Philippa Kaur

Animal culture and mediumFive different clonal lineages of the water flea Daphnia magna were sampled from two ponds on agricultural land in Belgium (Vleteren: 50°55′06.7″ N, 2°43′27.0″ E and De Haan 51°13′53.8″ N, 3°01′49.2″). They were cultured separately in 210 ml glass jars under optimized laboratory conditions (20 ± 1 °C, 14:10 h light:dark cycle). Seed shrimp and rotifer resting eggs were obtained from a commercial supplier (MicroBioTests Inc., H. incongruens strain MBT/1999/10, product code TB36; B. calyciflorus, product code TK21, Belgium) and represent laboratory cultured, single clonal lineages. More details on animal culture are reported in the online supplementary methods (Appendix 3).Natural pond water was used as medium both in animal cultures and the experiment. It was extracted from a Belgian region (50°59′00.92″ N, 5°19′55.85″ E, Zonhoven) with soft, poorly buffered water (Alkalinity 3–8°d; pH 6.5–8.5) which is likely to be susceptible to acidification under elevated pCO2. More information on medium and mineral composition is reported in the online supplementary information (Appendix 3; Table S3, Appendix 1).Experimental set-upOrganisms were exposed to three pCO2 treatments, an ambient control (C; 1,520 ppm ± 702 SD), an elevated (T1; 25,609 ppm ± 4,541 SD) and an extreme pCO2 level (T2; 83,201 ppm ± 15,533 SD). The control pCO2 level represents the current global mean that is measured in lentic freshwaters considering most ponds and lakes are already supersaturated10,12. The T1 level is currently only observed in more extreme cases11. However, it reflects a pCO2 level that could be encountered more commonly in the field in the future. The T2 treatment represents an extreme test of the tolerance limits of extant species. These treatments are a necessary simplification of reality since pCO2 can experience strong fluctuations in ponds and lakes. An overview of freshwater pCO2 concentrations from literature can be found in Table S1 (Appendix 1).The elevated pCO2 concentrations were manipulated in the water by injecting pure CO2 (99.998% pure, ALPHAGAZ CO2 SFC * B50-N48, Airliquide, Belgium) from gas cylinders into the water (cf.49) at a constant flowrate, using a high-pressure regulator (HBS 200–10.2,5; AirLiquide, Belgium) and a flow controller (Sho-rate model 1350G, Brooks Instruments, USA). In the control treatment, ambient air was supplied at a similar rate as the CO2 to ensure equal perturbation levels across all containers. Water of all experimental containers (including control) were also injected with ambient air to keep the water oxygenated. A relatively constant pCO2 was ensured by continuously monitoring pH and kept between a range of ~ 20,000–30,000 ppm (pH 6.9–6.7) for T1 and ~ 70,000–120,000 ppm (pH 6.4–6.1) for T2 (Figure S2, Appendix 2).Each treatment included 13 replicate 210 mL glass jars per species, resulting in a total of 117 experimental units. Per replicate, one mature water flea (8–11 days old) was inoculated in a jar containing aerated pond water. The five clonal lineages were distributed evenly over the experimental conditions so that each condition had the same number of replicates per clone. Seed shrimp replicates each contained one newly hatched ( More

Deterministic logistic modelThe following population dynamics model was applied to reconstruct the initial dugong population size in 1894 from fishery statistics between 1894 and 1914:$$N_{t + 1} = N_{t} left( {1 , + r{-}r , N_{t} /K} right) – C_{t} ,$$where r is the intrinsic rate of population increase, Nt is the population size in year t, K is the carrying capacity, and Ct is the number of individuals removed from the waters near the Ryukyu Islands in year t. The carrying capacity (K) in 1893 was sufficient to sustain the initial population of dugongs at that time (N1894). The intrinsic rate of population increase (r) was given between 1 and 5% within a range of natural one.Approximate Bayesian calculationWe conducted approximate Bayesian calculation (ABC)32 to estimate the number of individuals in 1979 based on bycatch data between 1979 and 2019, and the constraints of the numbers of individuals were 11 in 1997, three in 2007, and almost extinct in 2019. We denoted fecundity as f, the survival rate until 1 year old as s0, the annual survival rate after 1 year old as s, the age at maturity as am, and the physiological longevity as A. We assumed that the sex ratio at birth was 1:1 on average; the age at maturity am was eight years of age33, and the physiological longevity A was 73 years6. We ignored environmental stochasticity because no mass deaths caused by infectious diseases or changes in survival or mortality rates due to environmental fluctuations have not been recorded during this period. We also ignored density effects because the carrying capacity of the location was