Philippa Kaur

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In this section, we provide quantitative evaluation for both spatial coverage and temporal dynamics of ReaLSAT dataset.Spatial coverageSince the dataset was created using satellite imagery analysis, it can provide more comprehensive coverage than existing datasets. However, using an automated process also has its challenges. It can invariably lead to the detection of spurious waterbodies because of issues in data (e.g., due to errors in GSW maps used as inputs in ReaLSAT).To provide more insights into the types of lakes and potential issues in the spatial coverage of ReaLSAT, we randomly sampled 5,000 lakes out of 435,717 that are only present in ReaLSAT (i.e., not available in the HydroLAKES dataset). A human annotator used Google’s satellite imagery base layer to categorize these lakes. Figure 5a shows the geographical distribution of these lakes, and Fig. 5b shows the distribution of different lake types in the sample set. Out of the 5,000 lakes, the human annotator identified 2,019 traditional lakes and reservoirs where sufficient water was visible in the satellite imagery. Another 551 lakes in the sample set showed signs of a bowl-like depression but with no (or very little) water visible in the satellite imagery and were labeled as ephemeral. There were 861 other lakes that were tagged as farm ponds because they showed geometric patterns of farming in the imagery. This diversity of waterbody types discovered by ReaLSAT that were previously unreported by HydroLakes highlights one of the strengths of our approach. In limnology, the origin/type of lake is a very important regulator of ecosystem dynamics. For instance, reservoirs will have faster water flow/lower residence time than natural lakes, and therefore nutrient and carbon processing rates will differ; floodplain lakes may dry periodically, leading to the denudation of sediments; and farm ponds will likely have much higher rates of nutrient loading and methane production than non-agriculturally influenced lakes. Hence, capturing a more comprehensive range of waterbody categories can enable various scientific studies where knowing the origin/lake type could provide a critical understanding of the process.Fig. 5(a) Geographic location of 5000 randomly selected lakes used for manual evaluation of lake type. (b) Allocation of the 5000 manually referenced lakes to specific lake types. Regular implies a traditional lake or reservoir. Unverifiable implies that the lake type could not be identified based on the available Google Earth imagery.Full size imageAlong with the lentic water types discovered in the sampled set, we also found that ReaLSAT identified 603 river segments missed by our morphological score filter. As stated earlier, this is an inherent challenge with automated approaches that use a fixed score threshold for eliminating river segments. Another 239 lakes were tagged as wetlands because of significant vegetation inside and around the lake polygon. There were also 97 lakes that were adjacent to rivers, which were labeled as riverine or floodplain lakes that were formed as a result of river channels meandering over time. Furthermore, there were 59 lakes where the polygons represented only a small portion of a larger lake and were labeled as partial. Finally, for 571 polygons, there was not enough evidence to tag them in any of the above categories. Since Google imagery represents only a single snapshot in time, these 571 waterbodies could not be definitively labeled as spurious (hence, they were labeled as unverifiable), highlighting a limitation of this evaluation pipeline. In particular, a vast majority of these waterbodies appear to be ephemeral based on their surface area timeseries (completely dry for extended periods of time). Hence, if the satellite imagery layer is from one of these timesteps, the annotator would not be able to confirm the presence of the lake.To assess whether we would obtain a similar distribution of different waterbody categories in existing datasets, we performed a similar evaluation on another 5,000 lakes sampled from ReaLSAT where each polygon has some overlap (greater than 1 pixel) with a polygon from HydroLAKES. In this sampled set, the annotator identified 4,030 lakes as traditional lakes or reservoirs, 370 as ephemeral, 138 as farm ponds, 6 as river segments, 66 as wetlands, 95 as riverine or floodplain lakes, 20 as partial, and 275 as unverifiable.Compared to previous distribution, this set of 5,000 waterbodies contains relatively fewer river segments and wetlands polygons in HydroLAKES, because these categories were manually identified and removed during HydroLAKES database creation6. Similary, this set contains relatively few farm ponds because HydroLAKES was created by manual curation of existing static databases and hence does not contain new farm ponds that got created over the years.Temporal dynamicsTo assess the quality of surface extent maps, we