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The study area
The study was carried out on Egbokodo River (longitude 5° 38′ and 5° 41′ and latitude 5° 36′ and 5° 33′) in Warri South Local Government Area of Delta State, Southern Nigeria (Fig. 1), between the periods of September 2008–May 2009. The river is a brackish and tidal River that serves as a source of water for drinking, washing, and fishing to the communities in the catchment area. Three (3) Stations (tagged Stations A, B, and C) were selected about 150 m apart, based on distinct anthropogenic activities. Station A was located at a vandalized oil pipeline, while Stations B and C were located downstream at points of dredging and municipal waste disposal respectively. Station A was 6.3–9.3 m in depth, Station B was 11.4–16.1, and Station C was 7.5–19.5 m during the study duration.
Figure 1

Map of the study area showing sampled stations. Map designed using QGIS software version 3.10.1 ‘A Coruña’ (QGIS Development Team29). https://qgis.org/en/site/forusers/download.html#.

Full size image

The study area comprises of coarse and interspersed soil with lignite and patches of laterite and sandy clay soil. The climate of the study area is typically tropical. It is characterized by the humid tropical wet and dry climate which is primarily regulated by rainfall. The wet season lasts a period of 7 months (April to October). Rainfall ranged from 15 to 91 mm during this period. The driest months are December to January; with a mean monthly rainfall of 15 mm. The bank of the river was densely shaded by a thick canopy of vegetation, dominated by mangrove plants, Nypa palm, and Rhizophora sp.
Collection of samples (water and zooplankton)
Water samples were collected from the 3 stations using a 1 L sampling bottle which was pre-cleaned with the deionized water at each station. This sampling procedure was repeated monthly from September 2008 to May 2009. The samples were preserved in a cooler and transported to the laboratory where they were refrigerated at − 10 °C before the physiochemical analysis. Preservation and analysis of water samples were according to standard methods of the American Public Health Association (APHA).
Samples of zooplankton were collected at the 3 stations between 0800 and 1100 h by towing a hydrobios plankton net (mesh size 25 µm) with a speed boat at 2 knots, just below the water surface for 5 min at every station. At each station, the filtered zooplankton samples were condensed in a 25 mL plankton bottle and preserved using buffered 4% formalin. Each plankton bottle was properly labeled indicating the stations and dates of collection. This procedure was repeated for 9 months (September 2008–May 2009).
Analysis of water
Determination of pH
The pH was estimated using a PH meter—Orion Model 290A (ASTM D 1293B) and recorded accordingly every month.
Measurement of temperature (°C)
A mercury-in-glass thermometer was used to measure surface water temperature. A stable initial reading was ensured by shaking it the thermometer carefully. Afterward, the thermometer was left inside the water for about 3 min till a stable reading was observed and recorded.
Determination of phosphate
Five (5) mL antimony molybdate was added to 40 mL of water sample was in a 50 mL measuring cylinder. Afterward, 2 mL of Ascorbic acid was added to the mixture. It was left to stand for 30 min for full colour formation2. The absorbance was measured with a UV–visible spectrophotometer at 680 nm.
Phosphate was then calculated thus;

$${text{Phosphate}},({text{mg}}/{text{l}}) = frac{{{text{Y}} – {text{C}}}}{{text{M}}}$$
(1)

In Eq. (1) above, Y = absorbance of the sample.
C = absorbance of blank

$${text{M}} = {text{Gradient}}frac{{({text{B}} – {text{A}})}}{{text{X}}}$$
(2)