sufficiently greater than the initial population size, and our goal was to investigate the possibility of population recovery after a decrease in population using a population dynamics model and estimate the natural growth rate during this period. The detailed extinction risk depends on age structure.According to the life history parameters, except the physiological longevity compiled by (ref.33), the annual survival probability of an a year-old individual is s for a = 1, 2, …, 72; s0 for a = 0, and 0 for a = 73; the reproductive probability of an adult female  > 8 years old is 2f. As the number of years for a population to become extinct or recover depends on age composition, age-specific survival, and reproductive rates, we obtain the population growth rate by the maximum eigenvalue of the following Leslie matrix, L = {Lij} (i = 1,…73, j = 1,…,73) as:$$L_{i1} = s_{0} f/2quad {text{for}}quad i ge a_{m} ,L_{i+ 1,i} = squad {text{for}}quad i = 1, ldots ,72,quad {text{and}}quad L_{ij} = 0,{text{otherwise}}{.}$$We used the population growth rate λ, defined by the maximum eigenvalue of L, as an indicator of the population growth rate.We assumed that the sex of each individual in 1979 was randomly sampled by the 1:1 sex ratio, and its age was randomly sampled by the stable age structure that is given by the eigenvector of the Leslie matrix with the maximum eigenvalue. We assumed that the number of individuals at age 1 year in year t + 1, denoted by N1,t+1, is determined by the binomial distribution:$$Prleft[ {N_{1,t + 1} = x} right] = left( {begin{array}{*{20}c} {N_{f} } \ x \ end{array} } right)left( {s_{0} f} right)^{x} left[ {1 – left( {s_{0} f} right)} right]^{{N_{f} – x}} ,$$where Nf represents the number of adult females in year t. We assumed that no twins were born. We assumed that the probability that an individual with age x survived in the next year is s if x = 1 or s0 if x = 0. We also assumed that Ct individuals who died by bycatch were randomly chosen from any sex and age because the age of individuals caught by bycatch is rarely known. We do not know the sex of some individuals.We assumed the following prior distributions for N1997, f, and s: N1979 (in) U(11, 80), f (in) U(1/14, 1/6) if at least one adult male existed in the population, s0 (in) U(0.1, 0.85); and s (in) U(0.8, 0.97), where U(a, b) is the uniform random variable between a and b. These probabilities were constant for each simulation trial from 1997 to 2019. We selected the set of parameters with the population growth rate (λ) obtained when the maximum eigenvalue of the Leslie matrix was between 0.96 and 1.01.We rejected trials that did not satisfy the following summary statistics: N1997 ≥ 11 (intensive survey in 1997), Nt ≥ 3 during 2004–2017 (monitoring), and N2019 ≤ 1 (“local extinction”). We obtained the prior distributions of N1997, f, s0, s, and N2004, and of the  > 130,000 trials in the prior distribution with natural population growth rates λ of 96.1–98.8%, 99.3% were rejected. For 95% of the 1000 adopted trials, N1979 ranged from 14 to 58. If λ  > 98%, N1997 was ≤ 45 for the adopted trials (Extended Data Fig. 7. Even if all the stranding deaths were due to anthropogenic factors, such as the release of dugongs after bycatch or boat strike, the range of N1997 changed to  98%, with only a slight upward shift, but positive natural growth rate (or λ  > 1) was again very unlikely (0.3%) among the adopted trials.Population viability analysis to assess the impact of bycatch on the extinction riskWe re-evaluated the extinction risk with and without bycatch using the 1000 parameter sets of N1979, f, s0, and s that satisfied the summary statistics in the ABC and stochastic individual-based model, beginning from N1979 for the corresponding parameters. For each parameter set, 100 trials were conducted for each scenario to compare the extinction risks. More

From the popular algorithms, we chose the Random forest model as the most suitable for our case. The data required for predictions can be divided into plant occurrence records and environmental features. Bioclimatic variables and soil properties were selected as the main environmental features. All of the data were obtained from open sources.Heracleum Sosnowskiy plant descriptionHeracleum sosnowskyi is a monocarpic perennial plant of the Apiaceae family. The height is up to 3–5 m with a straight stem up to 12 cm in diameter. HS compound steam leaves can reach 150 cm, both long and wide38. The blooming period starts in July and continues until the end of September. Plant reproduction is performed by seeds only. The seeds’ depth of germination is reported as mainly in the upper 5 cm down to 15 cm of soil. One plant can produce 10–20,000 seeds39,40. Seeds germinate in the early spring, while some have reported that a period of cold stratification for the dormancy break is