from the Paleobiology Database (PBDB, http://paleobiodb.org), which was downloaded on 4 January 2021. The fossil dataset includes all metazoans except for Arachnida, Insecta, Ostracoda, and Tetrapoda and consists of 850,840 fossil occurrences of 37,134 genera. These four groups that are excluded (in keeping with some previous studies64,69) were not used because many of them (Arachnida, Insecta, and Tetrapoda) are terrestrial and appear in marine strata. Ostracoda also has a record in terrestrial rocks. Similar to most studies using data from PBDB, we use genus-level occurrences because genus-level taxonomy is better standardized than the taxonomy of species. GF extinction rate (({r}_{{GF}})) is calculated by using the equation:$${r}_{{GF}}={{{{{rm{log }}}}}}left(frac{2T+{pT}}{3T+{pT}+{GF}}right)$$
(5)
Three-timer extinction rate (({r}_{3T})) is calculated by using the equation:$${r}_{3T}={{{{{rm{log }}}}}}left(frac{2{T}_{i}}{3T}right)+{{{{{rm{log }}}}}}left(frac{3T}{3T+{pT}}right)$$
(6)
where ({GF}) represents gap-fillers, i.e., the number of genera in time bins (i-1) and (i+2) and not within bin (i+1); (2{T}_{i}) represents the number of genera sampled before and within the ith time bin; (3T) represents the number of genera sampled in three consecutive time bins; and ({pT}) represents the number of genera sampled before and after but not within a time bin.Autocorrelation analysisWe tested for serial correlation in the time series of extinction rate, ∆T, and log R, because autocorrelation could impact the calculated correlation coefficients between these data. The autocorrelation functions (ACFs) of extinction rate, ∆T, and log R are very low at all lags >1 (Supplementary Fig. 6), indicating that all-time series lack significant serial correlation.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

Temperature profiles, sampling methods, and physical descriptions were compiled for 153 globally-distributed lakes. We selected vertical temperature profiles measured in the same location for multiple years, typically the deepest part of the lake, using a single mid-summer profile for each lake and year. Data are presented for all years for which profiles were available, with both raw vertical temperature profiles and interpolated vertical temperature profiles presented. Depth increments for raw profiles vary by lake based on sampling methodology, as described for each lake below. For interpolation, the selected temperature profiles were linearly interpolated or binned to 0.5 m increments throughout the water column. Mid-summer profiles for each lake were selected as described in (7). Briefly, we calculated relative thermal resistance to mixing28,29 (RTR) for all profiles for each lake for each year. The day of year with the maximum RTR value for each year was selected for each lake, and the median day of year across all years per lake was considered the target day for single mid-summer profile selection. For each lake and year, the temperature profile nearest to this median day of year ± 21 days was selected. If no profile spanning surface to near the maximum lake depth was available, then no profile was selected. Details on sampling methods from each contributing research group follows, organized by major geographic region and alphabetized within geographic region. Additional details for water chemistry sampling methods can be found in the Supplementary Information (Supplementary Table S1).Western North AmericaCastle Lake in California, USA has been sampled annually since 1959. Temperature data were measured between 12:00 and 14:00 at the deepest part of the lake in 1 m depth intervals. Measurements were taken using a reversing thermometer (1959–1975), Hydrolab (1975–1982; 2011–2013), and Yellow Springs Instruments (YSI) model 85 (1999–2011) (sampling instrument is unknown for most of the 1980s and 90s).Crater Lake is located in the northwestern USA at the crest of the Cascade Mountains in Oregon. Temperature data were collected between 8:00 and 17:00 from the middle of the lake in the deepest basin. From 1988 to present, temperature profiles were measured by lowering a Seabird Instruments SBE19 CTD through the water column at a rate of approximately 0.5 m per second, collecting two readings per second. Data were binned to 1 m increments. Prior to 1988, temperature profiles were measured manually with a Montedoro-Whitney thermistor with a 250 m cable.Emerald Lake, California, USA has temperature profiles measured from the deepest part of the lake beginning in 1984. Temperature was measured in 1 m increments throughout the water column from 1984 to 2006 using a YSI 58, after which a thermistor chain was deployed with Onset Water Temp Pro spaced at 0.5 to 1 m intervals.Flathead Lake in northwest Montana, USA is the largest freshwater lake west of the Mississippi River by surface area. Temperature data were collected fortnightly in June, July, and August 1978–2013 with various instruments. From 1978–1984, data were collected with a Hydrolab I; from 1985–1993, with a Hydrolab Surveyor II; from 1993–2003, with a Hydrolab Surveyor III; and from 2003–2001, with a Hydrolab Surveyor IV. Since 2011, data have been collected with a Hydrolab DS5 unit. Measurements were taken between 10:00 and 14:00 at the deepest point of the lake in 1 to 10 m depth intervals throughout the water column, with greater intervals between measurements at deeper depths. The intervals between depths on a given sampling date also changed periodically early in the data set before consistent intervals were established. Instrumentation was calibrated before each sampling event to account for the elevation of the sampling site and highly variable barometric pressure.Washington Lake is located in Washington, USA. Temperature profiles were measured between 9:00 and 16:00 in the main trench of the lake. Temperature was measured using various bathythermographs between 1933–1986, a Kahl digital temperature meter 202WA510 beginning in 1974, and a YSI 6600 V2 sonde beginning in 2012. For measurements with the bathythermographs, temperature was measured every 1 m close to the thermocline and every 5 m elsewhere. When the Kahl temperature meter was used, temperature was recorded for every meter through 20 m and in 5 m increments after 20 m depth. Data from the YSI sonde were recorded continuously, so temperature readings were used at similar intervals to those from the previous instruments.Central North AmericaActon Lake is a eutrophic reservoir in southeastern Ohio, USA. Temperature was measured in the deepest part of the lake at 0.5 m depth increments from 0 to 5 m and 1 m increments for the remainder of the water column. Temperature data were typically collected in the mid-morning and occasionally in the early afternoon in mid-July of each year from 1992–2012 using a handheld YSI temperature/dissolved oxygen sensor or a YSI sonde.Lakes Bighorn, Harrison, Pipit, and Snowflake are located along the eastern front range of the Rocky Mountains in the Cascade