B = absorbance of standard (Eq. 2)
A = absorbance of blank
X = concentration of the standard.
Determination of nitrate
Nitrate was tested using the diazotization method—Alpha 419 C/ASTM D3867. 0.5 mL of (0.1% W/V) NaN3 was added to the water sample to remove any NO2 present. 3.0 mL of (2.6% W/V) NH4Cl solution was added. One (1) mL of (2.1% W/V) Borax solution was added. 0.5–0.6 g of spongy cadmium was added. It was then covered and shaken for some 15 to 20 min. Afterward, 7 mL of the solution was transferred to a 25 mL measuring cylinder. 1 mL of (1.0% W/V in 10% HCl) sulphanilamide reagent and was mixed by swirling. After about 3 min, 1.0 mL N-1—naphthalene diamine dihydrochloride (0.1% W/V) was added and mixed thoroughly2. The mark was made-up with distilled water. The blank solution was also subjected to the same treatment as the sample. After about 10–20 min, the absorbance of both the water sample and the blank solutions were measured with a UV–visible spectrophotometer at a wavelength of 543 nm.
Analysis of total petroleum hydrocarbons (TPH)
HP-5 capillary column coated with 5% phenyl methyl siloxane (30 m length × 0.32 mm diameter × 0.25 µm film thickness) (Agilent Technologies) was used as a stationary phase of separation of hydrocarbons from water samples. 1µL of the samples was injected in splitless mode at an injection temperature of 300 °C, and pressure of 13.74psi and a total flow of 21.364 mL/min. Purge flow to split vent was set at 15 mL/min at 0.75 min. The oven was initially programmed at 40 °C (1 min) then ramped at 12 °C/min to 300 °C for 10 min. The temperature of the flame ionization detector was regulated to 300 °C using hydrogen gas. Airflow was at 30 mL/min while nitrogen was used as makeup gas at a flow of 22 mL/min. Agilent 7890B gas chromatography coupled to flame ionization detector (GC-FID) was used for the determination of TPH at 254 nm. After calibration, water samples were analyzed and corresponding TPH concentrations were obtained3,10.
Analysis of oil and grease (OG)
One (1) L separating funnels with retort stand, 100 mL volumetric flask, glass jar, xylene, and anhydrous sodium sulfate were used in determining the concentrations of oil and grease (OG) in the water.
Extraction
Twenty (20) mL xylene was put in a glass jar containing a water sample. The content of the jar was shaken, poured into the separating funnel and shaken again. It was allowed for phase separation and the bottom layer xylene was drained into a 100 mL volumetric flask through a funnel with a plug of glass wool and about 2/3 full with anhydrous Na2SO4.
Another 20 mL xylene was added to the content in the separating funnel, agitated thoroughly and xylene layer was again drained into the same flask. Water was drained into a measuring cylinder and the volume was noted. Separating funnel was rinsed with 20 mL xylene into the same flask as done earlier. It was made up to mark of the extract in the 100 mL volumetric flask with pure xylene.
The oil and grease (OG) was calculated thus:
The concentration of oil reported as OG (mg/L)

$$= frac{{{text{Conc}}. , left( {{text{mg}}/{text{L}},{text{extract}}} right) times {text{DF}} times {text{EV}},{text{(mL)}}}}{{{text{The}},{text{volume}},{text{of}},{text{water}},,{text{(mL)}}}}$$
(3)

In Eq. (3), DF = Dilution factor
CF = Conversion factor from absorbance to mg/L extract
EV = Extraction volume of solvent in (mL).
Analysis of trace metals
Ten (10) mL of water sample was put in a beaker and 2 mL concentrated nitric acid was added to the sample. The mixture was then heated to evaporation and allowed to cool afterward and then transferred into a volumetric flask. It was then allowed to stand for 24 h, after when it was centrifuged at 3000 rpm until clear. The sample was screened for suspended solids which were filtered off before further analysis. The trace metals in the mixture were then read using an atomic absorption spectrophotometer (AAS, Philips model PU 9100) at a wavelength range of 250–350 V using the ABS knob10.
The experimental procedures were conducted as described by Estefan et al.30 and modified by Jones Jr.31.
Quality control and quality assurance
Validation of trace metals
The precision of the AAS was validated by repeating every experimental procedure 3 times. Certified reference materials (CRM) and standard reference materials (SRM) published by the Federal Environmental Protection Agency32 were employed as a guide. The recovery rates ranged from 87 to 95%. The calculated relative standard deviation (SD) was  More