obligatory for germination development. Suitable conditions for HS include a temperate climate with warm humid summers and cold winters, while it is probably not drought resistant. Plants of HS tend to neutral soils with a pH range from 6 to 7, rich in nutrients, and being reported as nitrophilous, so the eutrophication of the environment favours HS development. HS plants do not tolerate shade conditions in the first growing period.HS is mostly spread in artificial and semi-natural habitats, including grasslands, pastures, parks, roadsides, agricultural fields, riverbanks or canal sides, and other distributed habitats. Currently, the main pathways of spread include an involuntary entry with soil on vehicles, machinery, footwear or the use of soil as a commodity (as the growing medium rich in organic matter)39.Study areaThe area for modelling extends from approximately 41(^{circ }) to 70(^{circ }) N and from 27(^{circ }) to 60(^{circ }) E, and Kaliningrad region, it equals to approximately 4 mln km2 (Fig. 4).Figure 4Map of the study area: white colour represents the territory used for prediction, red points correspond to the dataset of HS occurrence, collected from the available sources.Full size imageThe European part of Russia is the most inhabited part of the country, and it is the home of approximately 80% of the total population of Russia. It includes the East European Plain, Caucasus mountains and Ural mountains, with the predominance of the East European Plain. Environmental characteristics across the territory of study vary significantly. The climate is changing from semi-arid in the south to subarctic in the north, including humid continental climate conditions. Natural vegetation is represented by almost all types of biomes with the prevalence of different types of forests: broadleaf and mixed forests, coniferous forests, and boreal forests (taiga), while the area of arable lands is reported to be approximately 650,000 km241,42. The territory is subjected to the constant land-use types and cover changes due to the urbanization and switch of the status of arable lands—i.e. reduction of croplands and development of fallows and forests, and, vice versa, returning of some of them into the cultivation process43. The soil cover is represented by the contrast by their physicochemical properties groups, in the northern part of Luvisols, Podzols, Histosols, while of the southern part—by Chernozems, Kastanozems, Solonetz44.Collection of the input dataPlant occurrence dataPlant occurrence coordinates were collected from several publicly available sources related to citizen science projects: the Global Biodiversity Information Facility database45, iNaturalist database46, and the database of the “Antiborschevik” community47. Records were documented by human observation and collected from 2000 to 2021. The overall number of initial occurrence points from combined sources is 7637.Environmental predictorsClimate data Modelling was performed for current and future climate conditions at its two scenarios, selected year ranges were 2000–2018 and 2040–2060 respectively.Climatic variables were collected from the Worldclim database48, containing the average seasonal information relevant to the physiological characteristics of species and available at different resolutions. We chose 10 arc-minutes spatial resolution taking into account the size of the studied area. Table 1 provides a short description of the used bioclimatic features, and we refer the reader to the Worldclim project for detailed information on the variables’ calculation.For the future climate scenarios, we used two Shared Socioeconomic Pathways (SSPs)49—1-2.6 and 5-8.5, corresponding to the lowest (keeping global mean temperature increase below 2 (^{circ })C) and the highest (at the increase of population without technological change) predicted future greenhouse gases emission scenarios. For these data, we took the same resolution (10 arc-minutes) as discussed above.We used the Equilibrium Climate Sensitivity to select the climate model to model future HS distribution. Equilibrium climate sensitivity (ECS) is defined as the global mean surface air temperature change due to a rapid doubling of carbon dioxide concentrations as soon as the associated ocean-atmosphere-sea ice system reaches equilibrium. As the ECS value increases, the model’s sensitivity to the CO(_2) concentration in the atmosphere increases. We have chosen CanESM5 model (ECS—5.6), CNRM-CM6-1 model (ECS—4.3) and BCC-CSM2-MR model (ECS—3.0)50.Table 1 Description of used bioclimatic variables.Full size tableFor the future climate scenarios we selected three climate models:

BCC-CSM2-MR Beijing Climate Center climate system model developed in Beijing Climate Center, China Meteorological Administration51. Model has horizontal resolution 1.125(^{circ }) by 1.125(^{circ }).