performed a quantitative evaluation on a random selection of extent maps. These extent maps were compared against reference maps created by a human annotator using a semi-automated pixel classification procedure. This strategy of creating reference maps is used extensively in the remote sensing literature (e.g. see36,37,38,39). Next, we describe our evaluation process in detail.Sample selectionThere are 462,574 lakes out of 681,137 total lakes where the label updates (corrections and imputations) by the ORBIT approach have trust scores within our chosen thresholds (as described in the methods section). To evaluate these candidate lakes effectively, we focus on lake extent maps where the ORBIT approach resulted in a different map than the underlying GSW extent based map. Hence, we remove maps where no updates were made by the ORBIT approach (neither corrections nor imputations) from the candidate pool of extent maps used for evaluation. We also remove maps where the percentage of missing labels was more than 90% because these maps tend to suffer from significant cloud coverage. Hence, it would be challenging to generate reference maps. Since the GSW dataset has a significant amount of missing data for most places in the world before 2000, we evaluated maps only from 2000 onwards. These three filters left us with a total of 51,077,278 water extent maps considered for selection. Figure 6a shows the distribution of percentage pixels updated made by the ORBIT approach in these water extent maps. To evaluate the robustness of our approach in comparison to GSW maps, we randomly selected 10,000 water extent maps such that extents with significant updates are given higher weight to reduce the skew in distribution towards extents with relative less updates (Fig. 6b).Fig. 6Distribution of updates made by the ORBIT approach. (a) distribution using candidate water extents (b) distribution using randomly selected 10000 water extent maps for evaluation.Full size imageSample pruningFrom these randomly selected water extent maps, we removed maps for which a reference map could not be generated due to clouds or the inability of the annotator to distinguish between land and water. A final set of 2,095 water extent maps were considered for evaluation. Figure 7a shows the distribution of percentage updates in the final set of evaluation extents and Fig. 7b shows the geographical distribution of these extent maps.Fig. 7Summary of the dataset used for evaluating water extent maps. (a) Distribution of updates made by the ORBIT approach in the water extent maps selected for evaluation. (b) Geographical location of the lakes in the evaluation set.Full size imageReference map generationFor these water extent maps, we created ground truth reference maps using a semi-automatic labeling process37,38,39. Specifically, the annotator selects land and water samples to train an SVM (Support Vector Machine) classification model for each image. The annotator keeps adding samples until a stable map is generated. As a final step, the annotator masks out pixels affected by clouds, cloud shadows, and any other region where the annotator is not confident about the accuracy of the reference labels. This process enables a quick and robust generation of reference maps. Supplementary Fig. S7 shows one of the reference maps in the evaluation set. While this strategy of comparing maps is different from the traditional approach of comparing pixels (often selected using stratified sampling), it provides a much more exhaustive evaluation of surface extent maps. The reference maps used for evaluation in this study are also available as part of the dataset.ComparisonTo compare the extent maps generated by ReaLSAT with the reference maps, we used accuracy as the evaluation metric, a widely used metric to measure the quality of classification maps. Accuracy is simply defined as the ratio of pixels with correct labels over a total number of pixels. Specifically, we assign 1 to water pixels and 0 to land pixels. Since GSW based extent maps contain missing labels, they are assigned a value of 0.5 to reflect the uncertainty between land and water. Accuracy is then calculated as follows:$$Accuracy=1-frac{1}{Rast C}mathop{sum }limits_{i=1}^{R}mathop{sum }limits_{j=1}^{C}left|ReferenceMap[i,j]-PredictedMap[i,j]right|$$
(2)