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SARS-CoV-2 genome data sets and associated travel history
To focus on the early stage of COVID-19 spread, we analyzed SARS-CoV-2 genome sequences and metadata available in GISAID on March 10th8. We curated a data set of 305 genomes by removing error-prone sequences, keeping only genomes with appropriate metadata, and a single genome from patients with multiple genomes available. We assigned each genome a global lineage designation based on the nomenclature scheme outlined in Rambaut et al.28 using pangolin v1.1.14 (https://github.com/hCoV-2019/pangolin), lineages data release 2020-05-19 (https://github.com/hCoV-2019/lineages). We aligned the remaining genomes using MAFFT v.7.45329 and partially trimmed the 5′ and 3′ ends. All sequences were associated with exact sampling dates in their meta-information, except for one genome from Anhui with known month of sampling. Upon visualizing root-to-tip divergence as a function of sampling time, using TempEst v.1.5.330 based on an ML tree inferred with IQ-TREE v.2.0-rc131, we removed one potential outlier. The root-to-tip plots without the outlier are shown in Supplementary Fig. S3. We formally tested for temporal signal using BETS32. The final 282 genomes were sampled from 28 different countries, with Chinese samples originating from 13 provinces, one municipality (Beijing), and one special administrative area (Hong Kong), which we considered as separate locations in our (discrete) phylogeographic analyses. Phylogenetic signal in the data set was explored through likelihood mapping analysis33 (Supplementary Fig. S4).
We searched for travel history data associated with the genomes in the GISAID records, media reports, and publications and retrieved recent travel locations for 64 genomes (22.5%, Supplementary Table 2): 43 traveled/returned from Hubei (Wuhan), 1 from Beijing, 3 from China without further detail (which we associated with an appropriate ambiguity code in our phylogeographic analysis that represents all sampled Chinese locations), 2 from Singapore, 1 from Southeast Asia (which we also associated with an ambiguity code that represents all sampled Southeast Asian locations), 7 from Italy, and 7 from Iran. In this data set, Italy is better represented by recent travel locations than actual samples (n = 4) and Iran is exclusively represented by travelers returning from this country. For 46 out of the 64 genomes, we retrieved the date of travel, which represents the most recent time point at which the ancestral lineage circulated in the travel location.
In order to examine (i) to what extent our reconstructions could be updated by the genome data that has become available retrospectively for the same locations and the same time period before March 10 ~4 months after this date, and (ii) how sampling bias can be mitigated by downsampling from the larger collection of available genomes, we assembled an additional data set of 500 genomes. For this purpose, SARS-CoV-2 genomes were downloaded from GISAID on June 23, 2020 and processed according to the COG-UK pre-analysis pipeline (https://github.com/COG-UK/grapevine). Briefly, sequences were aligned to the reference sequence Wuhan-Hu-1 (Genbank accession number NC_045512) using Minimap2 v.2.1734. Problematic sites were masked (https://virological.org/t/issues-with-sars-cov-2-sequencing-data/473), and sequences with More

SCALE model
Combining the processes of biogeochemical transformation, soil erosion/deposition and resultant landscape evolution, and bioturbation by soil fauna, the governing equations, which conserves SOC mass, can be expressed as24:

$$begin{aligned} frac{ partial }{ partial t } int _{0}^{Z} mathbf{C } dz =int _{0}^{Z} mathbf{g } dz – nabla cdot mathbf{q }_C + int _{0}^{Z} nabla cdot big [D(z) nabla mathbf{C } big ] dz end{aligned}$$
(1)

where (mathbf{C }) is the SOC concentration ([M L^{-3}]), (mathbf{C } = [ C_l,C_h,C_b]^T) represents the fast (or litter), slow (or humus), and microbial biomass pool, respectively; (mathbf{g }) is the rate of the biogeochemical transformation process, which is a function of the fast (or litter), slow (or humus), and microbial biomass pool; (nabla cdot mathbf{q }_C) is the surface SOC flux associated with soil transport; and (D(z)) is the bioturbation diffusivity. When the vertical column is discretized into several layers, the equations can be written as:

$$begin{aligned} text {Top soil layer: }&frac{ partial big ( mathbf{C }_1 z_1 big ) }{ partial t } = mathbf{g }_1 z_1 – nabla cdot mathbf{q }_C end{aligned}$$
(2a)

$$begin{aligned} text {Sublayers: }&frac{ partial mathbf{C }_n }{ partial t } = mathbf{g }_n + nabla cdot big [D(z) nabla mathbf{C } big ] dz end{aligned}$$
(2b)

where the subscripts 1 and n denote the surface soil layer and the (n{mathrm{^{th}}}) layer below-surface, respectively. Other details of these equations can be found in24.
The mechanisms of soil transport and the resultant landscape evolution can be categorized into two groups: overland flow-driven transport and diffusion-driven transport from other disturbances (e.g., wind, animal, and raindrop splash). The 2-D mass conservation equation of soil transport and the resultant landscape evolution follows Exner equation:

$$begin{aligned} frac{partial eta }{partial t} = U -nabla cdot q_d – nabla cdot q_s end{aligned}$$
(3)

where (eta) is soil surface elevation [L]; U is the rate of tectonic uplift or glacial rebound ([L T^{-1}]); (q_d) is the volume flux of sediment per unit width by hillslope diffusion ([L^2 T^{-1}]); (q_s) is the volume flux of sediment per unit width by overland flow ([L^2 T^{-1}]).
The diffusion-driven transport (nabla cdot q_d) is a combination of wind erosion, animal disturbance, soil creep, raindrop splash, and biogenic transport. The 2-D equation of (q_d) is expressed as a linear relationship with slope47:

