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Study areaThe NCA encompasses an area of 8292 km2 and in 2020 had approximately 87,000 inhabitants23, who are primarily dependent on livestock for their livelihoods. It is a multiple-use area where people coexist with wildlife and livestock, and practise pastoralism with transhumance, characterised by seasonal movements of livestock for accessing resources such as grazing areas and water. The NCA comprises eleven administrative wards: Alailelai, Endulen, Eyasi, Laitole, Kakesio, Misigiyo, Ngorongoro, Naiyobi, Nainokanoka, Ngoile and Olbalbal (Fig. 1). The NCA was chosen for our study as it is known to be hyperendemic for anthrax4,17,20. In addition, informal consultations we held prior to the study, as well as tailored data collection at the community and household level, indicated that local communities have a good understanding of the disease in humans and animals, and of practices around carcass and livestock management that increase risks, particularly in certain locations and periods of the year24.Figure 1Locations of participatory mapping. Map showing the 11 administrative wards of the Ngorongoro Conservation Area in northern Tanzania and the locations where participatory mapping sessions took place (red dots). The maps were produced in QGIS 2.18.2 using data from the National Bureau of Statistics, Tanzania (http://www.nbs.go.tz/).Full size imageEthics approval and consent to participateThe study received approval from the National Institute for Medical Research, Tanzania, with reference number NIMRJHQ/R.8a/Vol. IX/2660; the Tanzania Commission for Science and Technology (numbers 2016-94-NA-2016-88 (O. R. Aminu), 2016-95-NA-2016-45 (T. L. Forde) and 2018-377-NA-2016-45 (T. Lembo)); Kilimanjaro Christian Medical University College Ethics Review Committee (certificate No. 2050); and the University of Glasgow College of Medical Veterinary & Life Sciences Ethics Committee (application number 200150152). Approval and permission to access communities and participants were also obtained from relevant local authorities. Written informed consent was obtained from all participants involved in the study. All data collected were analysed anonymously, ensuring the confidentiality of participants. All research activities were performed in accordance with relevant guidelines and regulations.Participatory mappingA participatory mapping approach based on methodology previously tested in East Africa25 was employed to define areas of anthrax risk for animals in the NCA based on community knowledge. Georeferenced maps of the NCA were produced using data from Google and DigitalGlobe (2016). The maps used datum Arc 1960/UTM zone 36S and grid intervals of 1000 km and were produced at 1:10,000 and 1:50,000 scales, in order to provide participants with a choice. Ten participatory mapping focus groups were held at ward administrative level (Fig. 1) in order to identify areas in the NCA that communities perceive as posing a high risk of anthrax. One mapping exercise was held in each ward. Ngoile and Olbalbal wards were covered at the same time and treated as one, as they had only recently (in 2015) been split from one ward (Olbalbal). Each session had between ten and thirteen participants, who consisted of village and ward administrators, animal health professionals (including community animal health workers and livestock field officers), community leaders, and selected community members. These participants represented members of the community concerned with animal health and owning livestock and, as such, were likely to hold in-depth knowledge relating to community experience of animal health and disease, including anthrax. Participants were recruited by consulting with animal health professionals as well as village and ward administrators, who gave permission to conduct the mapping sessions.The mapping sessions were conducted in Swahili and translated into English by an interpreter. Participants’ general knowledge of the area was first verified by testing whether they could correctly identify popular locations such as health centres, places of worship, markets and schools. Subsequently, participants discussed among themselves and came to a consensus about areas they considered to be at high risk of anthrax. Specifically, we asked them to identify locations they perceived as areas where they considered their animals to be at risk of being exposed to anthrax. These areas were drawn on the maps provided (Fig. 2). While they did not locate areas where the animals had succumbed to disease, we also asked for generic information on locations where anthrax outbreaks had occurred in the past to define areas that could be targeted for active surveillance of cases. In order to improve the fidelity of the data, participants defined risk areas in relation to their own locality (ward) and locations where their animals access resources. Therefore, the areas were not defined by administrative boundaries, as communities may access locations outside their wards, for instance for grazing or watering. The resulting maps were scanned, digitised and analysed as detailed in the following sections. Further detail on the participatory mapping process is provided in the Supplementary Methods (Additional File 1).Figure 2Participatory mapping of anthrax risk areas in the Ngorongoro Conservation Area. Images show (A) the set-up of a mapping session, (B) participants engaged during a session and (C) an example of a 1:50,000 scale map annotated by participants. The map was created with QGIS opensource mapping software. The basemap used was a scanned and geo-referenced full colour 1:50,000 scale topographic map produced by the Surveys & Mapping Division, Ministry of Lands, Housing & Human Settlements, Dar es Salaam, Tanzania. The grid is based on the Arc1960 UTM 36S projection