where, R is the number of rows and C is the number of columns of the map.When the accuracy of RealSAT and GSW labels are compared, a vast majority of points lie above the diagonal 1:1 line, which implies that ReaLSAT labels were more accurate overall (Fig. 8a). In Fig. 8 the points are colored based on % of pixels where GSW labels were missing. To better show the improvement in RealSAT labeling, we plot the distribution of the difference in accuracy values between the two datasets as shown in Fig. 8b. A positive value indicates that the surface extent map from the ReaLSAT dataset had better accuracy than the map from the GSW dataset and vice versa. For ease of visualization, we plot this distribution after excluding cases where the accuracy from both datasets was equal. The positively skewed distribution demonstrates the efficacy of the ORBIT approach.Fig. 8Comparison of accuracy values using GSW labels vs ReaLSAT labels. (a) Scatter plot of accuracy values using GSW labels vs ReaLSAT labels. (b) Histogram of difference in accuracy between ReaLSAT labels vs GSW labels. Positive value represents cases where ReaLSAT labels were more accurate than GSW labels. (c) Histogram of difference in accuracy values for the scenario where pixels labelled as land by both products as well as ground truth were removed to reduce the skew of surrounding land pixel on the accuracy values.Full size imageNote that the shape of a lake will influence the number of land pixels surrounding it, which might bias the accuracy values. For example, the reference map shown in Supplementary Fig. S7 contains more than 70% of land pixels. To address this bias, we also calculated accuracy values after removing pixels that were labeled as land by both datasets as well as the ground truth. This variation allows a more strict evaluation of water extent maps. Figure 8c shows the distribution of the difference in accuracy values under this scenario (after excluding cases with equal accuracy). As shown, a vast majority of the distribution is still towards positive values. Furthermore, the distribution has a larger spread towards high positive values, suggesting significant improvement made by the ORBIT approach.From Fig. 8, we can see that for some cases ReaLSAT based extent maps are less accurate relative to GSW. As described earlier, violation of assumptions made by the ORBIT approach could lead to the observed poor performance. Out of 2,095 extent maps, GSW labels show better accuracy than ReaLSAT for 323 of them. On visual analysis of errors in these maps, we found that 165 maps are slightly different only at the lake’s boundary. We categorized the remaining extent maps based on the reason behind the observed poor performance. In particular, 45 maps have poor performance due to occlusion of water surface by algae, 18 maps contain farm ponds, 8 contain mining lakes, 27 maps have unreliable bathymetry, 30 maps have issues due to the weighting factor used by ORBIT approach, and 30 maps have class conditional missing data. All the reference maps and corresponding maps from GSW and ReaLSAT are provided with the dataset.Next, we describe some of these cases in detail.Impact of algae: It can be difficult to visually differentiate surface algae or floating aquatic plants from terrestrial vegetation40, as they have similar reflectance spectra. Therefore, surface algal blooms often get incorrectly labeled as land in the reference maps. However, in most cases, the appearance and disappearance of algae on a lake are independent of the bathymetry. Thus, algae pixels get detected as physically inconsistent by the ORBIT approach, and consequently, these pixels are updated based on the labels of other pixels without algae. In many cases, while the accuracy with respect to the reference map is poor (because algae get labeled as land), ReaLSAT based extent maps are closer to the true extent of the lake. For example, Supplementary Fig. S8 illustrates the impact of algae on the extent mapping of Center Lake, Texas. In this example, the bimodal distribution of fraction values (either low or high) reveals high confidence in lake persistence (Supplementary Fig. S8b). On Oct 22, 2008, false-color composite processing of LANDSAT-5 imagery reveals a strong vegetative signal on the west side of the lake (Supplementary Fig. S8c). Since we know that this is a lake, we can assume that the west side of the lake is experiencing a large surface algal bloom with a similar reflectance to the surrounding terrestrial landscape. Because of the strong vegetative reflectance signal, the semi-automated reference mapping labels the west side of the lake as land (Supplementary Fig. S8d), as does most GSW labels (Supplementary Fig. S8e). Conversely, the ReaLSAT extent map labels the west side of the lake as water (Supplementary Fig. S8f). However, we calculate accuracy based on the semi-automated reference map (Supplementary Fig. S8d). Due to this, the GSW extent map is considered more accurate than the ReaLSAT map, even though this is not true because the reference map is incorrectly labeled. Therefore, some negative accuracy values may be a misrepresentation of reality due to surface algal blooms.Impact of variable bathymetry: Even though we tried to remove lakes with unreliable bathymetry by using score-based filters defined in an earlier section, not all cases were removed. For example, agricultural ponds often have small sections that are connected and change shape based on agricultural needs. Supplementary Fig. S9 highlights an example of labeling issues on agricultural ponds in Mexico. In this area, satellite imagery and the GSW fraction map confirm the presence