Philippa Kaur

Identification of a regime shift in Beaufort summer sea ice characteristicsFigure 2 shows time series of Beaufort Sea summer sea ice concentration, sea ice age, and sea ice thickness, as well as the ratio of Beaufort ice volume to that of the entire Arctic27. We define the summer to be the months of July, August, and September. A step function has been fit to the time series with a breakpoint determined by a minimization of the root-mean square fit to the data with a significance test of the difference of the means that takes into account the temporal autocorrelation of geophysical time series28; see Methods Section for further information. The first three metrics (Fig. 2a–c) indicate a transition toward less extensive, thinner and younger ice pack occurred around 2007. Furthermore, the Beaufort’s contribution to total Arctic ice volume decreased in 2007 from approximately 10% to 5% (Fig. 2d). We will refer in this study to the period from 2007-present as the “young ice regime,” while the period prior to 2007 will be referred to as the “old ice regime.” All metrics indicate that the summers of 2020 and 2021 (as well as 2013), stand out with ice characteristics above the mean for this new ice regime. This is especially true for the ice volume ratio where values for these past two summers approach those typical of conditions prior to the 2007 transition.Fig. 2: Characteristics of summer (July–September) Beaufort Sea ice.Time series of the: a sea ice concentration (%) from the NSIDC CDR dataset 1979–2021; b sea ice age (years) from the NSIDC dataset 1984–2021; c sea ice thickness (m) from PIOMAS 1979–2021 and d ratio of the volume of Beaufort sea ice to Arctic sea ice from PIOMAS 1979–2021. In all cases, the red lines represent the step function fit with the specified breakpoint that minimizes the root-mean square error in the fit. The statistical significance of the step is indicated in the legend.Full size imageSea ice conditions during 2019/2020 and 2020/2021Figure 3 shows time series of Beaufort Sea ice concentration and thickness for the 2-year period October 2019–September 2021, as well as climatological values for the first 13 years of the young ice regime (2007–2019) and anomalies with respect to these 13 years. The results show that starting in May of both years, concentration and thickness were both higher than the climatology by at least 1 standard deviation. The area-mean thickness anomaly was larger in 2020, while the sea ice concentration anomaly was larger in 2021.Fig. 3: Monthly mean Beaufort Sea ice characteristics from October 1 2019–September 30 2021.Time series (red curves) of the (a) monthly mean sea ice concentration (%) from the NSIDC CDR dataset and (b) monthly mean sea ice thickness (m) from PIOMAS with the climatological monthly mean values shown in black with one standard deviation above/below the mean indicated by the shading. The climatology is based on 2007–2019. In (c) and (d), the corresponding anomalies are shown with the shading representing +/− one standard deviation.Full size imageFigure 4 provides the Beaufort Sea ice thickness and age distributions in summer for (i) the first 13 years of the old ice regime (1979–1991) when the region was dominated by multi-year ice, (ii) the first 13 years of the young ice regime (2007–2019), (iii) the year 2020, and (iv) the year 2021. A kernel smoothing technique29 was used to fit the distributions to the data. The old ice regime was dominated by thick, old ice, with smaller contributions from thin, young ice. In contrast, the young ice regime is dominated by thin, young ice with a long “tail” of thick, old ice. The years 2020 and 2021 are representative of this young ice regime, although with thick and old ice generally ≥1 standard deviation above the mean (An exception is the amount of ice older than ~2 years in 2020, which is very close to the mean). Further analysis (Supplementary Fig. S1) indicates that many years in the young ice regime show small secondary peaks of thick or old ice (such as seen in the 2021 thickness distribution between 1.5 and 2 m). These “long-tailed” thickness and age distributions are similar to that found in summer 2020 in the Wandel Sea1. Thus, it seems that the Beaufort Sea is now dominated by thin, young ice, but a substantial component of thick, old ice remains. In the following sections, we examine the advective origins of this thick, old ice.Fig. 4: Frequency distribution of summer (July–September) sea ice characteristics in the region of interest.a PIOMAS sea ice thickness distribution and b NSIDC sea ice age distribution. Climatological distributions for 1979–1991 (1984–1991 for ice age) and 2007–2019 are shown as well as distributions for 2020 and 2021. The shading represents one standard deviation above/below the 2007–2019 mean.Full size imageImpact of sea ice transport on the observed anomalies during the summers of 2020 and 2021Recent work23,24,25 has emphasized the role that sea ice mass transport plays in determining the characteristics of pack ice in the Beaufort Sea. This