Valley of Banff National Park (Alberta, Canada). Temperature data were collected in the deepest point of the lake in 1 m depth intervals between 10:00 and 15:00 using a MK II Thermistor (Flett Research Ltd, Winnipeg, Canada) for measurements taken beginning in the 1990s. Earlier measurements (1960s and 1970s) were taken using a YSI Model 425 C thermistor thermometer, which was calibrated against a mercury thermometer.Douglas Lake is a mildly eutrophic glacially formed multiple ice-block kettle lake in northwestern Cheboygan County, Michigan, USA. Temperature data were collected from 1913–2014 at 1 m intervals from 0 to 24 m in the center of South Fishtail Bay Kettle with a reversing thermometer at approximately 1 m increments (1913–1970), a YSI model 54 A oxygen electrode-thermistor thermometer (1971–1982), a Hydrolab MS-5 multiprobe (1983–2009), and an 8-node MHL thermistor string (2010–2014).Lakes Eucha, Grand Lake O’ the Cherokees, Spavinaw, Texoma, and Thunderbird are reservoirs located in Oklahoma, USA. Temperature profiles were measured at the deepest point in these reservoirs between morning and mid-day with various sensors. YSI temperature probes were used for all samples collected in Lake Thunderbird, through 1991 in Grand Lake O’ the Cherokees, through 1995 in Lakes Eucha and Spavinaw, and through 2000 in Lake Texoma. Beginning in 2000 in Lake Texoma, Hydrolabs were used for temperature measurements, with a Hydrolab H2O through 2008, and Hydrolab DSX5 thereafter. A Hydrolab was used for measurements in Grand Lake O’ the Cherokees from 2011–2013. Lakes Eucha and Spavinaw were sampled with a Hydrolab H2O through 2005, a YSI 6930 V2 for samples from 2006–2012, and a YSI EXO1 for samples after 2012.Katepwa Lake is a eutrophic, riverine site located in the Qu’Appelle River drainage basin in southern Saskatchewan, Canada. The lake has been sampled since 1994 as part of the Qu’Appelle Long-term Ecological Research network (QU-LTER). Temperature data were collected from a standard deep site in 1 m intervals between 10:30 and 13:00 using a YSI 85 or similar multi-parameter probe.Northeastern North AmericaTemperature profiles were collected beginning in 1969 for six lakes (Lakes 222, 224, 239, 240, 373, and 442) at the IISD Experimental Lakes Area (International Institute for Sustainable Development, Northwestern Ontario, Canada). Profiles were measured in the deepest part of the lake in 1 m intervals, except in the thermocline where temperature was measured every 0.25 m. Montedoro-Whitney thermistors (models TC-5A and TC-5C) were used through 1983, a Flett Research Mark II digital telethermometer was used for temperature measurements from 1984–2009, and a RBR XRX620 multifunction probe with integrated temperature sensor for measurements beginning in 2010.The Dorset “A lakes”, Blue Chalk, Chub, Crosson, Dickie, Harp, Heney, Plastic, and Red Chalk Lakes are located in the Muskoka-Haliburton region of south-central Ontario, Canada. The study sites are primarily small headwater lakes, with the exception of Red Chalk Lake which is located downstream of Blue Chalk Lake. Temperature data were collected from the deepest point in each lake using a YSI 58 temperature/dissolved oxygen meter (or occasionally a digital YSI 95 meter) beginning in the late 1970s. Measurements were collected between 9:00 and 16:00, with readings taken every 1 m from the lake surface (0.1 m depth) to within approximately 1 m of lake sediments.Bubble Pond, Eagle Lake, and Jordan Pond are located on Mount Desert Island off the coast of Maine, USA. Temperature data were collected from the location of the maximum depth at 1 m increments using a YSI 600XL multiparameter water-quality monitor (sonde) from 2006 to present and a YSI 54ARC before this time.Lake Champlain is located in the northeastern USA, on the border of Vermont and New York state and partly extending into Quebéc, Canada. Temperature data were collected from a sampling station in Mallet’s Bay beginning in 1992. Measurements of temperature were taken at 1 m intervals between 8:00 and 17:00 using a Hydrolab MS-5 multi-probe sonde.Clearwater, Sans Chambre, and Whitepine Lakes are located in northeastern Ontario, Canada, and Hawley Lake is in the Hudson Bay Lowlands area of subarctic Ontario, Canada. Temperature profiles were measured from near the area of maximum depth, beginning in the 1970s in Clearwater and Hawley, and beginning in the 1980s in Sans Chambre and Whitepine. Measurements were taken in 1 m intervals through the water column between 12:00 and 17:00. Clearwater, Sans Chambre, and Whitepine Lakes’ temperature profiles were measured using various YSI temperature/dissolved oxygen meters (models 50B, 51B, 52, 54, or 58) beginning in 1998, with a YSI model 54 temperature/dissolved oxygen meter used for Sans Chambre and Whitepine for most earlier years’ measurements. In Clearwater Lake, a YSI model 432D telethermometer was used through 1975, a Montedoro-Whitney TC-5C thermistor from 1976–1981, and a Mark II Telethermometer from 1982–1998. In Hawley Lake, various YSI temperature probes were used in earlier years, and since 2009, a YSI Pro ODO meter was used to measure water temperature.Lakes Giles and Lacawac are located in the Pocono Mountains region of Pennsylvania, USA. Temperature data were collected from the deepest point in the lake in late July or early August each year beginning in 1988, between 9:00 and 16:00. Temperature measurements were measured in 1 m increments using a YSI 58 temperature/dissolved oxygen meter (1988–1992), or with a rapid recording Biospherical Instruments PUV 500 (1993–2003) or BIC 2104 P (2004-present) recording at 4 Hz while being lowered through the water column. Profiles taken with the YSI were linearly interpolated to 0.5 m depth increments, and profiles taken with the PUV and BIC were binned to 0.5 m depth increments.Lake Lillinonah is a reservoir on the Housatonic River located in western Connecticut, USA. Temperature data were collected at the deepest point of the lake between 9:00 and 17:00 beginning in 1996 by First Light Power. Measurements were collected every 5 m using a YSI 58 temperature/dissolved oxygen meter.Mohonk Lake is a small glacial lake with a single deep basin located at the Shawangunk Ridge, New York, USA. Temperature data were collected from the northern end of the lake from 1984–2013 during daytime. Temperature measurements were measured manually in 1 m increments using Digi-sense Economical Thermistor 400 series (Model #93210-00). Additional weekly temperature profiles are publicly available30.Temperature profiles were compiled from 11 lakes from the North Temperate Long Term Ecological Research Network in Wisconsin, USA. Fish, Mendota, Monona, and Wingra Lakes are located in southern Wisconsin, and Allequash, Big Muskellunge, Crystal Bog, Crystal Lake, Sparkling, Trout Bog, and Trout Lake are in northern Wisconsin. Temperature profiles were measured in 1 m increments from the surface to lake bottom at the deepest