of agricultural ponds (Supplementary Fig. S9a,b). These individual ponds are filled and drained based on operational decisions and do not follow a consistent pattern of growing or shrinking. Thus, the ORBIT approach can introduce spurious updates in water extent maps for these farms. In the Landsat-5 imagery from 2009–10–08, some of the ponds are dry, while others are filled (Supplementary Fig. S9c). This distribution of water is evident from a visual inspection and is confirmed in the semi-automated reference map (Supplementary Fig. S9d). Due to the similar elevations between the individual pond sections, the ORBIT approach spuriously fills the remaining sections with water based on the incorrectly learned bathymetry (Supplementary Fig. S9f). While quantification of such uncertainties is outside the scope of this paper, we hope that the wider research community can use RealSAT to address such questions. In particular, changes in bathymetry of a lake can be identified using spatial-temporal patterns in the label corrections. Specifically, if the elevation of some pixels in a lake increases after a certain time (e.g., sediment deposits leading to increase the elevation of a pixel), they will appear as physically inconsistent to the ORBIT framework, and hence the labels for these locations will be changed from land to water much more frequently after this increase in elevation.Impact of bias in errors and missing data: As mentioned earlier in the methods section, based on our observation, the confidence of water labels is higher than land labels in the GSW dataset. To account for this bias, we used a weighting factor of 3 for the water class. While this weighting factor improves the ORBIT approach’s performance in most cases, this assumption leads to an overestimation of water for some lakes. For example, Supplementary Fig. S10 compares the water extent maps with and without the weighting factor for a small reservoir in eastern Brazil. As we can see, the GSW labels contain false positives, and due to the weighting factor of 3, ORBIT prefers to update the land labels to water which further increases the number of false positives, as shown in Supplementary Fig. S10e. However, if we use a weighting factor of 1 for this example, the ORBIT approach can effectively remove many of the false positives in the GSW map, as shown in Supplementary Fig. S10f.Similarly, apart from missing data due to clouds in the GSW dataset, there can also be missing values on pixels where the GSW classification model is not confident. Hence, for some water extent maps, class-dependent missing data (compared to missing data which is class independent) adversely impact the ORBIT approach. For example, Supplementary Fig. S11 shows a water extent map for Zhongleng Reservoir in China, where missing data along the eastern edges is not independent but has resulted from ambiguous pixels around the lake where the GSW’s approach was not confident. In such a scenario, the ORBIT approach heavily relies on information from nearby timesteps to infer labels for missing pixels, leading to errors in ReaLSAT maps if there is a significant variation in lake extent in nearby timesteps, as shown in Supplementary Fig. S11e. More

Current scientific definitions of pondsWe compiled existing scientific definitions of ponds by conducting a backwards and forwards search of papers referenced in or subsequently referencing three seminal pond papers8,17,18 (see “Methods”). We ultimately compiled 54 pond definitions from scientific literature (data available19). The variables most often included in definitions were surface area (91% of definitions), depth (48%), permanence (48%), origin (i.e., natural or human-made; 33%), and standing water (33%; Fig. 2a). When surface area or depth were included in definitions, they were often mentioned qualitatively (e.g., “small” and “shallow”). Of the 61% of definitions that included a maximum pond surface area, the range was 0.1 to 100 ha, the median was 2 ha, and all but two definitions were ≤ 10 ha (Fig. 2b). For depth, only 17% of studies provided a maximum depth cutoff, which ranged 2 to 8 m (Fig. 2c). Of the 26 definitions mentioning permanence, 22 stated that ponds could be temporary or permanent and only three indicated that ponds are exclusively permanent waterbodies. Of the 18 definitions mentioning origin, 17 mentioned that ponds could be natural or human-made with the remaining study indicating ponds can have diverse origins.Figure 2Summary of “pond” definitions from scientific literature including (a) presence of various morphological, biological, and physical characteristics in the definition as blue bars (n = 54 definitions total). Bold black lines indicate the number of definitions with surface area and depth values. Histograms of the upper limits from “pond” definitions for (b) surface area and (c) maximum depth.Full size imageOther important factors included in definitions related to morphometry. For example, 30% of definitions