transport can be decomposed into contributions from ice motion and from ice thickness; the seasonal climatology of these constituents as well as conditions during 2019/2020 and 2020/2021 are shown in Fig. 5. The climatology (Fig. 5a–d) indicates the presence of a seasonally varying anticyclonic Beaufort Gyre in the western Arctic as well as the presence of the thickest ice along the northern coast of Greenland and the Canadian Arctic Archipelago, i.e., the LIA. The spatial extent of the Beaufort Gyre is largest during the cool season, defined as fall (OND), winter (JFM) and spring (AMJ) when there is transport of ice from the LIA into the Beaufort Sea as well as transport of ice out of the Beaufort Sea into the Chukchi Sea. During summer (JAS), the Beaufort Gyre shrinks to only fill the Beaufort Sea.Fig. 5: Annual cycle in seasonal mean (OND: October–December; JFM: January–March; AMJ: April–June; JAS: July–September) sea ice thickness (shading – m) and sea ice motion (vectors- km/day).Results are shown for climatology (a–d) as well as 2019/2020 (e–h) and 2020/2021 (i–l). The polygon indicates the region along the Beaufort Coast over which statistics were computed. All fields are from PIOMAS.Full size imageThe situation during 2019/2020 (Fig. 5e–h) differs markedly from the climatology. During fall 2019 (Fig. 5e), the Beaufort Gyre was displaced southwestward with a small region of cyclonic ice motion at the boundary between the Chukchi and Beaufort Seas. Consistent with the collapse of the Beaufort High during winter 202026, ice motion during this period (Fig. 5f) is generally eastward in the Beaufort Sea and largely cyclonic over the entire Arctic Ocean. This results in ice transport from the Chukchi Sea into the Beaufort Sea and even beyond, i.e., into the LIA. In spring 2020 (Fig. 5g), transport continued from the Beaufort Sea to the LIA, although the Chukchi-to-Beaufort transport abated. By summer 2020, ice motion had reverted toward climatology (Fig. 5h).Conditions during 2020/2021 (Fig. 5i–l) were closer to climatology as compared to 2019/2020, although with some differences. Most notably during fall 2020 (Fig. 5i), the transport of thick, old ice from the LIA was restricted to a narrow region along the coast of the Canadian Arctic Archipelago, which appears to be linked to the presence of thick ice in the eastern Beaufort Sea. As discussed previously23, this strong transport continued into winter 2021 (Fig. 5j), although its width increased and thus broadly impacted the northeastern Beaufort Sea. There was also strong westward transport out of the Beaufort into the Chukchi Sea.Figure 6 shows the anomalies in sea ice motion, mass convergence, and thickness for the winters of 2020 and 2021 as well as the anomalies in sea-level pressure and 10 m wind fields for the same periods. The contrast in ice motion and sea ice thickness between the two winters is striking. During winter 2020 (Fig. 6a), anomalous cyclonic ice motion is evident as well as anomalously thick sea ice against Banks Island caused by convergence forced by eastward motion at this time (Fig. 6c, which actually started in fall 2019, Fig. 5f). Convergence also extends from the eastern Beaufort into the western LIA, where it acts to counter the long-term thinning trend; the result is enhanced negative ice thickness anomalies. This is supported by a comparison with winter 2021 (Fig. 6b, d), when thickness anomalies were much more negative and ice motion in the western LIA was closer to climatology, i.e., weakly divergent. Comparison of ice motion and thickness fields in the winters of 2020 and 2021 (Supplementary Fig. S2) demonstrates that the differences between these 2 years extend all the way from the Chukchi and Beaufort Seas into the western LIA.Fig. 6: Anomalous nature of the winter (JFM) sea ice and atmospheric circulation during 2020 and 2021.Sea ice thickness (shading – m) and sea ice motion (vectors- km/day) anomalies with respect to climatology (2007–2019) for: a 2020 and b 2021. Sea ice mass convergence (shading – m/month) and sea ice motion (vectors- km/day) anomalies with respect to climatology (2007–2019) for: c 2020 and d 2021. Sea-level pressure (contours – mb), 10 m wind (vectors- m/s) and 10 m wind speed (shading-m/s) anomalies with respect to climatology (1979–2021) for: e 2020 and f 2021. The polygon indicates the region along the Beaufort Coast over which statistics were computed. Sea ice fields are from PIOMAS. Atmospheric fields are from ERA5.Full size imageThe atmospheric circulation anomalies for these two winters highlight the role that sea-level pressure plays in forcing ice motion. During winter 2020 (Fig. 6e), the collapse of the Beaufort High26 resulted in lower sea-level pressures across the Arctic Ocean associated with a minimum 16 mb lower than climatology centered over the Barents Sea. Associated with this collapse, a cyclonic surface wind anomaly was present across the Arctic Ocean with a particularly high amplitude across the western boundary of the Beaufort Sea. In contrast, winter 2021 (Fig. 6f) was characterized