and datum. The map was exported from QGIS in Acrobat Pdf format to enable it to be printed at suitable sizes for using in the fieldwork and to be manually annotated during the participatory mapping.Full size imageDigitisation of maps and generation of random pointsScanned maps were saved as PDF files and converted to high resolution TIFF files for digitisation in QGIS 2.18.2-Las Palmas free OpenSource software26. All maps were georeferenced with geographical coordinates during production and reference points were available to enable the precise mapping of all locations. The digitization was carried out using the QGIS digitizing tools and by creating polygon layers of the defined risk areas.Sourcing data on the environmental predictors of anthraxAvailable soil and environmental data (250 m grid) for Tanzania were obtained from various sources (Table 1). From the available data, we selected the following seven variables which have previously been shown to contribute to or explain the risk of anthrax based on the biology of B. anthracis (Table 1).Table 1 Environmental factors with potential to influence anthrax occurrence.Full size tableCation exchange capacity (CEC)Measured in cmol/kg, CEC is the total capacity of the soil to retain exchangeable cations such as Ca2+, Mg2+ etc. It is an inherent soil characteristic and is difficult to alter significantly. It influences the soil’s ability to hold on to essential nutrients and provides a buffer against soil acidification27. CEC has been reported to be positively correlated with anthrax risk. In addition, CEC is a proxy for calcium content, which may contribute to anthrax risk in a pH-dependent manner as explained below19,22.Predicted topsoil pH (pH)Soil pH below 6.0 (acidic soil) is thought to inhibit the viability of spores19 thus a positive effect of higher pH on the risk of anthrax is expected. It has been suggested that the exosporium of B. anthracis is negatively charged in soils with neutral to slightly alkaline pH. This negative charge attracts positively charged cations in soil, mainly calcium, enabling the spores to be firmly attached to soil particles and calcium to be maintained within the spore core, thereby promoting the viability of B. anthracis19,28.Distance to inland water bodies (DOWS)Both the distance from water and proximity to water may increase anthrax risk. Distance to inland water may indicate the degree to which an area is dry/arid. Anthrax outbreaks have been shown to occur in areas with very dry conditions19. Although anthrax occurrence has also been associated with high soil moisture, this relates more to the spore germination in the environment (a mechanism that is disputed) and the concentration of spores in moist humus that amount to an infectious dose18,29. Spores will survive much longer in soils with low moisture content19. Low moisture may also be associated with low vegetation which results in animals grazing close to the soil, increasing the risk of ingesting soil with spores. Hampson et al. reported that anthrax outbreaks occurred close to water sources in the Serengeti ecosystem of Tanzania in periods of heavy rainfall20, and Steenkamp et al. found that close proximity to water bodies was key to the transmission of B. anthracis spores in Kruger National Park, South Africa22. Water is an important resource for livestock and a large number of animals may congregate at water sources during dry seasons. The close proximity of a water source to a risk area may increase the chance of infection, particularly during periods of high precipitation which might unearth buried spores.Average enhanced vegetation index (EVI)Vegetation density may influence the likelihood of an animal ingesting soil or inhaling dust that may be contaminated with spores. Grazing animals are more likely to encounter bacteria in soil with low vegetation density20, although there is a possibility that spores can be washed onto higher vegetation by the action of water19. Vegetation index may also reflect the moisture content of soil. Arid/dry conditions favour the formation and resistance of spores in the environment, thus lower vegetation may be associated with the occurrence of anthrax.Average daytime land surface temperature (LSTD)Anthrax has been more commonly reported to occur in regions with warmer climates worldwide. Minett observed that under generally favourable conditions and at 32 °C to 37 °C, sporulation of B. anthracis occurs readily but vegetative cells are more likely to disintegrate at temperatures below 21 °C30. Another hypothesis for the association of high temperature with anthrax occurrence is altered host immune response to disease due to stress caused by elevated temperatures19. In addition, elevated temperatures are usually associated with arid areas where vegetation is low, limiting access to adequate nutrition, which in turn affects immunity. Similarly, in hotter climates where infectious diseases occur more often, host interactions with other pathogens may modulate immune response to anthrax31. In this case, a lower infectious and lethal dose of spores would be sufficient to cause infection and death, respectively19. Contact with and ingestion of soil, spores and abrasive pasture is also higher with low vegetation in hot and arid areas19,32. In boreal regions such as in northern Canada, where anthrax occurs in wood bison, and Siberia, the disease is more commonly reported in the summer19. We therefore hypothesised a positive effect of LSTD on the risk of anthrax.SlopeSpores of B. anthracis are hypothesized to persist more easily in flat landscapes that are characterised by shallow slopes19, as it is thought that wind and water may disperse spores more easily along areas with a higher slope gradient, thereby decreasing the density of spores to levels that may be insufficient to cause infection in a susceptible