location in each lake since 1981 (northern lakes) and 1995–1996 (southern lakes), and since 1894 in Mendota31. Various temperature/dissolved oxygen probes were used to collect these data, and were calibrated in the field prior to data collection. For the northern lakes, a Montedoro Whitney CTU-3B sensor was used for some data collected between 1981–1986, a Whitney TC-5C for 1982–1983, Whitney DOR-2A in 1984, YSI-57 in 1983 and 1985, and a YSI-58 for 1985 onward. Temperature data in the southern lakes were collected with a YSI-58 temperature/dissolved oxygen sensor.Lake Opeongo is the largest oligotrophic deepwater lake in Algonquin Provincial Park, Ontario, Canada. Temperature data were collected by Ontario Ministry of Natural Resources and Forestry staff at Harkness Fisheries Research Station in the west basin of Opeongo’s South Arm in midsummer in 1958–1965, and between 9:00 and 16:00 in 1998–2014. Temperature loggers (Hobo Tidbits, 1998–2004; Onset Water Temperature Pro V1, 2004–2008; and Onset Pro V2, 2009–2014) were installed after ice out in the same location at approximately 1.5 m intervals to a 15 m depth, with a deepwater thermistor placed at approximately 20 m. Temperatures were recorded at high temporal resolution (10 to 15 minute intervals) throughout the summer using these temperature strings. Earlier (1958–1965) temperature profiles were measured using handheld thermistors.Lake Sunapee is the fifth largest lake located within New Hampshire, USA. Temperature data were collected in the morning from the deepest point of the lake in the central basin beginning in 1986. Temperature measurements were measured manually in 1 m increments using a YSI 52 temperature meter.Lake Wallenpaupack is a reservoir in the Pocono Mountains region of Pennsylvania, USA. Temperature profiles were measured in the center of the lake during daytime hours, with measurements taken every 0.5 to 1 m. Temperature was measured with various YSI instruments including a YSI 610-DM/600XL (2002–2005), YSI 85 (2008–2010), YSI 600XL sonde with 600D datalogger (2011–2012), and various temperature sensors prior to 2002.Southeastern North AmericaLake Annie is located in the Lake Wales Ridge region, Florida, USA. Temperature data were collected from the deepest point in the lake on a monthly basis between 9:00 and 16:00 beginning in 1984. Temperature measurements were measured manually in 1 m increments using a Montedoro Corporation Thermistor Model TC-5c (1984–2008) and a YSI Pro Plus (2009–2014) with values measured while lowering the meter to the bottom of the lake and again when raising it to the surface. The data reported are the means of the two depth-specific values.Temperature data were measured in Lake Okeechobee in Florida, USA beginning in 1973. Temperature data were measured near the surface at approximately 0.2 m between 8:00 and 12:00 using a Hydrolab through 1995 and a YSI 58 temperature sensor from 1996–2014.SubarcticAleknagik, Beverley, Chignik, Hidden, Kulik, Little Togiak, Lynx, and Nerka Lakes are located in Alaska, USA. Temperature profiles have been measured on these lakes since the 1960s (Aleknagik, Beverley, Kulik, Little Togiak, and Nerka Lakes), and since the early 2000s (Chignik, Hidden, and Lynx Lakes). Measurements were taken during the day, typically between 10:00 and 19:00. A bathythermograph was used for temperature measurements through 1967, a digital thermistor from 1968–1998, a YSI 660 sonde for samples from 1999–2012, and a YSI Castaway beginning in 2013, with the exception of samples from Chignik Lake, for which a handheld thermometer was used to measure the water temperature from Van Dorn casts.Toolik Lake is a kettle lake located in Alaska, USA. Temperature data were collected between June and August in the south basin of the lake between 9:00 and 11:00. Temperature was recorded in 1 m increments throughout the water column with a Hydrolab profiler sampling Surveyor 4a datalogger and a datasonde 4a multiprobe32,33,34,35.Vulture Lake is located in the sub-Arctic region of the Northwest Territories, Canada. Temperature data were collected from one of the deeper parts of the lake in late July or early August during the open-water season from 1997–2014. Temperature measurements were recorded using a multi-probe sonde (e.g., YSI 6820) at 0.5 m or 1 m increments (every 1 m for most years, except 1999 [every 0.5 m] and 2009 [every 0.2 m]). The corresponding depth was measured using the readings from the depth sensor. Measurements were collected continuously from just below the lake’s surface to approximately 0.5 m above the water-sediment interface. Sensors equipped on profiling instruments were calibrated following the manufacturer’s recommended frequency and methods to ensure accurate and reliable operation of the sensors in the field.Central and South AmericaLake Atitlán is a deep tropical mountain lake in the Guatemalan highlands, sampled in 1968–1969 and 2010–2011. Temperature profiles at the lake center were measured manually between 7:00 and 13:00 in variable increments (1 to 5 m) in the first 30 m using a YSI 51 or YSI 95 temperature/dissolved oxygen meter. For depths below 30 m, samples were collected using a Van Dorn bottle and temperature was measured immediately upon the sample reaching the surface.Lake Mascardi is located in the North Patagonian Andes in Argentina. Temperature data were collected near the deepest point in the Catedral arm of the lake in mid-summer (January-February) between 12:00 and 14:00 beginning in 1994. Temperature measurements were taken using rapid recording Biospherical Instruments PUV 500 (1994) or PUV 500B (1996–2014) recording at 4 Hz while being lowered through the water column.AfricaLake Kivu is located on the border between Rwanda and the Democratic Republic of Congo, and is one of the seven African Great Lakes. Temperature data were collected in the Ishungu basin beginning in 2002 between 9:00 and 16:00. Temperature was measured in 5 m increments using a YSI 55 temperature/dissolved oxygen meter (2002–2005), or with a suite of instruments (YSI 6600 V2, Hydrolab DS4a, DataSonde 4a 42071, Sea and Sun 725 and 257) recording at high frequency while being lowered through the water column. All temperature profiles were vertically interpolated to a regular vertical grid with 1 m increments using piecewise cubic Hermite interpolation18,26.Lake Nkugute is located in Uganda. Temperature data for Lake Nkugute were collected in the deepest part of the lake at a depth interval of 1 to 5 m using a liquid-in-glass thermometer set in a Ruttner sampler in 1964 with the contemporary (2002-present) data being measured with a YSI sonde.Lake Nkuruba is located in western Uganda, in the vicinity of Kibale National Park (northern sector). Temperature data were collected beginning in 1992 from the deepest part of the lake in 1 m increments through a depth of approximately 30 m. Temperature was measured manually using a YSI 50 or 51B temperature/dissolved oxygen meter.Lake Tanganyika has