by higher sea-level pressure over the Arctic Ocean with a maximum anomaly of 8 mb over the Barents Sea. As a result of this pressure perturbation, wind speeds were higher over the Arctic Ocean but did not reach the magnitudes observed during winter 2020.Quantifying the role of ice transport in anomalous Beaufort Sea ice conditions during the winters of 2019/2020 and 2020/2021Ice area and volume fluxes provide a way to quantify the transport of sea ice30. Figure 7 shows the cumulative fluxes across the boundaries of the Beaufort Sea (as defined in Fig. 1) from October 1 through the following June 1 for 2019/2020, 2020/2021, as well as a climatology for the first 13 years of the young ice period 2007–2019. Positive values indicate a flux into the region. Daily PIOMAS ice motion and ice thickness data were used to calculate these fluxes. The ice area fluxes were also computed using the NSIDC ice motion data31 with similar results obtained (Supplementary Fig. S3).Fig. 7: Variability in the PIOMAS sea ice fluxes into the region of interest.Cumulative: a ice area (105km2) flux and b ice volume flux (102km3) through the northern boundary of the region of interest. Cumulative: c ice area (105km2) flux and d ice volume flux (102km3) through the western boundary of the region of interest. The net cumulative: e ice area (105km2) flux and f ice volume flux (102km3) through the northern and western boundaries of the region of interest. Results are shown for climatology (2007–2019) as well as for 2019–2020 and 2020–2021 with the shading representing +/− one standard deviation above/below the climatological mean. Positive fluxes are into the region of interest.Full size imageWe first consider the northern boundary. The ice area and volume fluxes across this boundary are relatively small in the climatology (Fig. 7a, b), with interannual variability that includes some years in which the fluxes are negative, i.e., from the Beaufort Sea toward the LIA. During the period from October to January in 2019/2020 as well as in 2020/2021, ice area and volume fluxes are positive and growing, at rates near or above the climatological mean, indicating intensifying ice transport into the Beaufort Sea. After January, the 2 years differ. In winter 2020 the cumulative fluxes plateau, indicating near-zero values in contrast to the climatology which continues to grow. Then in spring 2020 the fluxes turn strongly negative, with values of one or more standard deviation below the mean, implying an export of ice from the Beaufort Sea into the LIA. In fact, the cool season 2019/2020 ends with an unusually large net export of ice volume from the Beaufort into the LIA. The following year, we see that the fluxes in winter 2021 continue to intensify at about 1 standard deviation above the mean. Then in spring the cumulative fluxes decline back toward the climatological mean, with end-of-cool-season values near the climatological young ice regime mean of net transport from the LIA into the Beaufort.At the western boundary, climatological ice area and ice volume fluxes are both directed out of the Beaufort Sea and into the Chukchi Sea (Fig. 7c, d). Although interannual variability is higher than that at the northern boundary, the fluxes are typically always negative. This is what makes the 2019/2020 area and volume fluxes so remarkable, in that they are nearly zero through winter 2020, and then turn strongly positive in spring, with values at or exceeding the mean by more than one standard deviation throughout the entire period. These positive fluxes reflect strong ice import from the west (Fig. 5f). In contrast, the fluxes in 2020/2021 became large and negative by early winter (greater than 1 standard deviation from the climatological mean), although this moderates later in the winter and spring. In this year, fluxes were strongly westward, from the Beaufort into the Chukchi Sea.The sum of the fluxes across the two boundaries provides a measure of the net transport into the Beaufort Sea (Fig. 7e, f). The climatology indicates that the net ice area and volume fluxes are negative, indicating a loss of ice from the Beaufort Sea. This reflects the fact that the transport out of the region through the western boundary usually exceeds the transport into the region through the northern boundary. In this context, the net fluxes during 2019/2020 again stand out as remarkable, since they are strongly positive (i.e., net transport into the Beaufort), especially for ice area flux. The net fluxes during 2020/2021 are closer to climatological values, and are well within the range of climatological variability.Impact of cool season ice fluxes on Beaufort Sea summer ice conditionsIn this section, we seek to quantify how the cool-season sea ice transport into the Beaufort Sea impacts ice conditions in the following summer using two metrics. The first metric is Beaufort Sea ice volume (i.e., the product of ice thickness and ice concentration27) on June 1. Even though melt can occur in parts of the study region prior to the beginning of June32, it is nevertheless