mentioned the potential for plants to colonize the entire basin, which relates to high light penetration (mentioned in 11% of definitions) and/or shallow depths. For example, Wetzel11 defines ponds as having enough light penetration that macrophyte photosynthesis can occur over the entire waterbody. As such, these conditions may be comparable to the littoral region of lakes (11% of definitions). Lastly, 7% of pond definitions mentioned mixing versus stratification, whereby ponds mix more than lakes20 yet less than shallow lakes due to a smaller fetch16.To assess if there was agreement in pond definitions among papers, we examined the number of times each definition was cited. Across the 54 definitions, there were 89 citations of 48 unique papers. Ultimately, most papers (75%) were only cited only once, indicating no consensus in pond definition. The most cited paper was Biggs et al.21, which accounted for 15% of citations. The next two most cited papers were Oertli et al.17 and Sondergaard et al.18, which were seminal papers included in our backwards-forwards search, and each comprised 8% of citations.International definitionsAt an international level, there is no consensus on how to discriminate among ponds, lakes, and wetlands. In North America, wetlands are generally considered to be shallow:  More

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Mathematical theoryWe consider a biological response (e.g. body size, survival, biodiversity) to two environmental drivers (i.e. any abiotic or biotic factor) but the same idea may be applied to a larger number of drivers. The response depends of a set of predictors consisting in the magnitudes (m1 and m2) and time scales of fluctuation of two drivers (i = 1, 2); in addition, the response is quantified at least once after the fluctuations have been experienced (Fig. 1a).Time is defined using two different frames; chronological (= clock) time (measured by clocks) and biological time. For the “clock” time scales of the fluctuations (t1, t2) there are associated biological times (τ1, τ2). Likewise, for the clock time at which the response is quantified (({t}^{^*})) there is an associated biological time (τ({^*})).Biological time is the proportion of (clock) time needed to reach a life history event (e.g. moulting, maturity). Hence, for t1, t2 and t({^*}) we obtain τi = ti/D and τ({^*}) = t({^*})/D, (D = clock time needed to reach such life history event). We express the τi and τ({^*}) in terms of a function L = 1/D. For instance, for t({^*}) we obtain:$${tau }^{^*}={t}^{^*}cdot{L}$$
(1)
where L = L(ω) = D−1(ω) characterises the timing of a life history event (with units as the inverse of clock time units). L depends on the set of predictors ω associated to the fluctuations; an important set of predictors will be defined by thermal fluctuations (the amplitude and time scales), which in ectotherm species have a strong influence on developmental time32,33. We find by differentiation that L provides the transform function between clock and biological time frames; for instance, if L does not depend on any ti we have L = dτ/dti.The response is expressed as a function of the predictors defined above, as R(m1, m2, t1, t2, t({^*})) = r[m1, m2, τ1 (t1), τ2(t2), τ({^*})]. The contribution of each predictor to the response is better understood by the partial derivatives with respect to each predictor; this defines a system of partial differential equations (PDE; Supplementary note 1) which expressed in matrix form give the following matrix equation.$$left[begin{array}{c}frac{dR}{d{m}_{1}}\ frac{dR}{d{m}_{2}}\ frac{dR}{d{t}_{1}}\ frac{dR}{d{t}_{2}}\ frac{dR}{d{t}^{^*}}end{array}right]=left[begin{array}{ccccc}1& frac{d{m}_{2}}{d{m}_{1}}& frac{d{tau }_{1}}{d{m}_{1}}& frac{d{tau }_{2}}{d{m}_{1}}& frac{d{tau }^{^*}}{d{m}_{1}}\ frac{d{m}_{1}}{d{m}_{2}}& 1& frac{d{tau }_{1}}{d{m}_{2}}& frac{d{tau }_{2}}{d{m}_{2}}& frac{d{tau }^{^*}}{d{m}_{2}}\ frac{d{m}_{1}}{d{t}_{1}}& frac{d{m}_{2}}{d{t}_{1}}& frac{d{tau }_{1}}{d{t}_{1}}& frac{d{tau }_{2}}{d{t}_{1}}& 0\ frac{d{m}_{1}}{d{t}_{2}}& frac{d{m}_{2}}{d{t}_{2}}& frac{d{tau }_{1}}{d{t}_{2}}& frac{d{tau }_{2}}{d{t}_{2}}& 0\ 0& 0& 0& 0& frac{d{tau }^{^*}}{d{t}^{^*}}end{array}right]cdot left[begin{array}{c}frac{dr}{d{m}_{1}}\ frac{dr}{d{m}_{2}}\ frac{dr}{d{tau }_{1}}\ frac{dr}{d{tau }_{2}}\ frac{dr}{d{tau }^{^*}}end{array}right]$$
(2)