CanESM5 Canadian Earth System Model version 5 developed in Canadian Center for Climate Modelling and Analysis, Canada52. Horizontal resolution 2.81(^{circ }) by 2.81(^{circ }).

CNRM-CM6-1 Climate model developed in National Center of Meteorological Research, France53. Horizontal resolution 1.4(^{circ }) by 1.4(^{circ }).

Authors of the WorldClim project prepared historical and future climate data to a uniform spatial (10 arc-minutes) and temporal resolution.Soil data Soil data were downloaded from the SoilGrids database54—a system for global digital soil mapping. SoilGrids provides continuous data at several depths of the spatial distribution of soil properties across the globe with selected resolution. It uses a machine learning approach to reconstruct continuous data from 230,000 soil profile observations from the WoSIS (The World Soil Information Service) database and a series of environmental covariates.From the whole set of the data provided by SoilGrids several properties were chosen for the forecasting: relative percentage of silt (Silt, %), sand (Sand, %), a volumetric fraction of coarse fragments (CF, %), cation exchange capacity (CEC, ({text{cmol}}_{c}/{text{kg}})) and soil organic carbon (SOC, g/kg) at the depth 5–15 cm, where the HS seeds are assumed to be located. These variables are expected to be more stable over time than bioclimatic predictors; thus, chosen soil properties could be implemented for the future time the same as in the present.Data pre-processingAll the data were transformed to the ASCII format by R script and using software DIVA-GIS following the tutorial for the preparation of WorldClim files for use in SDM (http://www.lep-net.org/wp-content/uploads/2016/08/WorldClim_to_MaxEnt_Tutorial.pdf) with unified selected resolution 340 sq.km.Optimization of the occurrence points amountThe general problem in using the available data collected from the databases of the citizen science projects is that the points of observation are distributed non-uniformly. For instance, the frequency of the records depends on the density of the population directly. The spatial filtering of the data (reducing the number of points) can be performed to reduce the sampling bias55. We prepared three datasets with a distance between points of 4, 7 and 10 km with 2402, 1846 and 1504 occurrence points correspondingly filtering the initial dataset. For the thinning step thin() function was used within the R package spThin with 100 iterations for each of chosen thinning distances. To understand how much data we could lose, we used the analysis of feature distribution and evaluated the general fairness of the model performance.Pseudo-absence generationDue to the availability only of the presence points, it is important to generate the absence points for further implementation of the selected algorithm. Although the generation of pseudo-absence points in SDM research is a widespread solution, a closer look at the literature reveals several gaps and shortcomings. Since the raw dataset of the HS distribution demonstrates strong sampling bias, the generation of pseudo-absence points using the usual ‘random’ strategy can aggravate the sampling bias problem. Thus, the combination of the ‘disk’ and ‘random’ strategies was applied for the generation of the pseudo-absence points using the biomod R package17.

The ‘disk’ strategy is established on the geographic distance works as separation from truth presence and possible absence points. The optimal geographic distance for HS was chosen as 25 km. This distance was chosen empirically by trial-and-error. We started with 18 km (because the size of the cell is   9–18 km depending on location) and finished with 50 km. Using distances such as 30–50 km lead to a positive spatial autocorrelation. Thus, we decided to set 25 km which finally provided both optimal model performance and reduced spatial autocorrelation.

The second part of the generation was based on the ‘random’ strategy with filtration: according to the different range of climate conditions on the territory of Russia, there are several places where HS is not detected, thus not growing. The selection of unsuitable places for HS related to the north of Russia, where it is might be too cold for plant species. From all amount of randomly generated generated points we selected points with condition latitude ( > 64^{circ }), according to tundra board line.