$$begin{aligned} q_d = – D_{x} frac{ partial eta }{partial x} – D_{y} frac{ partial eta }{partial y} end{aligned}$$
(4)

where (D_x) and (D_y) are the soil diffusion coefficient in x and y direction, respectively ([L^2 T^{-1}]). The overland flow-driven transport (nabla cdot q_s) is a combined form of the divergence of stream-power but limited by the detachment capacity48:

$$begin{aligned} nabla cdot q_s = min Bigg ( D_c, frac{q_{s,out} – sum q_{s, in}}{d_s} Bigg ) end{aligned}$$
(5)

where (D_c) is the detachment capacity, which is the upper limit of of local erosion rate ([L/S]); (q_{s,out}) is the sediment flux out of a cell and (sum q_{s, in}) is the total sediment flux into a cell assumed at sediment transport capacity.
The rate of change of SOC on the surface driven by soil transport is (nabla cdot q_s), which has a linear relationship with soil transport flux:

$$begin{aligned} nabla cdot mathbf{q }_{C} = nabla cdot ( k_{soc} mathbf{C }_{1} q_d) + nabla cdot ( k_{soc} mathbf{C }_{1} q_s) end{aligned}$$
(6)

where (mathbf{C } = [C_l , C_h , C_b ]^T), and the subscript 1 denotes the surface soil layer; (q_d) and (q_s) are soil transport flux of diffusion and overland flow; (k_{soc}) is an enrichment ratio, which represents a preferential transport (mobilization and deposition) of SOC. The SOC fluxes driven by diffusion and overland flow sediment transport are:

$$begin{aligned} nabla cdot ( k_{soc} mathbf{C }_{1} q_d)= & {} – frac{partial }{partial x} bigg (k_{soc} mathbf{C }_{1} D_{x} frac{ partial eta }{partial x} bigg ) – frac{partial }{partial y} bigg (k_{soc} mathbf{C }_{1} D_{y} frac{ partial eta }{partial y}bigg ) end{aligned}$$
(7)