In the PDE (Eq. 2), the left-hand side is a vector column of the derivatives of the response in clock time (R), with respect to each predictor; the right-hand side is the standard (= inner) product of a matrix (M) by a vector of the derivatives of the response in biological time (r), i.e. R = Mr. The matrix contains the derivatives of the predictors with respect to each other, with time both expressed in clock or biological scales; one can think of M as an object containing coefficients that transform r into R in the same way as a constant (= 1000) would transform kilometres into meters of distance. The large number of terms in M highlights the considerable diversity and the challenges in quantifying responses to multivariate environmental fluctuations. We show below how to use Eq. (2) to quantify the effect of fluctuating environmental drivers on biological responses, as mediated by biological time.First, we note that M contains three groups of terms: (1) Terms accounting for situations where the magnitude of a driver affects the magnitude of the second driver (e.g. temperature drives oxygen concentration in aquatic habitats): these are dmi/dmj for any i, j = 1, 2. (2) Terms accounting for cases where the magnitudes and time scales of stressors are related: dmi/dtj and dmi/dti. (3) Terms where biological time depends on the magnitude or time scale of the environmental fluctuation dτi/dtj and dτi/dmj. The terms of groups (1) and (2) are zero when they are mutually independent, such as in a factorial experiment with orthogonal manipulation. We will set those to zero in the rest of this analysis.Second, we note that for group (3) there are three scenarios: (3a) biological time does not depend on any environmental driver. This is the trivial case where biological time is proportional to clock time, not considered here; M is simplified to a diagonal matrix, i.e. with constants in the diagonal, and zero’s otherwise leading to a single constant term per equation (3b). Biological time depends on the magnitudes of any or both drivers. In such case, τ1 τ2, and τ({^*}) will be driven by the same equation: if τi = ti · L (m1, m2) we obtain dτi/dtj = dτi/dti = L (m1, m2). (3c) Biological time depends on the time scale of the fluctuations: in such case, differentiating Eq. (1) with respect to time, we obtain dτi/dti = L + ti dL/dti.Here, we explore four special cases where the equations are simplified to highlight the importance of biological time in modifying the responses as compared to clock time. We start with the simplest case where there is a single environmental variable and then we consider cases with two variables. We focus on cases representing the most frequent experiments carried out on multiple driver research, i.e. factorial manipulations where terms of the groups 1 and 2 are zero.Case 1: responses to the magnitude of a single variableWe start with the simplest case i.e. where the response is driven by the magnitude of a single driver, e.g. temperature (= m). Examples of this case are laboratory experiments quantifying the effect of temperature on body mass or survival of a given species, or mesocosm experiments quantifying effects of temperature on species richness where thermal treatments are kept constant over time. Here, the response is quantified at different times, both in the clock and biological frames. In such case we have R(m, t({^*})) = r[m, τ({^*})(m, t({^*}))] and the PDEs simplify to.$$frac{dR}{dm}=frac{partial r}{partial m}+frac{partial r}{partial {tau }^{^*}}cdot frac{d{tau }^{^*}}{dm}$$
(3)
From Eq. (3), and because dR/dm ≠ dr/dm, we see that the response to the magnitude of the driver depends on a component quantifying the effect biological time: as long as dτ({^*})/dm ≠ 0 the time reference frame affects the observed effect of m on the response. The simulation illustrated in Fig. 2 shows a case where there are differences between the observed responses at clock vs biological times. In the simulated experiment, there is a strong effect of the magnitude of the driver on the response at clock time, but such effect is much less pronounced at biological time. By contrast, there is no effect when the response is measured in the biological time frame.Figure 2Case 1: Response to the magnitude of a single variable (m). Horizontal line: measurement taken at clock time t({^*}) = t({^*})c; note that, along the line, the response increase with m (it crosses the colour gradient). Curve with yellow circles: measurements taken at a constant biological time (τ({^*})c = 100); along the curve, the response does not vary with m. The equations used were: R = m(0.5t({^*})), τ({^*}) = t. m giving r = 0.5. τ({^*}) not depending on m.Full size imageEquation (3) (details in Supplementary code 1) captures an obvious but important feature of experiments manipulating temperature over the development of ectotherms, for instance, from birth to metamorphosis; namely that there is no consistent definition of a simultaneous event across the different time frames. Experiments are usually stopped at different clock times because organisms must be sampled at the same biological time. All points located in the horizonal line in Fig. 3 represent simultaneous events, as defined in clock time occurring at different temperatures (e.g. whether an animal is dead or alive); however, simultaneous events occurring in biological time are represented by the points on the curve. Hence, Fig. 2 gives a geometric representation of such fact. Temperature as a driver of developmental rates32 is a central candidate to produce responses that differ at clock vs biological time.