temperature profile data dating back to 1912. Temperature profiles in this lake were typically measured in the morning, between 9:00 and 12:00 from the north basin of the lake near Kigoma, Tanzania. Temperature profiles over this century of data collection have been taken using a multitude of instruments. Since 1993, temperature profiles have been measured using a YSI 6600 V2 sonde, titanium RBRduo TD, Seacat Profiler V3.1b, Onset HOBO U22 temperature loggers, CTD Seabird 19, STD-12 Plus CTP profiler, and a YSI 58 temperature/dissolved oxygen meter. Prior to 1993 various methods were used, including measurement of water temperature from Van Dorn casts with a mercury thermometer, various data loggers, reversing thermometers, and Niskin bottles and a bathythermograph36.Lake Victoria is shared between Tanzania, Uganda, and Kenya. Temperature data were collected from stations across the lake during acoustic surveys each year in 2000–2001 and in 2008 using a submersible Conductivity Temperature-Depth profiling system (CTD, Sea-bird Electronics, Sea Cat SBE 19).Scandinavia and Northern EuropeLakes Allgjuttern, Brunnsjön, Fiolen, Fracksjön, Övre Skärjön, Remmarsjön, Rotehogstjärnen, St. Skärsjön, Stensjön, and Stora Envättern are relatively small, boreal lakes in Sweden. In contrast, Lake Vänern is the largest lake in Sweden. Temperature data have been collected since 1988 (since 1973 for Lake Vänern) between morning and mid-afternoon. Temperature measurements were taken from the deepest point in each lake from the surface through 1 m above the lake bottom at depth intervals varying between 1 m and up to 10 m (in Lake Vänern).Lakes Byglandsfjorden, Hornindalsvatnet, Mjøsa, Øyeren, Selbusjøen, and Strynevatnet are all large and deep lakes located in the central region of western Norway. Temperature profiles were measured in the deepest part of the lake between 9:00 and 16:00 beginning in the mid-1990s. Temperature was measured using an Aanderaa 4060 every meter in the upper part of the water column, with greater than 1 m intervals between measurements in depths below 20 m. For the past 2 to 3 years of data collection, a Castaway CTD recorded temperatures while lowered. Data collection was made by The Norwegian Water Resources and Energy Directorate (NVE).Temperature data were collected from Sweden’s Lake Erken from the deepest point in the lake in 1 m intervals between 7:30 and 9:30 beginning in 1940. Temperature was measured at 1 m intervals using a variety of instruments. In early years, a thermometer inside a transparent Ruttner sampler was read to obtain the temperature of water collected from different depths. Later, underwater thermistors were used from a variety of manufacturers. In recent years, combined temperature and dissolved oxygen sensors have been used to collect water temperature measurements: a YSI model 52 (1996–2006), WTW Oxi 340i (2006–2012), and Hach HQ40d sensor system (2012-present).Lakes Inarijärvi, Kallavesi, Konnevesi, Näsijärvi, Päijänne, Pielinen, and Pyhäjärvi are generally large lakes located throughout Finland, from southern Finland to the northern-most part of the country (Lapland). Lake Pesiöjärvi in the same region has a significantly smaller surface area than others. Lakes Konnevesi and Päijänne have two different temperature profile sites (Konnevesi: Näreselkä and Pynnölänniemi, Päijänne: Linnasaari and Päijätsalo). Temperature data were typically collected in the deepest part of the lake, or in case of large fragmented lakes, the deepest part of the respective basin. Reversing mercury thermometers (with or without Ruttner water samplers) were used until the digital temperature measurements were introduced in the early 1970s. HL Hydrolab Ab PT77A (approximately 1975–1995), DeltaOhm HD8601P (1995–2005), and HT Hydrotechnik Type 110 (2005–2014) have been used for measuring water temperature profiles. However, all device types have been used at each station as long as they have worked. Unfortunately, site-specific documentation of devices used during different years are not available. Before the 1980s, measurements were collected every 5 m before and every 10 m after a depth of 20 m. Since 1980–1981, measurements have been made in 1 m intervals from the surface through 20 m, every 2 m from 20 to 50 m, and every 5 m past 50 m.Lake Pyhäselkä (Pyhaselka) is a large, humic lake located in North Karelia, Finland. It is the northernmost basin of the Saimaa lake system. Temperature data were collected from the deepest point in the lake (Kokonluoto) in 5 m depth intervals beginning in 1962 between 8:00 and 16:00 using a thermometer in a Ruttner water sampler.Central EuropeLakes Annecy, Bourget, and Geneva are located in eastern France37. Temperature data were collected from the deepest point in the lake between 9:30 and 11:00, beginning in 1991 for Lake Annecy, in 1984 for Lake Bourget, and in 1974 for Lake Geneva. In Lake Annecy, various multiparameter probes, including Meerestechnik Elektronik (1991–2001), CTD 90 (2003–2005), CTD 90 M (2008–2011, 2013), and RBR (2012) were used for temperature measurements at depth intervals between 0.01 and 1.8 m. Temperature profiles were measured in Lake Bourget at depth intervals between 0.01 and 10 m using various multiparameter probes, including ISMA probe DNTC (1984–1985), Meerestechnik Elektronik ECO 236 (1986–1998), CTD SBE 19SeaCAT Profiler (1992–2002), and CTD SBE 19plus V2 SeaCAT (2003–2013). In Lake Geneva, water temperature was measured from water samples with a thermometer until 1990 from discrete depths of 5 to 10 m intervals through 50 m depth and approximately 50 m intervals for the rest of the water column. After 1990 various multiparameter probes were used for temperature measurements, including Meerestechnik Elektronik (1991–2001), CTD 90 (2002–2007), CTD 90 M (2008–2011, 2013), and RBR (2012). Data for Annecy, Bourget, and Geneva are available38 at https://data.inrae.fr/dataset.xhtml?persistentId=doi:10.15454/YOLA0Y.The Cumbrian lakes in the English Lake District, Bassenthwaite Lake, Blelham Tarn, Derwent Water, Esthwaite Water, Grasmere, and Windermere North Basin, are located in northwest England. Temperature data were collected in midsummer at the deepest point of each between 9:00 and 14:00 beginning in 1991. Temperature profiles were measured at depth intervals judged in the field depending on the stratification pattern and depth of the lake. Temperature was measured with various combined temperature/oxygen sensors, including a YSI 58 (1991–2002) and WTW Oxi 340 (2002–2013) in Windermere North Basin, and a YSI 58 (1991–2002), WTW Oxi 340i (2002–2010), and Hach HQ 30d and LDO probe (2010–2013) in all other lakes.Lake Constance is located on the border of Germany, Switzerland, and Austria. Temperature profiles were measured at the deepest part of the central basin (Upper Lake Constance) of the lake beginning in 1964, with measurements taken between 9:00 and 10:00 using a thermometer. Depth intervals between samples increased with