$$begin{aligned} nabla cdot big (k_{soc} mathbf{C }_{1} q_s big )= & {} left{ begin{array}{ll} k_{soc} mathbf{C }_{1} D_c , &{}quad mathrm{if} D_c < frac{q_{s,out} - sum q_{s, in}}{d_s} \ frac{ k_{soc} mathbf{C }_{1,out} q_{s,out} - sum big ( k_{soc}mathbf{C }_{1,in} q_{s, in} big ) }{d_s} , &{}quad mathrm{otherwise} end{array} right. end{aligned}$$ (8) The initial conditions include land surface elevation, SOC profiles, soil moisture, and surface water depth for each grid box. Table S1 (in Supplementary Information) lists variables associated with initial conditions. We simulate the SOC surface transport and vertical transformation at a daily time step with 2 (times) 2 (mathrm{m}^2) spatial resolution on the surface and a range of vertical grid sizes varying from 5 to 60 (text{cm}). The initial elevation is from lidar (Light Detection and Ranging) DEM (digital elevation model) with a 2-m resolution, collected by a drone in 2015. The SOC profiles at each 2-D grid are estimated by combining the surface SOC contents from the field survey in 2016 and SOC profiles from soil cores sampled at a nearby site 2-km away (see Section below). We assume that the initial soil thickness is 1-m because the most active soil thickness49 regarding SOC and soil moisture are within the top 1 (text{m}), and the deeper SOC contents are quite uniform19,20, indicating a relatively stable condition. Also, we neglect bedrock weathering, and therefore, the soil thickness change is only from the surface soil erosion or deposition. The SOC stock and soil thickness change in the results are compared with the initial SOC stock in the top 1 m. Other initial values such as soil moisture profile and surface water depth (i.e., initial values assigned spatially uniform at each grid point) have a short-time memory in that the impacts only last for a few days to weeks and are predominantly determined by the external forcing and resulting dynamics (more information in Supplementary Information Table S1). Initial soil organic carbon profiles SOC in a natural setting exponentially decreases with soil depth. Here we assume the following relationship between SOC content and soil depth: $$begin{aligned} SOC(Z) = a e^{-bZ} + c end{aligned}$$ (9) where (text{SOC}(text{Z})) represents SOC content at depth (text{Z}); (text{Z}) is zero at the surface, and positive downward; (a), (b), and (c) are positive and constant parameters, where (a+c) represents surface SOC content, (b) represents the decay rate, and (c) represents the relative stable or immobile SOC at depth. Parameters (b) and (c) in this study are estimated from twenty deep cores—ten sites at hills of the Reference Watershed, eight sites at GLC Watershed in the hills, and two sites in the consolidated gully land (Figure S7a). The SOC contents are shown as dots in Figure S6 (in Supplementary Information). We assume the exponential decay rate ((b)) and the immobile SOC ((c)) are spatially uniform in the cornfield in the consolidated gully area and natural field, respectively. We use the least square non-linear method to fit the sample points, and the fitted curves are solid lines in Figure S6 (in Supplementary Information) with the corresponding relationships obtained as: $$begin{aligned} text {For trees/shrubs: }&SOC(Z) = 8.04 e^{-9.63Z}+2.55 end{aligned}$$ (10a) $$begin{aligned} text {For crops: }&SOC(Z) = 3.71 e^{-7.06Z}+2.27 end{aligned}$$ (10b) hence in the natural area, (b= 9.63) and (c = 2.55); and in the cornfield, (b = 7.06) and (c = 2.27). The parameter (a) varies spatially. Soil samples near the soil surface were collected across the whole area of both the GLC and Reference Watershed in 2016. To ensure that the sampling sites are uniformly distributed and represent all land cover types in each watershed, we superimposed an 80-m (times) 80-m grid on the DEM map. In each grid cell, we selected one representative site to collect soil samples; in consolidated gullies, adjacent sampling sites were spaced at an interval of  40-m because of the relatively narrow width. SOC was measured in a laboratory by using the dichromate oxidation method50. In the GLC Watershed, soil samples were collected at 89 locations with 178 samples (0–10 cm and 10–20 cm); in the Reference Watershed, soil samples were collected at 72 locations with 144 samples (Figure 2a, in Supplementary Information). We used Kriging51 to interpolate the SOC content in each 2-D grid box in the two watersheds. Then (a) is back-calculated with the given values of the two layers at 5 cm and 15 cm (the middle point of the two layers, respectively). The final surface SOC (which equals (a+c)) is shown in Figure S7a (in Supplementary Information). Forcing data To explore the co-evolution of the landscape and the vertical profiles of SOC over the decadal time scale, we target a 50-years simulation. The meteorological data is collected from the China National Field Observation Station in An’sai ((36^{circ } 51^{prime } 30^{prime prime } N), (109^{circ } 19^{prime } 23^{prime } E); data source: http://asa.cern.ac.cn), 44 km away to the northwest from the two watersheds, with a 10 years record from 2008 to 2017, which is the best available data for the simulation. The mean annual precipitation is 560 mm. These data are used to train a stochastic Weather Generator52, which is used to create an ensemble of another 40 years of data (Figure S7c in Supplementary Information). Landcover is also obtained from the field survey in 2015 (Figure S7b in Supplementary Information). It is essential for simulating surface water runoff and the input of SOC from plant residues. Different types of landcover provide different fractions that control the surface water runoff velocity, and such fraction is represented by Manning’s coefficient (Table S2 in Supplementary Information). The plant residues include dead leaves, roots, stems, and corn stover after harvest. Here, we estimate plant residues as a function of the Normalized Difference Vegetation Index (NDVI) (Figure S3a) which is obtained from Landsat satellite data (see more information in the Section below). Litter input estimation The NDVI is collected from Landsat satellite data for a full 2 years period, 2016 and 2017. It is spatially divided into three areas, the consolidated gully land, the GLC Watershed, and the Reference Watershed. The spatial distribution of NDVIs for the two watersheds are nearly identical, so we took the spatial means of NDVI for the two watersheds excluding the consolidated gully to represent the natural area (Figure S3a in Supplementary Information). During the growing season, the NDVIs in a natural area (Figure S3(a1) in Supplementary Information) are smaller than the one in the consolidated gully (Figure S3a2 in Supplementary Information). This is because the crop inside the consolidated gully has higher vegetation density. We assume the rate of litterfall has an exponentially increasing relationship with NDVI (Figure S3b in Supplementary Information). As the NDVI increases, the plant residues increase in general. When a plant’s growth slows down, the NDVI increase also slows down and is close to the maximum, but on the other hand, the litterfall rate increases much faster near the end of the growing season. This characterization allows us to fill in the gap due to the lack of litterfall data about the various land vegetation types, including the cornfield in the consolidated gully. Governing equations of SOC transformation The equation below shows the SOC transformation, which is directly affected by litter input and decomposition rate29: $$begin{aligned} frac{partial big ( C_l + C_h + C_b big ) }{partial t} = I_{litter} - big (r_r K_l C_l+r_r K_h C_h big ) end{aligned}$$ (11) where (C_l), (C_h), and (C_b) are defined in Eq. 1; (I_{litter}) is the litter input from the sum of above-ground litter fall and below-ground root-litter ([M L^{-2}T^{-1} ]); (r_r) defines the fraction of decomposed organic carbon to (hbox {CO}_2) ([-]) ((0 le r_r le 1-r_h)), which typically ranges between 0.6 and 0.8; (K_l) and (K_h) are rates of carbon decomposition in fast and slow pool, respectively ([T^{-1}]). They are regulated by soil moisture and carbon-nitrogen ((C/N)) ratio as shown below29: $$begin{aligned}&K_l = varphi (C/N) f_d(theta ) k_l C_b end{aligned}$$ (12a) $$begin{aligned}&K_h =varphi (C/N) f_d(theta ) k_h C_b end{aligned}$$ (12b) where (k_l) and (k_h) represent the rate of decomposition as a simplified term that encompasses different organic components in the litter and humus pool, respectively ([L^3 T^{-1} M^{-1}]); (varphi (C/N)) is a ratio that is from the reduction of the decomposition rate if the immobilization (controlled by nitrogen content) fails to meet the nitrogen demand by the microbes ([-]). (varphi approx 1) in agricultural fields where nitrogen supply is usually sufficient from fertilizers; (f_d(theta )) ([-]) represents the soil moisture effects on decomposition29. The optimal soil moisture condition is the field capacity which provides the highest (f_d)29, meaning that very dry or very wet conditions will result in a smaller (f_d), and hence reduce the decomposition rate. The decomposition rates for litter (or fast) pool and humus (or slow) pool from the equations are (r_r K_l) and (r_r K_h), respectively. In this study, we test the different mean residence times on the surface soil layer (5-cm) by assigning a new decay rate of the decomposition parameter (k_h) in humus (or slow) pool. A complete list of parameters can be found in Supplementary Information Table S2. More