We explore further this case with an example where the response is expressed as a function of time and an instantaneous rate μ(m) quantifying for instance mortality, growth or biomass loss. For this example, we obtain R(m, t({^*})) = r[μ(m), τ({^*})(m, t({^*}))]. By differentiating in both sides, we get:$$frac{dR}{dm}=frac{partial r}{partial mu }cdot frac{dmu }{dm}+frac{partial r}{partial {tau }^{^*}}cdot frac{d{tau }^{^*}}{dm}$$
(4)
Equation (4) shows that m affects the response through two components: the instantaneous rate (dμ/dm) and the biological time (dτ({^*})/dm). We call the first component “eco-physiological” and the second component “phenological” (m drives the timing of a biological event, e.g. time to maturation). Those components are not evident if the response is expressed in clock time; otherwise we would obtain dR/dm = ∂R/∂μ · dμ/dm.In order to better understand Eq. (4), consider an example where the response is biomass loss experienced by an organism during the process of migration (e.g. towards a feeding or reproductive ground); when the access to food during migration is very limited the result should be a decrease in body mass reserves through time. Let biomass (B) be modelled as an exponential decaying function of time and an instantaneous rate of biomass loss μ; let μ depend on temperature (= m) such that, μ = μ(m). In such case we obtain:$$B(m,t)={e}^{-mu left(mright)cdot {t}^{^*}}={e}^{-mu left(mright)cdot {tau }^{^*}left(m,{t}^{^*}right)}$$
(5)
By differentiation in both sides of Eq. (5) we get:$$frac{dB}{dm}={-e}^{-mu left(mright)cdot {tau }^{^*}left(m,{t}^{^*}right)}left{{tau }^{^*}cdot frac{dmu }{dm}+mu cdot frac{d{tau }^{^*}}{dm}right}$$
(6)
Equation (6) shows the eco-physiological (dμ/dm) and phenological components (dτ({^*})/dm) within the brackets. If μ responds linearly to temperature, then dμ/dm would be represented by a constant quantifying the thermal sensitivity of biomass loss; the value of such constant would depend on physiological processes associated to use of reserves to sustain movement and the basal metabolic rate. Likewise, if τ({^*}) responds linearly to temperature, the dτ({^*})/dm would be driven by a constant controlling the sensitivity of developmental time to temperature.Because biomass is a trait that is central to fitness, Eq. (6) gives the indirect contribution of phenological and physiological responses to fitness. Assuming that fitness should be maximised, adaptive responses should involve the mitigation of negative effect of m on both components of Eq. (5), represented by the partial derivative of the right-hand term. For instance, organisms with the ability to minimise the eco-physiological effect (through e.g. a compensatory physiological mechanisms) or the phenological effect (e.g. shortening the exposure time) would complete the migration minimal loss of reserves.By generalization, Eqs. (4–6) help us to provide biological meaning to the terms of the matrix M: any term of the form dτ({^*})/dmj, dτi/dmj or dτi/dtj represents the effect of an environmental driver on the timing of a phenological event; hence, they are phenological components. Terms that contain the effect of an environmental variable on an instantaneous rate are eco-physiological components. By substitution we find that the terms of the matrix in Eq. (2) can be classified in two categories according to whether the component is eco-physiological (E) or phenological (P):$$left[begin{array}{ccccc}E& 0& P& P& P\ 0& E& P& P& P\ 0& 0& P& P& 0\ 0& 0& P& P& 0\ 0& 0& 0& 0& Pend{array}right]$$
(7)
Case 2: multiple driver responsesHere we expand the previous case by looking at a response to the magnitude of two different drivers; i.e. keeping the levels of each driver constant over the duration of the experiment. Examples of this case are experiments quantifying the effect of temperature and nutrient load on body mass (e.g. in a rearing containers) or species richness (e.g. in mesocosms). This case is represented by the terms of first two rows of the matrix and the vectors of Eq. (2), with the terms of the remaining rows set to zero. Here, there are different scenarios, but we focus on the one highlighting the importance of biological time.Consider a case where biological time depends on the magnitude of the first driver while the response is explicitly driven by the magnitude of the second driver (Fig. 4). For instance, the response may be the survival rate of a host organism exposed to different temperature and parasitic load. The response in clock time is described as R(mP, t({^*})). The driver controlling the biological time is temperature (mT) while the parasitic load (mP) controls survival. In such case, dτ({^*})/∂mP = 0, dR/dmP ≠ 0 and dR/dmT = 0. Although by definition the response in clock time does not depend on mT , it will do so in biological time. This is because, applying the matrix multiplication in Eq. (2), we obtain:
$$frac{partial R}{partial {m}_{T}}=frac{partial r}{partial {m}_{T}}+frac{partial r}{partial tau *}cdot frac{dtau *}{d{m}_{T}}$$
(8a)
$$0=frac{partial r}{partial {m}_{T}}+frac{partial r}{partial tau *}cdot frac{dtau *}{d{m}_{T}}$$
(8b)
$$frac{partial r}{partial {m}_{T}}=-frac{partial r}{partial tau *}cdot frac{dtau *}{d{m}_{T}}$$
(8c)