depth, with measurements taken every 2.5 to 5 m through 20 m, every 10 m through 50 m, and every 50 m down to a depth of 250 m39.Lake Mondsee is located in the Lake District “Salzkammergut” of Austria. Temperature data were collected from the deepest point of the lake approximately monthly near noon beginning in 1968. Temperature measurements were usually measured at depth intervals of 1 to 2 m in the epilimnion and 5 to 10 m intervals in the hypolimnion using a thermometer housed in a Schindler sampler prior to 1998. Beginning in 1998, data were extracted from continuous YSI 6920 profiler readings, with a YSI 6600 used beginning in 2008 and a thermistor chain from 2010–2013.Lake Müggelsee, located in Berlin, Germany was sampled weekly beginning in 1978 between 8:00 and 9:00. Temperature was measured every 0.5 to 1 m at the deepest part of the lake using a Hydrolab H2O sensor beginning in 1992, and a thermistor probe for years prior to 1992.Lake Piburgersee is located in Tyrol, Austria. Temperature was measured using a calibrated thermometer every 3 m throughout the water column in the deepest part of the lake, beginning in 1970. Additional information and analyses are available elsewhere40.Plusssee (Plußsee) is located in northern Germany (Schleswig-Holstein). Temperature data were collected from the deepest point of the lake between 9:00 and 15:00 beginning in 1971. Temperature was measured manually in 1 m increments from 0 to 15 m and in 5 m increments from 15 to 25 m using a thermometer mounted into a Ruttner sampler prior to 1976, and after 1976 with a WTW temperature/dissolved oxygen probe.Traunsee is a large and deep oligotrophic lake in the Salzkammergut lake district of Austria. Temperature data were collected at the deepest point of the lake between 9:00 and 12:00 beginning in 1965. Temperature was measured at 2 to 5 m intervals through 20 m, and at 20 m intervals through the rest of the water column using a mercury thermometer mounted in a 5-liter water sampler.Lower Lake Zurich is located in Switzerland. Temperature profiles were measured at the deepest part of this lake between 8:30 and 12:00 beginning in 1936. Measurements were taken at the deepest point in the lake using a range of sensors, mainly NTC thermistors (1936–2000). Beginning in 2001 various sondes were used, including FLP-10 multisonde (2001–2008), multisonde Hydrolab DS5 (2008–2015) at 0.5 to 1 m intervals through 30 m, 5 m intervals through 50 m, and 10 m intervals throughout the rest of the water column.Southern EuropeLake Garda is one of the largest lakes in Europe, and the largest Italian lake. Owing to its deep depth, Lake Garda is characterized by long periods of incomplete vertical winter water circulation, which are interrupted by full mixing of the water column after the occurrence of harsh winters. Limnological investigations have been carried out since 1991 in a pelagic station located at the point of maximum depth of the northwest basin. Profiles of water temperature were recorded during the summer months using Idronaut Ocean Seven 401 (1991–1997), Seacat SBE 19–03 (1998–2008), and Idronaut Ocean Seven 316Plus since 200925.Lake Iseo is located in northern Italy (Lombardy Region). It is a deep mesotrophic lake, characterized by long periods of incomplete vertical winter water circulation41. Temperature was measured at the deepest point in the lake using an automatic thermistor probe coupled with an oxygen sensor from 1993 to 2011 with Microprocessor Oximeter WTW OXI 320 and from 2012 to 2016 with Microprocessor WTW multi 3410. Temperature was measured for at least ten discrete depths and all measurements were regularly checked with a mercury-filled Celsius reversing thermometer.Lake Lugano is located in the foothills of the Central Alps, on the border between Switzerland and Italy. The lake is divided into a northern and southern basin, which are separated by a causeway (built on a natural moraine). Due to reduced connectivity42 (flow of approximately 0.38 km3 year−1 from north to south) and different morphometric characteristics, the two basins were considered separately in the data set. Temperature profiles were collected at sites near the deepest point of each basin. Temperature was measured using reversing thermometers from 1974–1979 and multiparameter probes thereafter (Hydropolyester HTP 77 during 1980–1985, Ocean Seven 401 during 1986–1993, Ocean Seven 316 during 1994–2015). As an exception, temperatures at depths greater than 100 m were also measured using a reversing thermometer between 1980–1985. Temperature was measured for at least nine or seven discrete depths for the northern and southern basins, respectively, from 1974–1986, whereas full temperature profiles with vertical resolution of 0.5 to 1 m were measured between 1987–2015.Lake Maggiore is a deep lake located in northwestern Italy and Switzerland, south of the Alps. Lake Maggiore can be classified as holo-oligomictic, with complete overturns only occurring at the end of particularly cold and windy winters. Temperature data have been collected at the deepest point of the lake since 1981, usually between 10:00 and 12:00 at the Ghiffa station. Temperature has been measured at discrete depths of 0, 5, 10, 20, 30, 50 m, and every 50 m through 360 m using mercury-filled thermometers connected to the bottle used for water sampling43.Eastern EuropeLakes Batorino, Myastro, and Naroch are located in the northwest part of Belarus, in the glacial landscape. Temperature data were collected monthly in the center of the lake during the vegetative season of May to October beginning in the 1950s and 60s. Measurements of water temperature were taken every 2 to 4 m through the water column between 9:00 and 14:00 with a mercury deepwater thermometer with a scale resolution of 0.1°C mounted in a metal frame, or in a Ruttner sampler.RussiaLake Baikal is located in Siberia, Russia. Temperature data were collected from a station situated 2.8 km from the shoreline near the Bolshie Koty settlement at depths of 0 and 50 m between 9:00 and 12:00 from 1948–2009. Temperature was measured with a mercury thermometer inside a Van Dorn bottle.Lake Glubokoe is located in Central European Russia, Moscow Province. Temperature profile data were collected from the deepest point of the lake from the surface through 10 m. Since 1982, water was taken from the required depth with a 10 L bathometer and its temperature was measured with a mercury thermometer; instrumentation prior to 1982 is unknown.Kurilskoye Lake in Kamchatka, Russia was sampled beginning in 1942. Temperature profiles were measured in the deepest part of the lake between 8:00 and 15:00. Various temperature sensors were used over time, including a reversing thermometer (1942–1965), bathythermograph (1980–2003), Hydrolab (2004–2008), and RINKO profiler (2009–2014).Lake Shira is located in the south of Siberia, Russia. Temperature profiles were measured in the deepest part of the lake between 11:00 and 15:00 from the surface to the