a useful date for the start of the melt season. Our second metric is Beaufort Sea area-mean ice concentration during September, a measure of ice conditions at the end of the melt season and a closely observed indicator of climate change33,34.In Fig. 8, we correlate the net ice volume flux over the cool season, i.e., the period from October 1 to June 1 of the following year, against the Beaufort Sea June 1 ice volume anomaly, calculated by detrending the time series using a step function in 2007 that takes into account the changes between the new and old ice regimes (Fig. 2). The ice volume flux does not exhibit any trend and so no detrending was done for this time series. The correlation was done for both the old and young ice regimes. For both periods, there is a statistically significant linear relationship showing that larger net cool season ice transport into the Beaufort Sea leads to larger ice volume anomalies on June 1. However, the larger spread in the data for the old ice regime leads to a smaller percentage of the variance explained by ice transport, ~14%, as compared to ~45% for the young ice regime. The statistics are similar if May 1 is used as the end of the cool season, although using April 1 degrades the relationship to statistical insignificance, consistent with the springtime “predictability barrier”35 that arises from late-winter variability in ice-dynamics and ice growth.Fig. 8: Relationship between cumulative cool season ice volume flux into the Beaufort Sea region and June 1 Beaufort Sea ice volume 1980–2021.Scatterplot of the cool season (October 1 – June 1 following year) PIOMAS ice volume flux and June 1 PIOMAS ice volume anomaly. Linear least squares fit to the data for the two regimes are also shown as are the percentage of the variance explained. The ice volume time series has been detrended by step functions with a breakpoint in 2007.Full size imageRegarding conditions at the end of the summer, it seems intuitive that ice retreat might be slowed by the presence of thick ice. Indeed, discussions in the popular press5 have speculated that thick ice contributed to the relatively moderate September 2021 sea ice extent (12th lowest on record and the highest since 2014). On the other hand, it has also been suggested that cool atmospheric conditions during the summer of 2021 contributed to this relative maxima in sea ice extent36.To explore this question, we correlate PIOMAS-derived Beaufort Sea ice volume on June 1 with NSIDC CDR-derived September-mean sea ice concentration37. Given the nature of the underlying time series (Fig. 2), we have used step functions with a breakpoint in 2007 to detrend the data (see Materials and Methods). Although there is considerable spread, Fig. 9 indicates that there is a statistically significant linear relationship, with June 1 ice volume accounting for just under 40% of the variability in ice concentration during September for both the old and new ice regimes. Similar results are obtained if one uses the PIOMAS sea ice concentration during September (Supplementary Fig. S4).Fig. 9: Relationship between June 1 ice conditions and September mean ice concentration 1979–2021.Scatterplot of the June 1 PIOMAS ice volume anomaly and the NSIDC CDR September monthly mean ice concentration anomaly. Linear least squares fit to the data for the two regimes are also shown as are the percentage of the variance explained. Both time series have been detrended by step functions with a breakpoint in 2007.Full size imageA next logical step might be to link these two correlations together and ask, How does cool season ice transport impact end-of-summer ice concentration? Given the above results, assuming that there are no other factors related to cool season transport that impact summer ice melt and the cascade of probabilities, one would expect that the former would explain ~5% and ~16% of the variability in the latter for the old and new ice regimes. The results shown in Supplementary Fig. S5 confirm these assumptions; however, we also find that this relationship is not statistically significant in either regime. More

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Modelling complex evolved food websOur interest here is to develop a conceptual comparison between the eigenvalue spectrum of a complex, evolved food web and a random matrix analog. We therefore focus on the widely-used generalised Lotka–Volterra equations for consumer-resource interactions. For simplicity, we further restrict to a single basic nutrient source, and require that species feeding on the basic nutrient source are never omnivorous31, e.g., plants do not consume other plants. The original Lotka–Volterra equations32,33 describe spatially and temporally homogeneous, consumer-resource relations. The generalised Lotka–Volterra equations34,35,36 can be used to describe the dynamics of larger, more complex food webs, and encode the dynamics of primary producers as$$begin{aligned} frac{dot{S_i}}{S_i} = k_i left( 1 – sum _{j=1}^{n_1} S_j right) – alpha _i – sum _{k=n_1+1}^{n} eta _{ki} S_k, end{aligned}$$