Features selection procedureTo avoid over-fitting and to choose the most conscientious set of parameters for final modelling, two approaches were combined. We searched features that are not correlated with others by a selected threshold is equal to 0.8 in absolute values56 and estimated variable importance using the Mean Decrease Gini (MDG) and the Mean Decrease Accuracy (MDA) as the result of modelling on enumerated parameters’ combinations. MDG score is related to the homogeneity of the nodes and leaves coefficient. With the rise of the MDG score the importance of the corresponding feature is also increasing. MDA describes how much accuracy decrease by removing the feature. We selected the most important features according to the MDG and MDA scores by the highest values of both metrics using a sequential search from an initial set of variables.Modelling approachRandom forestChoosing the appropriate method for creating the tool for accurate SDM is crucial because the overall performance could vary dramatically, depending on the selected model and particular use case. There is a limited amount of acceptable machine learning methods that can be used in SDM. Several popular methods demonstrated high performance in modelling on large areas: GBM, RF, and GLM. In particular, for modelling and prediction of the potential distribution of invasive species, GLM and RF were used57. We decided to use RF because this model was successfully implemented for solving a variety of tasks such as predictions of animal and plant distributions, and also was used for making predictions on a large territory58. The other important advantage that should be noticed is the straightforward interpretability of RF, which means that it is possible to evaluate the impact of each environmental parameter on the occurrence of the invasive species.Approach to the cross-validation of the modelA unique approach for the model calibration is needed to reduce spatial autocorrelation caused by the absence of a strict sampling design. In our case, the data was split into training and testing folds using the spatial blocks technique in a scheme of 13-fold cross-validation. Random spatial splitting was performed 20 times to calibrate the model, with a distance between blocks set as 100 km. To calibrate the model we used a spatial blocks approach with random type from R package blockCV.Evaluation of the model performanceTo evaluate the performance of the model a classic approach for ecology was used—Area Under Curve (AUC) or Receiver operating characteristic (ROC), related to the independent threshold techniques16. The principle of methods lies in the standard confusion matrix, where rows and columns represent actual and predicted classes. The construction of ROC curves uses all possible thresholds to obtain different confusion matrices which leads to the reproduction of the curve with two-dimensional space: (1) on y-axis is True Positive Rate (sensitivity, recall); (2) on x-axis is False Positive Rate (equal to 1 − specificity). In our case true positive (TP, sensitivity) rate means that predicted places where HS grows correspond to actual. Similarly, true negative rate (TN, specificity) indicates correctly classified locations as absence points. In contrast, the missteps when the model predicted places as presence points for plants that are incorrect are False Positive, FP, and places where HS is absent, according to the model, while this is not true are recognised as False Negative, FN. More

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The GUTS modelFor completeness, we will prove the result for the most general GUTS model, which will also prove the result for all reduced forms.Theorem 1
For any model version within the GUTS framework, let (S(t; alpha )) denote the survival probability at time t for a given non-zero exposure profile (C_w(t)) scaled by some EMF value (alpha). For any chosen (x > 0) percentage effect (exposure-induced mortality), model end time (t_{E}) and background mortality (h_b) low enough such that (S(t_E; 0) > 0) there exists a unique EMF (alpha _*) such that$$begin{aligned} S(t_{E}; alpha _*) = left( 1 – frac{x}{100}right) S(t_E; 0), end{aligned}$$
(2)
(alpha _*) is the (hbox {LP}_{x}) for the exposure profile.
Baudrot and Charles10 calculated (LC_{50}) values for GUTS-RED-SD and GUTS-RED-IT. Their results implied the result of Theorem 1 for the main regulatory models. Our work makes the result explicit and generalises it to the whole GUTS framework. Another result of Theorem 1 is that the (hbox {LP}_{x}) is monotonically increasing with respect to x. For example, the (hbox {LP}_{50}) will always larger than the (hbox {LP}_{10}) for the same exposure profile. This result comes directly from (S4) in the SI.DEB modelsDue to the additional complexity of the DEB model we split the result into multiple theorems and proofs, starting by showing continuity and monotonicity of the damage ODE.
Theorem 2

Let
(C_w)
be some external concentration over time. Assume an effects model where the effects of higher exposure on growth and/or reproduction are always adverse (or zero) at all points in time. Then, defining the scaled damage ODE as
$$begin{aligned} begin{aligned} frac{D(t; alpha )}{dt} =&k_d(x_u alpha C_w – x_eD) – (x_G + x_R)D. end{aligned} end{aligned}$$
(3)

Then, for any combination of feedbacks (varvec{X} = [varvec{X}_u, 0, varvec{X}_G, varvec{X}_R]), damage is monotonically increasing with respect to (alpha), and is continuous with respect to (alpha) as long as changes to L and R are continuous. Moreover, damage is strictly monotonically increasing with respect to (alpha) whenever (D(t;1) > 0).