depth of 20 to 24 m with various temperature sensors including multisonde Hydrolab 4 A (2000–2008) and a YSI 6600 (2009–2014).Middle EastLake Kinneret is located in Israel (Jordan Valley). Temperature data were collected near the deepest point of the lake (Station A) between 7:00 and 16:00 from 1969–2013, measured every centimeter with an error of ± 0.005 °C, and averaged to every 1 m. Temperature measurements were taken from 1969 to 1986 using an underwater thermometer (Whitney-Montedoro), from 1987 to 2003 using a STD-12 Plus (Applied Microsystems), and from 2003 to 2013 using AML Oceanographic Minos•X.AsiaLake Biwa is located in the central part of the Japanese archipelago (Shiga Prefecture, Japan). Temperature data were collected from a station near the deepest part of the lake from 1958–2010. Temperature was mostly measured between 9:00 and 12:00 at 5 m intervals from 1959–2005, and at 1 m intervals beginning in 2006. Measurements were made using an electronic thermometer (Murayama Denki Ltd.) from 1958–1970, a thermistor thermometer (Shibaura electronics, HCB III) from 1970–1994, a CTD profiler (Alec electronics, ABT-1) from 1994–2006, and a CTD profiler (JFE Advantech, compact-CTD) from 2007–2010.AustraliaLake Burley Griffin is a reservoir constructed in 1963 by damming the Molonglo River. It is located in the geographic center of Canberra, the capital of Australia. Temperature data were collected from the deepest point of the reservoir, near the dam wall. Profiles were measured in 1 m depth intervals (reduced to 3 m intervals in 1992) between 8:45 and 16:15 from 1981–2010 by the National Capital Authority.Lakes Samsonvale (North Pine) and Somerset are located on the east coast of Australia, in southeast Queensland. Temperature in each lake was measured at a site approximately 100 m from the dam wall. Samsonvale’s (North Pine’s) temperature was measured using a YSI 6560 sensor on a YSI 6600 V2 sonde beginning in 2009 continuously over a 24-hour period at 1 m intervals. Prior to 2009, temperature was measured via a thermistor string with various unknown instruments. Somerset’s temperature profiles from 2000–2002 were measured using temperature sensors on a thermistor string, spaced at 0.5 m intervals through 3 m, 1 m intervals through 7 m, 2 m intervals through 17 m, and 3 m intervals for the rest of the water column.New ZealandLakes Brunner and Taupo are located in New Zealand, in the West Coast and Waikato regions, respectively. Temperature profiles were measured at the deepest part of these lakes between 9:00 and 16:00, beginning in the early 1990s. Temperature was measured by lowering CTD profilers through the water column, using an YSI EXO sonde (Brunner) and RBR profiler (Taupo).Lakes Okareka, Okaro, Okataina, Rerewhakaaitu, Rotoehu, Rotoiti, Rotoma, Rotorua, Tarawera, and Tikitapu are located in Rotorua, Bay of Plenty, New Zealand. Temperature profiles were measured between 10:00 and 14:00 in the central basin using Seatech CTD casts with a Seabird 19Plus or 19PlusV2 beginning in 2003. Temperature was measured at a frequency of 4 Hz during each cast and data were binned to 1 m depth intervals. Profiles before 2003 were measured in 1 m depth intervals with either a YSI Water Quality Logger 3800 or YSI Sonde model 3815.AntarcticaLakes Heywood (1962–1995), Moss (1972–2003), and Sombre (1973–2003) are located in the South Orkney Islands, Antarctica. Temperature data from these lakes were collected using a Mackareth-type probe at 1 to 2 m intervals in the deepest part of the lake. More

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Acoustic measurementsA set of acoustic echo sounder data was used to analyze fish density. This was collected within the Mycto-3D-MAP program using split-beam echo sounders at 38 and 120 kHz. The Mycto-3D-MAP program included multiple large-scale oceanographic surveys during 2 years and a dedicated cruise in the Kerguelen area. The dataset was collected during 4 large-scale surveys in 2013 and 2014, both in summer (including both northward and southward transects) and in winter, corresponding to 6 acoustic transects of 2860 linear kilometers (see Table 1 for more details). Note that all legs except summer 2014 (MYCTO-3D-Map cruise) were logistic operations, during which environmental in situ data (such as temperature or salinity profiles) could not be collected. The data were then treated with a bi-frequency algorithm, applied to the 38 and 120 kHz frequencies (details of data collection and processing are provided in37). This provides a quantitative estimation of the concentration of gas-bearing organisms, mostly attributed to fish containing a gas-filled swimbladder in the water column38. Most mesopelagic fish present swimbladders and several works pointed out that myctophids are the dominant mesopelagic fish in the region39. Therefore, we considered the acoustic signal as mainly representative of myctophids concentration. Data were organized in acoustic units, averaging acoustic data over 1.1 km along the ship trajectory on average. Myctophid school length is in the order of tens of meters40. For this reason, acoustic units were considered as not autocorrelated. Every acoustic unit is composed of 30 layers, ranging from 10 to 300 meters (30 layers in total).The dataset was used to infer the Acoustic Fish Concentration (AFC) in the water column. We considered as AFC of the point ((x_i), (y_i)) of the ship trajectory the average of the bifrequency acoustic backscattering of 29 out of 30 layers (the first layer, 0-10 m, was excluded due to surface noise) AFC quantity is dimensionless.As the previous measurements were performed through acoustic measurements, a non-invasive technique, fishes were not handled for this study.Table 1 Details of the acoustic transects analyzed.Full size tableRegional data selectionThe geographic area of interest of the present study is the Southern Ocean. To select the ship transects belonging to this region, we used the ecopartition of41. Only points falling in the Antarctic Southern Ocean region were considered. We highlight that this choice is consistent with the ecopartition of42 (group 5), which is specifically designed for myctophids, the reference fish family (Myctophidae) of this study. Furthermore, this choice allowed us to exclude large scale fronts (i.e., fronts that are visible on monthly or yearly averaged maps) which have been the subject of past research works43,44. In this way our analysis is focused specifically on fine-scale fronts.Day-night data separationSeveral species of myctophids present a diel vertical migration. They live at great depths during the day (between 500 and 1000 m), ascending at dusk in the upper euphotic layer, where they spend the night. Since the maximal depth reached by the echo sounder we used is 300 m, in the analysis reported in Figs. 2 and 3 we excluded data collected during the day. However, their analysis is reported in SI.1. A restriction of our acoustic analyses to the ocean upper layer