(1)
where (S_i), (iin {1,dots ,n_1}), denote the population densities of primary producers in units of biomass, normalised to the system carrying capacity and (n_1) denotes the total number of primary producers, (k_i >0) denote the growth rates of the corresponding primary producer (S_i), that is, the maximal reproduction rate at unlimited nutrient availability. We use (k_i=k) for all primary producers. The negative sum on the species (S_j) encodes logistic growth by accounting for nutrient depletion by all primary producers. For all other species, (S_k), (kin {n_1+1,dots ,n}) with n the total number of species in the food web, the equations read$$begin{aligned} frac{dot{S_k}}{S_k} = sum _{m=1}^{n} beta _{km}eta _{km} S_m – alpha _k – sum _{p=n_1+1}^{n} eta _{pk} S_p. end{aligned}$$
(2)
Here, (S_k) is again measured in units of normalised biomass. In Eqs. (1) and (2), (alpha _j >0) is the decay rate of a species (S_j), representing death not caused by consumption through other species. (eta _{ki}ge 0) is the link-specific interaction strength between consumer (S_k) and resource (S_i). On the RHS of either equation, note the final term representing the diminishing effects experienced by each resource species, which is caused by consumption. This term is mirrored by the first term in Eq. (2), which describes the strengthening effect on the consumer side. The coefficients (beta _{ki}le 1) encode link-specific consumption efficiency—that is, potentially incomplete use of energy removed from a resource species by its consumer. (beta _{ki}=1) would describe perfect consumption efficiency whereas in real food webs this value is estimated to lie considerably lower37. In our simulations we use (beta _{ki}=beta) for all interactions present.Equations (1) and (2) describe a simplified food web structure where consumption is modelled by the simple Holling type-I response38, where consumer resource fluxes scale proportional to the product of consumer and resource biomass density and there are no saturation effects. Moreover, Eqs. (1) and (2) assume that the food web is rigid in that species are incapable of adapting their consumption behaviour to changes within the food web, such as a decreasing population of resources or competition from an invasive species39. Yet, these equations allow for a coherent description of the energy fluxes between species and constitute an established framework for complex consumer-resource relations to evolve.To evolve food webs we simulate Eqs. (1) and (2) numerically. New species are added successively to an existing food web. We assume that invasion attempts occur on a slow timescale, such that equilibrium can be reached before the subsequent invasion attempt, though occasionally, the food web does not converge to its equilibrium state. After each invasion attempt the steady state species vector (mathbf {S^*}) is computed. In case of feasibility the eigenvalues of the community matrix are evaluated in order to determine the linear stability of the steady states. If feasibility is not obtained, that is, if (mathbf {S^*}) contain negative populations, Eqs. (1) and (2) are integrated numerically until extinctions occur and feasibility of the remaining species is reached (Details: “Materials and methods”). Examples of several invasion attempts are shown in Fig. 2.Figure 1Evolution of three food webs using different assembly rules. All main panels show decay rates of all species present plotted against invasion attempts, that is, evolutionary steps. The decay rates are plotted as (Delta alpha equiv alpha – alpha _{min}), where (alpha _{min}) denotes the lower limit on decay rates (compare: Table 1). The thin red line highlights the currently lowest producer decay rate. Grey symbols denote producers, yellow and magenta symbols denote consumers of one or two resources, respectively. Cyan symbols denote omnivores. (a) Food webs where only one resource per consumer is allowed, yielding a treelike food web without loops. (b) Consumers can have either one or two resources at the same trophic level. (c) Consumers are allowed one or two resources at any trophic level (lge 1). Note that both axes use logarithmic scaling. Insets: Normalised histograms of species richness, using all data. Note the logarithmic vertical axis scaling.Full size imageFigure 2Time series of a food web during several invasions. The panels (a–f) respectively correspond to invasion attempts 40312–40314, and 40316–40318 in Fig. 1c. Upper row: In each panel, orange circles and red “x”-symbols denote the invasive and extinct species, respectively. The vertical coordinate denotes trophic level, and node areas represent initial biomass densities. The green hexagon represents the basic nutrient source. (a) A species successfully invades the food web, but causes the extinction of two resident species, among these one of its own resources. (b) the invader is successful without causing any extinctions. (c) The invader is a primary producer and causes extinction of the invader from (b). (d) The invader replaces a resident species