The limitation that (varvec{X}_e = 0) will be discussed in greater detail later. However, depending on the pMoA of the stressor we can extend the result of Theorem 2 slightly.
Corollary 3.1
If the pMoA of a substance directly affects reproduction and does not affect growth, i.e. (varvec{S} = [0, 0, 0, s_{R}, s_{H}]) then the results of Theorem 2holds for any combination of feedbacks.
Finally, we can step from the results of Theorem 2 and Corollary 3.1 to show the existence and uniqueness of a critical multiplier ((hbox {EP}_{x}) or (hbox {LP}_{x})) for growth, reproduction and survival.
Theorem 3
Consider the DEB-TKTD model of Jager11 and a substance such that at least one of Theorem 2or Corollary 3.1hold. Further, let (C_w(t)) be a non-zero exposure profile where the time of first exposure is before (t_1) as defined in Table 1. Then, for any chosen percentage effect level (x > 0) there exists a unique EMF (alpha _* >0) such that$$begin{aligned} min left( frac{L(t_{E}; alpha _*)}{L_(t_{E}; 0)}, frac{R_c(t_{E}; alpha _*)}{R_c(t_{E};0)}, frac{S(t_{E}; alpha _*)}{S(t_{E}; 0)}right) = 1 – frac{x}{100} end{aligned}$$
(4)
this (alpha _*) is the (hbox {EP}_{x}) (or (hbox {LP}_{x})) for the exposure profile (C_w(t)).

Table 1 Table of the state variables and pMoAs (including combinations of pMoAs) in the DEB-TKTD model11.Full size table
The monotonicity of effects on all state variables in the DEB model means that, for the conditions described in Theorem 3, the (hbox {EP}_{x}) (or (hbox {LP}_{x})) is also monotonically increasing with respect to x.We should note here that one can either setup an algorithm to find the critical multiplier value for growth, reproduction and survival individually and then select the minimum or setup the algorithm to directly find the minimum critical multiplier as in (4). Both will produce the same result, but the second approach is likely to be faster.One could argue that ERA should consider the combined effects of lethal and sublethal stress on the individual’s fitness. This is possible using the continuous form of the Euler–Lotka equation24$$begin{aligned} B(t) = int _0^t B(t-a) l(a)b(a) da, end{aligned}$$
(5)
where B(t) is the number of births at time t, l(a) is the fraction of females which survive to age a and b(a) is the birth rate for mothers of age a. For the offspring of a test population which all have the same age (as is the standard in long-term toxicity experiments) this integral collapses to a single point, (B(t-a) = 1) when (t=a) and zero elsewhere. The DEB model provides exactly the values which we need to calculate B(t). Namely$$begin{aligned} l(a) = S(a), quad b(a) = frac{d}{dt}R_c(a). end{aligned}$$One can now find the births per individual per time predicted by the DEB model as$$begin{aligned} B(t) = S(t)frac{d}{dt}R_c(t). end{aligned}$$
(6)
Integrating (6) over the duration of the experiment gives the expected number of offspring produced per female alive at the start of the test.There are two clear options for how to proceed. Firstly, one could calculate (int _0^{t_E} B(t) dt) for each EMF and compare it to the control, similar to finding (hbox {EP}_{x}) values for individual endpoints. Alternatively, one can use B(t) as the basis to estimate the intrinsic population growth rate25. This quantity provides an estimation of population growth based on the survival and fecundity over time of individuals. Indeed, it is listed as a potential output value in the experimental guidelines for standard Daphnia magna reproduction tests26. For the first of these options we offer an extension to Theorem 3.