is also consistent with the fact that the fine-scale features we computed are derived in this work by satellite altimetry, thus representative of the upper part of the water column ((sim 50) m). Finally, we note that the choice of considering the echo sounder data in the first 300 m of the water columns is coherent with the fact that LCS may extend almost vertically in depth even at 600 m depth45,46 and with the fact that altimetry-derived velocity fields are consistent with subsurface currents in our region of interest down to 500 m20.Satellite dataVelocity current data and Finite-Size Lyapunov Exponent (FSLE) processing. Velocity currents are obtained from Sea Surface Height (SSH), which is measured by satellite altimetry, through geostrophic approximation. Data, which were downloaded from E.U. Copernicus Marine Environment Monitoring Service (CMEMS, http://marine.copernicus.eu/), were obtained from DUACS (Data Unification and Altimeter Combination System) delayed-time multi-mission altimeter, and displaced over a regular grid with spatial resolution of (frac{1}{4}times frac{1}{4}^circ) and a temporal resolution of 1 day.Trajectories were computed with a Runge-Kutta scheme of the 4th order with an integration time of 6 hours. Finite-size Lyapunov Exponents (FSLE) were computed following the methods in14, with initial and final separation of (0.04^circ) and (0.4^circ) respectively. This Lagrangian diagnostic is commonly used to identify Lagrangian Coherent Structures. This method determines the location of barriers to transport, and it is usually associated with oceanic fronts9. Details on the Lagrangian techniques applied to altimetry data, including a discussion of its limitation, can be found in10.Temperature field and gradient computation The Sea Surface Temperature (SST) field was produced from the OSTIA global foundation Sea Surface Temperature (product id: SST_GLO_ SST_L4_NRT_OBSERVATIONS_010_001) from both infrared and microwave radiometers, and downloaded from CMEMS website. The data are represented over a regular grid with spatial resolution of (0.05times 0.05^circ) and daily-mean maps. The SST gradient was obtained from:$$begin{aligned} Grad SST=sqrt{g_x^2+g_y^2} end{aligned}$$where (g_x) and (g_y) are the gradients along the west-east and the north-south direction, respectively. To compute (g_x), the following expression was used:$$begin{aligned} g_x=frac{1}{2 dx}cdot (SST_{i+1}-SST_{i-1}) end{aligned}$$where the SST values of the adjacent grid cells (along the west-east direction: cells (i+1) and (i-1)) were employed. dx identifies the kilometric distance between two grid points along the longitude (which varies with latitude). The definition is analog for (g_y), considering this time the north-south direction and (dysimeq 5) km (0.05(^circ)).Chlorophyll field Chlorophyll estimations were obtained from the Global Ocean Color product (OCEANCOLOUR_ GLO_CHL_L4_REP_OBSERVATIONS_009_082-TDS) produced by ACRI-ST. This was downloaded from CMEMS website. Daily observations were used, in order to match the temporal resolution of the acoustic and satellite observations. The spatial resolution of the product is 1/24(^{circ }).Estimation of satellite data along ship trajectory For each point ((x_i), (y_i)) of the ship trajectory, we extracted a local value of FSLE, SST gradient, and chlorophyll concentration. These were obtained by considering the respective average value in a circular around of radius (sigma) of each point ((x_i), (y_i)) . Different (sigma) were tested (ranging from 0.1(^circ) to 0.6(^circ)), and the best results were obtained with (sigma =0.2^circ), reference value reported in the present work. This value is consistent with the resolution of the altimetry data.Statistical processingFront identification FSLE and SST gradient were used as diagnostics to detect frontal features, since they are typically associated with front intensity and Lagrangian Coherent Structures10. Note that the two diagnostics provide similar but not identical information. FSLEs are used to analyze the horizontal transport and to identify material lines along which a confluence of waters with different origins occur. These lines typically display an enhanced SST gradient because water masses of different origin have often contrasted SST signature. However, this is not a necessary condition. FSLE ridges may be invisible in SST maps if transport occurs in a region of homogeneous SST. Conversely, SST gradient unveils structures separating waters of different temperatures, whose contrast is often – but not always – associated with horizontal transport. Therefore, even if they usually detect the same structures, these two metrics are complementary. Frontal features were identified by considering a local FSLE (or SST gradient, respectively) value larger than a given threshold. The threshold values have been chosen heuristically but consistently with the ones found in previous works. For the FSLEs, we used 0.08 days(^{-1}), a threshold value in the range of the ones chosen in18 and47. For the SST gradient, we considered representative of thermal front values greater than 0.009({^circ })C/km, which is about half the value chosen in47. However, in that work, the SST gradient was obtained from the advection of the SST field with satellite-derived currents for the previous 3 days, a procedure which is known to enhance tracer gradients.Bootstrap method The threshold value splits the AFC into two groups: AFC co-located with FSLE or SST gradient values over the threshold are considered as measured in proximity of a front (i.e., statistically associated with a front), while AFC values below the threshold are considered as not associated with a frontal structure. The statistical independence of the two groups was tested using a Mann-Whitney or U test. Finally, bootstrap analysis is applied following the methodologies used in47. This allows estimating the probability that the difference in the mean AFC values, over and under the threshold, is significant, and not the result of statistical fluctuations. Other diagnostics tested are reported in SI.1.Linear quantile regression Linear quantile regression method48 was employed as no significant correlation was found between the explanatory and the response variables. This can be due to the fact that multiple factors (such as prey or predator distributions) influence the fish distribution other than the frontal activity considered. The presence of these other factors can shadow the relationship of the explanatory variables (in this case, the FSLE and the SST gradient) with the mean value of the response variable (the AFC). A common method to address this problem is the use of the quantile regression48,49, that explores the influence of the explanatory variables on other parts of the response variable distribution. Previous studies, adopting this methodology, revealed the limiting role played by the explanatory variables in the processes considered50. The percentiles values used are 75th, 