of same niche as the invader. (e) The invader is unsuccessful in invading the food web as it shares a niche with one of the resident species. (f) The invader is a primary producer and causes the extinction of three resident species, among these the primary producer with lowest decay rate, corresponding to largest intrinsic fitness, which is highlighted by the black arrow. Lower row: Time series corresponding to each of the food webs above, where time is measured in units of the inverse primary producer growth rate, (k^{-1}). Blue and orange lines represent resident and invasive species, respectively, as the new steady state is approached. The black line in the last panel represents the producer with lowest decay rate. Note the double-log axis scaling.Full size imageLoops profoundly impact food web evolutionTo make sure our results do not depend on the details of the invasion process we allow for several qualitatively distinct evolutionary processes: (i) treelike food webs, where each consumer has a single resource; (ii) non-omnivorous food webs with loops; (iii) omnivorous food webs. Loops are known to be relevant for sustained limit cycles and chaotic attractors, thus widening the range of dynamical properties. Indeed, we find treelike food webs to stand out in that fitness, measured by species decay rates, indefinitely increases in the evolutionary process (Fig. 1a, dotted red line), a finding consistent with the recent literature30. This indefinite fitness improvement hinges on the absence of network loops: a given primary producer can only be replaced by an invading primary producer of greater intrinsic fitness, that is, lower decay rate.Allowing for network loops, evolved food web do not show indefinite fitness improvement (Fig. 1b,c) and mean species richness somewhat decreases (Fig. 1, insets). All histograms show a systematic difference in odd and even species richness, with food webs of odd species richness being the most frequent. This tendency is most pronounced for treelike food webs. We interpret this as a manifestation of the requirement of non-overlapping pairing28. Treelike food webs are feasible and stable if every species in the food web can be coincidentally paired with a connected species or nutrient that is not part of another pairing. In food webs of even species richness the nutrient is never included in such a pairing. Food webs consisting of several smaller trees that are connected through the nutrient source are therefore only feasible if every tree satisfies this requirement individually. On the contrary, the nutrient is always included in a pairing in food webs of odd species richness, and therefore odd food webs are more likely to be feasible. To a lesser extent this tendency is also found in the histograms representing food webs with network loops. We interpret this as resulting from the fact that 40-60% of the food webs from simulations allowing network loops are in fact treelike.Why do loops counteract indefinite fitness improvement? This can be seen as a manifestation of relative, rather than absolute, fitness, where a species can consume two resources and thereby can help eliminate even primary producers of high intrinsic fitness (Fig. 1b,c). An example of this is illustrated in Fig. 2f), where the intrinsically fittest producer is a node in a food web loop, and is driven to extinction during the invasion of a producer with lower intrinsic fitness.The evolution of intrinsic fitness in Fig. 1 implies that allowing for interaction loops makes resident species more vulnerable to extinction during invasions, because parameters that characterise high intrinsic fitness before an invasion might characterise low intrinsic fitness during the invasion. This is supported by the cumulative distribution of resident times (Fig. S1a), where residence times in food webs with network loops fall off faster than the residence times in treelike food webs. In Fig. S1b we observe that in accordance with this, the distribution of extinction event size falls off faster for treelike food webs (Fig. S1b), where the extinction event size is measured relative to the total number of species (species richness) in the food web. Fig. S1b therefore implies that interaction loops make food webs less robust to invasions, as invasive species tend to create larger extinction events here than in treelike food webs. Finally, we find invasive species to have higher success rates when invading food webs with interaction loops, and the success rate is found to increase with (beta). In simulations with (beta =0.75) we observe 11.5%, 27.2% and 29.8% for treelike, non-omnivorous, and omnivorous food webs with loops, respectively. The implications of this are twofold. On one hand, it is easier to assemble feasible food webs when multiple resources and omnivory are allowed. On the other hand, these food webs are more susceptible to invasions and their resident species are more vulnerable. If a food web contains two-resource species, removal of one of the two resources of a species (S_i) by an invader can already lead