The second right-hand term in Eq. (8a) quantifies the effect of temperature on the response mediated by biological time. In order to better understand the responses, consider a simple linear response: R = R0 − mP·t({^*}) and notice that, for a fixed clock time (t({^*})c) the effect of the magnitude of parasitism is constant (dR/dmP = −t({^*})c); hence, the response can be understood, geometrically, as a flat surface with slope not depending on temperature. Now, note that under the specific conditions of our example, r = R0 − mP·τ({^*})/L(mT). Hence, for a fixed biological time (τ({^*})c) we obtain ∂r/∂mP = −τ({^*})c/L(mT); i.e. the importance of the parasitic effect depends now on temperature. In addition, this example is valid for the case of additive effects of any two environmental drivers: assuming R = R0 − (a1·mP + a2· mT)·t({^*}) (a1, a2 are constants), we obtain dR/dmP = −a1t({^*}); however, ∂r/∂mP = −a1τ({^*})c/L(mT). In words, additive effects observed in clock time become interactive in biological time. This is illustrated in the simulation (Supplementary code 2) depicted in Fig. 4: the response in clock time depends on a single driver (parasite load); however, the response in biological time is interactive, i.e. the effect of parasite load depends on temperature.Figure 3Case 2: Multiple driver responses. (A) Modelled responses (colour scale) at a specific clock (t({^*}) = 40) and biological times (τ({^*}) = 1), showing an interactive effect only in the biological time frame. (B) Interaction plots of the responses for specific levels of temperature and a second driver showing that the effect high temperature mitigates the negative effect of the second driver on the response. The response was modelled with as a sigmoidal function R = exp(−t({^*})φ) with φ = 0.1[1 + exp(m2/2)]−1 to produce a strong gradient in the range of m2 = 25–30 units. The biological time was modelled based on the effect of temperature on the development of marine organisms33 as so that t({^*}) = τ({^*}) exp[−22.47 + 0.64/(k(m1 + 273)], i.e., using the Arrhenius equation with k: Boltzmann constant (≈ 8.617 10–5 eV K−1).Full size imageCase 3: role of clock and biological time scale of fluctuationPrevious examples did not consider, the time scale of the fluctuations as drivers of the response. Here we explore how a biological variable (= survival rate) responds to different levels of magnitude of a driver (= temperature) and to simultaneously changing the time scale of a fluctuation (from clock to biological time) of a second driver (= food limitation). As model, we use larval stages of a crab because there is sufficient information on the effect of temperature and food levels on survival and the timing of moulting33,34.We performed the so-called point-of-reserve-saturation experiment (PRS35), i.e. exposing groups of recently hatched larvae of the crab Hemigrapsus sanguineus to different initial feeding periods (= our time scale of fluctuation), after which they were starved until they either died or moulted to the second larval stage (Supplementary Fig. 1). H. sanguineus is originated from East Asia but has invaded the Atlantic shores of North America and North Europe36,37. This experiment was carried out at 4 temperature levels (15–21 °C), within the range of thermal tolerance of larvae of this species, i.e. where the magnitude of temperature does not affect survival38,39. In addition, because there is a single level of food limitation (= starvation), the magnitude of food limitation (mF) is not considered as a variable in the example.The response variable was the proportion of first stage larvae surviving the moulting event to the second stage, set to biological time τ({^*}) = 1. In response to different starvation periods (preceded by feeding), the survival shows a sigmoidal pattern35, characterised by a parameter, PRS50. This is the point of development where larval reserves are “saturated”; i.e. enough reserves have been accumulated during the previous feeding period to ensure survival and moulting to the next stage.Under the conditions of the experiment, the survival proportion (= R) is driven only by the time scale of a fluctuation (here t1 = t, τ1 = τ for simplicity), characterised by the starvation period; hence, R = R(t) = r[τ(t)] given that there is a single time of observation fixed to τ({^*}) = 1. Because biological time does not depend t, we get L = dτ/dt and:$$frac{dR}{dt}=frac{partial r}{partial tau }cdot mathcal{L}({m}_{2})$$
(9)
Equation (9) is represented in the PDE by the terms of row 3 and column 4 of M multiplied by the term of row 3 of the column vector r; dτ/dt = L(m), m represents the magnitude of temperature.The relationship between biological time and temperature was best explained by a power function D(T) = aTb (Fig. 4A, Supplementary Table 1, Supplementary Fig. 2), in consistence with previous studies36,40. The interaction between starvation time and temperature was weak (Supplementary Fig. 3); best models retained starvation time only at 21 °C where the percentage of explained variance was still low (R2  More

Genome’s collection and phylogenetic analysisThe study examined the genomes of 139 isolates, 61 of environmental samples (ENV) and 78 clinical (CLI) (Supplementary Table 1, Supplementary Fig. 1), with origin in 21 countries: USA (23/139, 17%), UK, Portugal and Spain (each 15/139, 33%), China (14/139, 10%), Germany (13/139, 9%), Thailand (11/139, 8%) and other countries (each  More

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