Corollary 3.2
Consider a DEB-TKTD model and exposure profile such that Theorem 3holds. The number of expected offspring per female, given by$$begin{aligned} mathrm {B}(t_E; alpha ) = int _0^{t_E} S(t; alpha )frac{d}{dt}R_c(t; alpha ) dt end{aligned}$$has a unique (hbox {EP}_{x}) (alpha _*) such that$$begin{aligned}frac{mathrm {B}(t_E; alpha _*)}{mathrm {B}(t_E; 0)} = 1 – frac{x}{100}end{aligned}$$
Our results provide a rigid boundary to the applicability domain of the EMF approach both in terms of existence and uniqueness. Existence relies on the initial time in the profile when external concentration is non-zero, as described in Table 1. While it is important to know about these conditions, they will rarely inhibit an ERA, since long initial periods with zero exposure are uncommon.Cases where uniqueness cannot be guaranteed require more caution and it is unwise to use root-finding algorithms. In the next subsection we explore what can happen outside of this domain and provide suggestions for how to still produce a single reliable (hbox {EP}_{x}) value.Surface:volume scaling of eliminationThere is a reason that in Theorem 2, (varvec{X}_e = 0) was specified. In some cases when (varvec{X}_e = 1) a higher multiplier is no guarantee of higher damage for all time. Consider a substance which acts on assimilation and has surface area:volume scaled elimination (i.e. (varvec{X} = [0, 1, 0, 0])). The damage ODE under some EMF (alpha) is then$$begin{aligned} frac{dD}{dt} = k_d left( alpha C_w – frac{L_m}{L} D right) , end{aligned}$$where (L_m) is the maximum length the organism can reach. The EMF has a positive direct effect on damage, but also an opposing indirect effect. Increasing damage decreases the size of the organism which, due to the surface area:volume elimination of damage, enables faster elimination of damage. As a result, not only does Theorem 2 no longer hold but in fact a larger multiplier value can cause lower damage at some points in an exposure profile. In other words, we observe a paradoxical result whereby more exposure translates to less effect some time after exposure.Figure 3 illustrates what we will refer to as the “more is less” scenario. The exposure consists of a single pulse early in the animal’s life, modelled for two multiplier values, (alpha _2 > alpha _1). During the exposure phase the direct effect of the higher exposure causes higher damage and greater effects on size. After the pulse, external exposure is zero, and therefore the external concentration and uptake remain zero regardless of (alpha). Regardless of the EMF, scaled damage can only decrease during this phase. However, the effects of the higher multiplier are still relevant. As Fig. 2 shows, the feedback processes still influence damage dynamics. The model organism exposed to (alpha _2C_w) is smaller and therefore able to eliminate damage more rapidly because (varvec{X}_e = 1). This eventually leads to lower damage for the model organism exposed to (alpha _2C_w) (i.e. (D(t; alpha _1) > D(t; alpha _2))). The more is less phenomenon can also impact growth and cumulative reproduction, as seen in Fig. 3b,c. Sometime after exposure (L(t;alpha _2) > L(t; alpha _1)) and (R(t;alpha _2) > R(t; alpha _1)). For survival, and any additional endpoints without recovery, this “crossover” is unlikely, mortality during the exposure phase (where (D(t; alpha _2) > D(t; alpha _1))) will almost certainly dominate any mortality during the recovery phase. Figure 3d shows that for certain (x%) effect levels (vertical axis) multiple (hbox {EP}_{x}) values exist.Figure 3An illustration of the issues which can occur using the EMF approach for substances with surface area:volume scaled elimination (i.e. (varvec{X} = [0, 1, 0, 0])). The (non-multiplied) exposure is a constant (1 mu g/L) for the first 14 days and zero thereafter and effects assimilation only ((varvec{S} = [1, 0,0, 0, 0])). (a) Scaled damage, (b) length over time, (c) cumulative reproduction. (d) Endpoint value as a proportion of control after 40 days. The shape of these curves show that certain effect levels can be caused by two distinct multiplier values. Parameter values are (L_0 = 0.1), (f = 1), (r_B = 0.1), (L_p = 0.6), (L_m = 1), (R_m = 15), (kappa = 0.8), (y_P = 0.64) (z_b = 0.1), (b_b = 1), (k_d = 0.05), (varvec{X} = [0, 1, 0, 0]). See the SI for the definitions of these parameter values.Full size imageIn practice, instances of non-uniqueness such as Fig. 3 will be rare since they rely on a sudden and significant decrease in external exposure. Moreover, EMF methods for DEB-TKTD models will include a moving time window method18 consisting of many exposures constructed sequentially and assessed. Each window will produce an (hbox {EP}_{x}) value, but only the lowest will be relevant for the ERA. A time window which starts slightly earlier in the broader exposure profile would feature the same pulse later in the model organism’s lifespan and thus not allow organism recovery. Depending on the exact endpoint used, one would expect those windows to have a lower (and unique) (hbox {EP}_{x}). However, the potential for multiple (hbox {EP}_{x}) values raises concerns