host. Therefore, we expected a negative relationship between slope and the risk of anthrax.Predicted topsoil organic carbon content (SOC)Organic matter (g/kg) may aid spore persistence by providing mechanical support. The negatively charged exosporium of spores is attracted to the positive charges on hummus-rich soil, thus anthrax is thought to persist in soil rich in organic matter18. Based on available evidence, we expected a positive effect of SOC on the risk of anthrax.Creating the datasetThe annotated and digitised maps yielded polygons of high-risk areas within the NCA (Fig. 3). After digitization, 5000 random points were generated33 to cover the 8292 km2 area of the NCA. This enabled us to obtain distinct points allowed by the 250 m grid resolution of the environmental variables. Points falling within the defined risk areas were selected to represent risk areas while those falling outside represented low-risk areas. Measures of the environmental characteristics associated with individual points were obtained with the ‘add Raster data to points’ feature in QGIS.Figure 3Ngorongoro Conservation Area map showing (A) defined risk areas (in red) and (B) distance to settlements. For analysis, 5000 random points were generated throughout the area; points falling within 4.26 km of human settlements (the average distance herds are moved from settlements in a day as determined through interviews of resident livestock owners) were retained for analysis (n = 2173, shown in blue in 3a). The maps were created in QGIS 2.18.2 using data from the National Bureau of Statistics, Tanzania (http://www.nbs.go.tz/).Full size imageIn order to focus on areas of greatest risk to humans and livestock and to exclude locations that are not accessible, only points within a certain range of distance from settlements were included (Fig. 3). On average, herders in the NCA move their livestock 4.26 km away from settlements for grazing and watering during the day (unpublished data obtained through a cross-sectional survey of 209 households). Thus, only points falling within this distance from settlements were selected, providing us with data on areas where infection is most likely to occur. Data on locations of settlements were obtained from satellite imagery and included permanent residences as well as temporary settlements (e.g. seasonal camps set up after long distance movement away from permanent settlements, typically in the dry season, in search of pasture and water). These data were collated from the Center for International Earth Science Information Network (CIESIN).After adjusting for accessibility of resource locations using the average distance moved by livestock, 2173 points were retained for analysis, of which 239 (11%) fell within high-risk areas.Data analysisAll statistical analyses were carried out in R (v 4.1.0) within the RStudio environment34. The aims of the statistical analysis were to infer the relationship between anthrax risk areas as determined through participatory mapping and the environmental factors identified in Table 1, and to use this relationship to make spatial predictions of anthrax risk across the study area. We achieved both aims by modelling the binary risk status (high or low) of the randomly generated points as a function of their environmental characteristics in a Bayesian spatial logit-binomial generalised linear mixed-effects model (GLMM), implemented in the package glmmfields35. Spatial autocorrelation (residual non-independence between nearby points) was accounted for by including spatial random effects in the GLMM. We chose relatively non-informative priors for the intercept and the covariates, using Student’s t-distributions centred at 0 and wide variances (intercept: df = 3, location = 0, scale = 10; betas: df = 3, location = 0, scale = 3). For the spatial Gaussian Process and the observation process scale parameters, we adopted the default glmmfields settings and used half-t priors (both gp_theta and gp_sigma: df = 3, location = 0, scale = 5), and 12 knots. To achieve convergence, the models were run for 5000 iterations35.First, univariable models were fitted to estimate unadjusted associations between each environmental factor (CEC, pH, DOWS, EVI, LSTD, slope, and SOC; Table 1; Supplementary Table S1) and high- and low-risk areas. Second, we constructed multivariable models by fitting multiple environmental variables (Supplementary Table S2). Three variables, SOC, slope and EVI showed a strongly right-skewed distribution and were therefore log-transformed prior to GLMM analysis to prevent excessive influence of outliers. All predictor variables were centred to zero mean and scaled to unit standard deviation for analysis, and odds ratios were rescaled back to the original units for ease of interpretation. Prior to fitting the multivariable GLMM, the presence of collinearity among the predictor variables—which were all continuous—was assessed using variance inflation factors (VIFs)36, calculated with the car package and illustrated using scatter plots (Supplementary Fig. S1)36. Three predictor variables showed a VIF greater than 3 (LSTD, ln EVI and pH with VIFs of 6.8, 4.2 and 3.5, respectively). Removal of LSTD and ln EVI reduced all VIFs to below 3, therefore these two variables were excluded from the multivariable regression analysis37.The model performance was assessed by calculating the area under the receiver operating characteristic curve. The predicted probability of being an anthrax high-risk area was determined and depicted on a map of the NCA using a regular grid of points generated throughout the NCA with one point sampled every 500 m.Consent for publicationPermission to publish was granted by the National Institute for Medical Research, Tanzania. 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

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