Quantifying structural sensitivity in models with uncertain component functions
In general, we consider a system of the form:

$$begin{aligned} dot{{mathbf{x }}}={mathbf{G}} left( h_1left( {mathbf{x}} right) ,h_2left( {mathbf{x}} right) ,ldots ,h_pleft( {mathbf{x}} right) , f_1left( {mathbf{x}} right) ,f_2left( {mathbf{x}} right) ,ldots ,f_{m-p}left( {mathbf{x}} right) right) , end{aligned}$$
(1)

where ({mathbf {x}}in {mathbb {R}}) is the vector of d state variables, (h_i, f_i:{mathbb {R}}^{d_i}rightarrow {mathbb {R}}) are the m different component functions describing the inflows and outflows of biomass, energy or individuals due to certain biological processes, with ({mathbf{G}} :{mathbb {R}}^mrightarrow {mathbb {R}}^d) being a composition function describing the general topology of the system. We consider that the precise mathematical formulation of the functions (f_i) are known (or at least postulated) with the only related uncertainty being the precise choice of their parameters. The functions (h_1,ldots h_p) are considered to have unspecified functional form. Instead, they are represented by bounds on their derivatives matching the qualitative properties we would expect from such a function. For example, the per-capita reproduction rate of a population is generally decreasing, at least at large population numbers, while a feeding term described by a Holling type II functional response of a predator should be increasing and decelerating. The (h_i) may also have quantitative bounds on their values:

$$begin{aligned} h_i^{text {low}}left( {mathbf{x}} right)1)), an important question remains: which of the unknown functions contribute the most to the degree of structural sensitivity in the system? The degree of structural sensitivity does not distinguish between the various sources of uncertainty and therefore cannot quantify the relative contributions of the unknown functions to the uncertainty in the model dynamics.
To determine the contribution of each unknown function (h_i), one can allow the error terms (left( varepsilon _1,varepsilon _2,ldots ,varepsilon _pright)) to vary with the goal of investigating how the degree of sensitivity varies with them. For the purpose of this section, let us denote the initial error terms by (varepsilon _i^0). We might be tempted to use the dependence on the (varepsilon _i) to perform global optimisation under certain constraints to find the best possible reduction of (left( varepsilon _1,varepsilon _2,ldots ,varepsilon _pright)). However, one should bear in mind that this analysis would depend on the base functions (hat{h}_i) considered. While these functions are ideally fitted to experimental data, they are only accurate within the error terms (varepsilon _i^0). Excessively reducing the (varepsilon _i) will force all admissible functions to conform strongly in their shape to these base functions far beyond their demonstrated accuracy of fit.
The dependence of the degree of sensitivity on (varepsilon _i) should therefore only be evaluated locally by calculating the gradient (left( frac{partial Delta }{partial varepsilon _1},ldots ,frac{partial Delta }{partial varepsilon _p}right) |_{left( varepsilon _1^0,ldots varepsilon _p^0right) }) giving the direction for the best local reduction of the errors. To adjust for the fact that the error terms may be of different orders of magnitude, when handling the vectors of error terms we should use the norm