across all areas which impose a multiplicative margin of safety. We cannot guarantee that a multiplier resulting in (x%) effects exists nor that any value found by the algorithm is unique.Although not pictured here, maintenance and growth pMoAs and combinations of feedbacks which include (varvec{X}_e = 1) can also produce the “crossover” in the damage values and the “more is less” phenomenon seen in Fig. 3. It can also arise for scenarios which do not feature a deviation from the standard rules for growth (e.g. a starvation phase) and for other DEB based models. The SI features a similar plot to Fig. 3 showing damage crossover for a standard DEB model.Knowing this, the obvious question is how to proceed? Certainly with caution when (varvec{X}_e = 1) is necessary in model calibration and validation. Under such circumstances algorithms must ensure that the (hbox {EP}_{x}) value found is the lowest multiplier which gives (x%) effects when there is a risk of non-uniqueness. The brute force approach, incrementing from zero until the desired effect level is met or exceeded, is one example. Whether it is realistic for higher EMF values to cause reduced effects in vivo then does not alter the conservatism of the approach for ERA.Table 2 summarises the domain where the margin of safety approach can be used in conjunction with a root-finding algorithm without concern in the DEB-TKTD model of Jager11. For model configurations where non-uniqueness could emerge using another method to find the (hbox {EP}_{x}) is advisable. For example, a brute-force approach starting from an EMF of 0 in small increments (e.g. by 0.1). Without good reason, calibration should first be attempted with no feedbacks. Under this guiding philosophy of pursuing model simplicity we expect that the problem cases will be rare.Table 2 A table to mark under which scenarios the EMF approach is and is not guaranteed to produce a unique (hbox {EP}_{x}).Full size tableOther issuesThe damage crossover illustrated in the previous subsection occurs more commonly, and to a greater extent, when the pMoA is assimilation effects. This is because, at least in this standard implementation, stress can cause (100%) effect and completely cease assimilation when (s_A ge 1) (see SI for details). When this is the case, higher exposure (even from an increased multiplier) does not translate to higher stress. This differs from other pMoAs, whose stress values are unbounded. Indeed, replacing (1 – s_A) with (1/(1 + s_A)) in the model ((S5) in the SI) reduces the occurrence and scale of “crossovers” such as Fig. 3. However, the formulation of the pMoA should not be based on how it might affect the algorithm or the EMF.Certain species require further deviations from the standard model. For instance, different life-stages, growth and/or reproduction rules might be introduced to explain observed phenomena. Before models featuring these deviations are used in an EMF approach one should consider the potential issues as we have done in this section. While a proof of existence and uniqueness of the (hbox {EP}_{x}) for each model variant is ideal it is also infeasible. However, modellers should ensure that their approach is robust enough to deal with issues around existence and uniqueness. Checking that the model endpoint is reduced by (x%) when the (hbox {EP}_{x}) is applied to the exposure profile is an easy way to check accuracy and existence. An argument (if not a full, formal proof) for uniqueness should also be considered. In cases where that is not possible, the algorithm must be set up to identify the lowest (hbox {EP}_{x}), or check that no lower values exist.One common addition is to DEB-TKTD models which feature starvation is to assume that there is some maximum amount of starvation/shrinking which an animal can survive. Once that point is met or exceeded death is instantaneous27. Such death mechanisms cause problems. They can introduce a discontinuity in the response versus multiplier value for a given time window (i.e. a “jump” in plots such as Fig. 3). For instance, if in the example given in Fig. 3 the animal was not allowed to shrink, and instead died, then the multiplier of 6 would result in (100%) effects on survival (and significant growth effects). In contrast, the exposure when the multiplier is 2 is survivable and the animal can recover. Presumably, for some critical (alpha _c in (2, 6)) the exact threshold for death is reached. This (alpha _c) is a discontinuity between partial and (100%) effects relative to control. Under some circumstances this will prohibit finding a multiplier which results in exactly (x%) effects, regardless of the method used.There are two readily apparent solutions to this at the individual level. One is to set (alpha _c) as the multiplier for the window, the second is to replace such discrete responses with graded responses. In this example for instance, shrinking could add to the lethal hazard h. It is not possible to universally recommend one approach over the other as it will depend on the species’ behaviour. Once that decision has been made these issues must be recognised and reported by the modellers. More