90th, 95th, and 99th. The analysis is performed using the statistical package QUANTREG in R (https://CRAN.R-project.org/package=quantreg, v.5.3848,51), using the default settings.Chlorophyll-rich waters selection The AFC observations were considered in chlorophyll-rich waters if the corresponding chlorophyll concentration was higher than a given threshold. All the other AFC measurements were excluded, and a linear regression performed between the remaining AFC and FSLE (or SST gradient) values. The corresponding thresholds (one for FSLE and one for SST gradient case) were selected though a sensitivity test reported in SI.1. These resulted in 0.22 and 0.17 mg/m(^3) for FSLE and for SST gradient, respectively. These values are consistent among them and, in addition, they are in coherence with previous estimates of chlorophyll concentration used to characterise productive waters in the Southern Ocean (0.26mg/m(^3)52).Gradient climbing modelAn individual-based mechanistic model is built to describe how fish could move along frontal features. We assume that the direction of fish movement along a frontal gradient is influenced by the noise of the prey field (SI. 2). Specifically, we consider a Markovian process along the (one dimensional) cross-front direction. Time is discretized in timesteps of length (varDelta tau). We presuppose that, at each timestep, the fish chooses between swimming in one of the two opposite cross-front directions (“left” and “right”). This decision depends on the comparison between the quantity of a tracer (a cue) present at its actual position and the one perceived at a distance (p_R) from it, where (p_R) is the perceptual range of the fish. We consider the fish immersed in a tracer whose concentration is described by the function T(x).An expression for the average velocity of the fish, (U_F(x)), can now be derived. We assume that the fish is able to observe simultaneously the tracer to its right and its left. Fish sensorial capacities are discussed in SI.2. The tracer observed is affected by noise. Noise distribution is considered uniform, with (-xi _{MAX}{tilde{T}}(x_0-varDelta x)), the fish will move to the right, and, vice versa, to the left. We hypothesize that the observational range is small enough to consider the tracer variation as linear, as illustrated in Fig. S7 (SI. 3). In this way:$$begin{aligned}&{tilde{T}}(x_0+varDelta x)=T(x_0)+ p_R,frac{partial T}{partial x}+xi _1 \&{tilde{T}}(x_0-varDelta x)=T(x_0)- p_R,frac{partial T}{partial x}+xi _2 ;. end{aligned}$$In case of absence of noise, or with (xi _{MAX}p_R,frac{partial T}{partial x}). If (T(x_0+varDelta x) >T(x_0-varDelta x)) (as in Fig. S7), and the fish will swim leftward if$$begin{aligned} xi _1-xi _2 >2p_R,frac{partial T}{partial x}; . end{aligned}$$Since (xi _1) and (xi _2) range both between (-xi _{MAX}) and (xi _{MAX}), we can obtain the probability of leftward moving P(L). This will be the probability that the difference between (xi _1) and (xi _2) is greater than (2p_R,frac{partial T}{partial x})$$begin{aligned} P(L)&=frac{1}{8xi _{MAX}^2} bigg (2 xi _{MAX} – 2 p_R,frac{partial T}{partial x}bigg )^2\&=frac{1}{2} bigg (1-frac{p_R}{xi _{MAX}},frac{partial T}{partial x}bigg )^2 end{aligned}$$.The probability of moving right will be$$begin{aligned} P(R)&=1-P(L) end{aligned}$$and their difference gives the frequency of rightward moving$$begin{aligned} P(R)-P(L)&=1-2P(L)=1-bigg (1-frac{p_R}{xi _{MAX}},frac{partial T}{partial x}bigg )^2\&=frac{p_R}{xi _{MAX}}frac{partial T}{partial x}bigg (2-frac{p_R}{xi _{MAX}}bigg |frac{partial T}{partial x}bigg |bigg ); , end{aligned}$$where the absolute value of (frac{partial T}{partial x}) has been added to preserve the correct climbing direction in case of negative gradient. The above expression leads to:$$begin{aligned} U_F(x)=frac{V p_R}{xi _{MAX}}frac{partial T}{partial x}bigg (2-frac{p_R}{xi _{MAX}}bigg |frac{partial T}{partial x}bigg |bigg );. end{aligned}$$
(1)
We then assume that, over a certain value of tracer gradient (frac{partial T}{partial x}_{MAX}), the fish are able to climb the gradient without being affected by the noise. This assumption, from a biological perspective, means that the fish does not live in a completely noisy environment, but that under favorable circumstances it is able to correctly identify the swimming direction that leads to higher prey availability. This means that$$begin{aligned} p_R*frac{partial T}{partial x}_{MAX}=xi _{MAX},. end{aligned}$$
(2)
Substituting then (2) into (1) gives:$$begin{aligned} U_F(x)=V frac{frac{partial T}{partial x}}{frac{partial T}{partial x}_{MAX}}bigg (2-frac{big |frac{partial T}{partial x}big |}{frac{partial T}{partial x}_{MAX}}bigg );. end{aligned}$$
(3)
Finally, we can include an eventual effect of transport by the ocean currents, considering that the tracer is advected passively by them, simply adding the current speed (U_C) to Expr. (3).The representations provided are individual based, with each individual representing a single fish. However, we note that all the considerations done are also valid if we consider an individual representing a fish school. (U_F) will then simply represent the average velocity of the fish schools. This aspect should be stressed since many fish species live and feed in groups, especially myctophids (see SI.2 for further details).Continuity equation in one dimension The solution of this model will now be discussed. The continuity equation is first considered in one dimension. The starting scenario is simply an initially homogeneous distribution of fish, that are moving in a one dimensional space with a velocity given by (U_{F}(x)).We assume that in the time scales considered (few days to some weeks), the fish biomass is conserved, so for instance fishing mortality or growing rates are neglected. In that case, we can express the evolution of the concentration of the fish (rho) by the continuity equation$$begin{aligned} frac{partial rho }{partial t}+nabla cdot mathbf{j },=,0 end{aligned}$$
(4)
in which (mathbf{j }=rho ;U_{F}(x)), so that Eq. (4) becomes$$begin{aligned} frac{partial rho }{partial t}+nabla cdot big (rho ;U_{F}(x)big ),=,0;. end{aligned}$$
(5)
In one dimension, the divergence is simply the partial derivate along the x-axis. Eq. (5) becomes$$begin{aligned} frac{partial rho }{partial t}=-frac{partial }{partial x} bigg (rho ;U_{F}bigg ) end{aligned}$$
(6)
Now, we decompose the fish concentration (rho) in two parts, a constant one and a variable one (rho ,=,rho _0+{tilde{rho }}). Eq. (6) will then become$$begin{aligned} frac{partial rho }{partial t}=-U_Ffrac{partial {tilde{rho }}}{partial x}-rho frac{partial U_F}{partial x};. end{aligned}$$
(7)
Using Expr. (3), Eq. (7) is numerically simulated with the Lax method. In Expr. (3) we impose that (U_F(x)=V) when (U_F(x) >V). This biological assumption means that fish maximal velocity is limited by a physiological constraint rather than by gradient steepness. Details of the physical and biological parameters are provided in SI.6. More