$$begin{aligned} left| varvec{varepsilon }right| = sqrt{sum _{i=1}^{p} left( frac{varepsilon _i }{varepsilon _i^0}right) ^2}. end{aligned}$$
(9)

Working in this norm, the gradient needs to be weighted by the initial error terms to provide the direction for the best local reduction of the error terms, this is described by the following structural sensitivity gradient.
Definition 2
The structural sensitivity gradient in a model with p unknown functions each having an error of magnitude (varepsilon ^0_i) is defined as

$$begin{aligned} left( -varepsilon _1^0cdot frac{partial Delta }{partial varepsilon _1},ldots ,-varepsilon _p^0cdot frac{partial Delta }{partial varepsilon _p}right) |_{left( varepsilon _1=varepsilon _1^0,ldots ,varepsilon _p=varepsilon _p^0right) }, end{aligned}$$
(10)

where (Delta left( varepsilon _1,ldots ,varepsilon _pright)) is the degree of structural sensitivity of the system considered as a function of the error terms (varepsilon _i).
One possible problem with the structural sensitivity gradient is that the degree of structural sensitivity in the system may not be an increasing function of the magnitude of the errors. Consider the case that the exact system is structurally unstable, e.g. at a bifurcation point. Then no matter how small the error terms are, there may still be very high levels of structural sensitivity, while larger error terms may cause the level of uncertainty to decrease5. In this case, the structural sensitivity gradient will indicate that one or more of the functions has a negative contribution to the uncertainty of the system, and cannot be taken as a basis for sensitivity analysis.
An alternative approach to quantifying the individual impact of unknown functions which avoids this issue is the computation of partial degrees of sensitivity with respect to each (h_k). To do this, we fix every unknown function except (h_k), a set which we denote ({mathbf{H}} _{sim k}), by fixing the (x_j^*), (h_ileft( {mathbf{x}} ^*right)), and (frac{partial h_i}{partial x_j} left( {mathbf{x}} ^* right)) that are consequently determined by the isocline equations. Denoting by (V_k) the cross-sections of V where only (h_k) varies, and the cross-sections for ({mathbf{H}} _{sim k}) by (V_{sim k}), the local partial degree of structural sensitivity can be defined as follows.
Definition 3
The local partial degree of structural sensitivity with respect to (h_k), is the degree of structural sensitivity in the model when (h_k) is unspecified and all other functions (h_i in {mathbf{H}} _{sim k}) are fixed:

$$begin{aligned} Delta _k({mathbf{H}} _{sim k}):= 4 cdot int _{V_{k_{text {stable}}}} rho _{mathbf{H} _kvert {mathbf{H}} _{sim k}} , d{mathbf{H}} _k cdot left( 1 – int _{V_{k_{text {stable}}}} rho _{mathbf{H} _kvert {mathbf{H}} _{sim k}} , d{mathbf{H}} _k right) , end{aligned}$$
(11)

where (rho _{mathbf{H} _kvert {mathbf{H}} _{sim k}}) is the conditional probability density function on ({mathbf{H}} _{sim k}):

$$begin{aligned} rho _{mathbf{H} _kvert {mathbf{H}} _{sim k}} = frac{rho }{int _V rho d{mathbf{H}} _{sim k }}, end{aligned}$$

with (rho) the (joint) probability distribution over V.
The local partial sensitivity (Delta _k({mathbf{H}} _{sim k})) is a function of ({mathbf{H}} _{sim k}) in that it depends upon the particular values at which the elements of (V_{sim k}) are fixed. As with the degree of structural sensitivity, it can be interpreted as either the probability that the stability of the given equilibrium will be different for two independent choices of the function (h_k) when the (h_{sim k}) are fixed at the given values, or in terms of variance as (Delta _k({mathbf{H}} _{sim k})=4cdot {text {Var}}_k(Yvert {mathbf{H}} _{sim k})) (Y is the Bernoulli variable for stability). If the joint probability distribution (rho) is uniform in V, then (Delta _k({mathbf{H}} _{sim k})) can be expressed purely in terms of the fraction of the volume of (V_k) which gives a stable equilibrium:

$$begin{aligned} Delta _k({mathbf{H}} _{sim k}) = 4 cdot frac{int _{V_{k_{text {stable}}}} d{mathbf{H}} _k}{int _{V_k} d{mathbf{H}} _k} cdot left( 1 – frac{int _{V_{k_{text {stable}}}} d{mathbf{H}} _k}{int _{V_k} d{mathbf{H}} _k} right) . end{aligned}$$
(12)

To obtain a global measure for the sensitivity of the model to (h_k), we can take the average of (Delta _k) over (V_{sim k}).
Definition 4
The partial degree of structural sensitivity with respect to (h_k) is given by

$$begin{aligned} bar{Delta }_k := int _{V_{sim k}} rho _{mathbf{H} _{sim k}} cdot Delta _k({mathbf{H}} _{sim k}) , d{mathbf{H}} _{sim k} end{aligned}$$
(13)

where (rho _{mathbf{H} _{sim k}}) is the marginal probability density function of ({mathbf{H}} _{sim k}).
Recalling the variance-based interpretation of the degree of sensitivity, we obtain (bar{Delta }_k = 4cdot E_{sim k}({text {Var}}_kleft( Yvert {mathbf{H}} _{sim k}right) )). In other words, (bar{Delta }_k) gives the scaled average variance when all functions except (h_k) are fixed. We can also relate the partial degree of structural sensitivity to indices used in conventional variance-based sensitivity analysis. Dividing (bar{Delta }_k) by the overall degree of structural sensitivity in the model gives us (frac{bar{Delta }_k}{Delta }=frac{E_{sim k}({text {Var}}_kleft( Yvert {mathbf{H}} _{sim k}right) )}{{text {Var}}(Y)}=S_{T_k}), the total effect index23 of (h_k) on the stability of the equilibrium. This is a measure of the total contribution of (h_k) to the sensitivity—both alone and in conjunction with the other functions ({mathbf{H}} _{sim k}). However, since the space of valid functions V is in general not a hypercube, the functions (h_i) are not independent factors, and a total decomposition of variance is not possible. Indeed, even if the joint probability distribution (rho) is uniform, the marginal probability distribution (rho _{mathbf{H} _{sim k}}) will generally not be: instead it will equal the volume of the corresponding cross-section (V_k) for ({mathbf{H}} _{sim k}), divided by the volume of V. An alternative to using the partial degrees of sensitivity would be to consider the first-order sensitivity indices (S_k=frac{{text {Var}}_kleft( E_{mathbf{H} _{sim k}}left( Yvert h_k right) right) }{{text {Var}}(Y)}). However, these do not take into account possible joint effects of the (h_i) on the structural sensitivity of the system, so a small (S_k) does not indicate that (h_k) is not a source of sensitivity, whereas (bar{Delta }_k=0) means that (h_k) does not contribute to the structural sensitivity in the system at all.
Similarly to the gradient of the total degree of sensitivity (Delta) as a function of the respective error tolerances, the vector (left( -bar{Delta }_{h_1},ldots , -bar{Delta }_{h_p}right)) needs to be scaled by the elements of (varepsilon ^0) to give us the optimal direction of decrease in (Delta) if the error terms (varepsilon _i) are subject to a proportional reduction. This is described by (left( -varepsilon _1^0 bar{Delta }_{h_1},ldots , -varepsilon _p^0bar{Delta }_{h_p}right)).
Outline of an iterative framework of experiments for reducing structural sensitivity
When dealing with partially specified models, an important practical task is the reduction of the overall uncertainty in the system by decreasing the uncertainty in the system processes (i.e. the unknown model functions). Here we propose an iterative process of such a reduction based on improving our empirical knowledge of the uncertain functions (h_k).
As a starting point, we assume that experiments have produced data on the unknown functions (h_1,ldots ,h_p), to which we can fit some base functions (hat{h}_1,ldots ,hat{h}_p) with initial errors (varepsilon _1^0,ldots ,varepsilon _p^0). We assume that it is possible to perform additional experiments on all uncertain processes in order to obtain more data such that the (varepsilon _i) can be decreased, but with the natural constraint that the total error can only be reduced by a magnitude of (0 More

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