to a cascading extinction of S, as exemplified by Fig. S2.Robustly bi-modal eigenvalue spectraWe now turn to the eigenvalue spectra of the evolved complex food webs, which we present as two-dimensional histograms in the complex plane (Fig. 3). Each simulation conducts (10^5) invasion attempts, yet the number of unique feasible food webs is considerably lower, that is, approximately equal to the aforementioned rates of successful invasions. Furthermore, the number of unique feasibly food webs drastically decreases with species richness. While the data shown represent relatively small networks, we find that key spectral features are very systematic as function of species richness. A generic feature is that spectra typically have many eigenvalues with small negative real parts. Further, the real parts scatter more and more closely at small negative values, as species richness increases beyond two. All spectra contain a considerable fraction of purely real eigenvalues, typically making up 15–30% of a spectrum.Figure 3Complex eigenvalue spectra of evolved food webs. Each panel represents the two-dimensional histogram in the complex plane. Species richness and invasion mechanism are as labelled in panels, that is, rows of panels represent treelike, non-omnivorous, and omnivorous food webs. Note that the colour scale is logarithmic, with green marking the areas with largest likelihood of eigenvalues (Details: “Materials and methods”). Eigenvalue spectra of omnivorous food webs of other species richnesses can be seen in Fig. S3.Full size imageThe origin of purely real eigenvaluesThe first column in Fig. 3 represents food webs with species richness two. These simple food webs only have one feasible configuration, namely that of one primary producer and one consumer. Any differences between spectra in the left column are therefore purely statistical. These food webs can be considered as isolated interactions between a consumer and its resource, hence the analytical eigenvalues of this food web can provide some insight on the dynamics underlying the eigenvalue spectra. From the analytical eigenvalues we obtain that an eigenvalue is purely real if the inequality$$begin{aligned} beta eta le frac{1}{2}left( gamma + sqrt{gamma ^2 + kgamma }right) , ,,,,,,,,text {with},, gamma equiv frac{alpha _2}{1-alpha _1/k}, end{aligned}$$
(3)
is fulfilled (Details: Sec. S3). Here, (alpha _1) and (alpha _2) are the decay rates of the resource and the consumer, respectively, and (beta eta) is short for (beta _{21}eta _{21}), the “consumption rate” of the consumer. (gamma) can be interpreted as the inverse intrinsic fitness of the food web.From feasibility, we have the additional requirement of (gamma < beta eta), hence, the consumer’s “consumption rate” is bounded also from below. As k decreases, the lower and upper boundaries on (beta eta) approach one-another until they are equal for (k=0). A food web with low producer growth rate is therefore likely to have complex eigenvalues. In the opposite limit, when (krightarrow infty), or equivalently (alpha _1 rightarrow 0), we see that (gamma) reduces to (alpha _2). In the first limit Eq. (3) reduces to (beta eta le infty) which will always be satisfied and all eigenvalues are therefore purely real in this limit. This corresponds to a food web where the consumer has infinite access to resources and there is no stress or constraints on the web that could cause oscillations. In the limit where (alpha _1 rightarrow 0), the eigenvalues pick up an imaginary component when (beta eta) is large compared to (alpha _2) and k. This occurs when the consumer population has a large intrinsic growth rate, thus heavily exploiting its resource.Overall, purely real eigenvalues characterise food webs where consumption of the resource is moderate compared to the intrinsic fitness of the resource. This corresponds to an over-damped limit where the consumer does not consume enough to cause any significant displacement of the resource population, hence a perturbation of the consumer population will not spread to its resource. For higher species richness the Jacobian quickly becomes too complicated to be solved analytically. Even so, we expect the dynamics between a consumer and its resources to be conceptually analogous, namely that “sustainable over-consumption” yields oscillating densities and complex eigenvalues.The set of smallest and largest real-valued eigenvalues is obtained when (beta eta) is only slightly larger than (gamma), hence barely satisfying the criterion of feasibility. The eigenvalues then reduce to (lambda _{pm } = -frac{k-alpha _1}{2} pm frac{k-alpha _1}{2}). (lambda _+) is always zero, that is, food webs of species richness two are always stable, and with our choice of parameters (lambda _- ge -0.95). We observe approximately the same range of real values in all numerical spectra of any species richness, thereby implying that the choice of parameters might be more important for the spectrum width than the structure of the food web.The overall shape is qualitatively similar for all food web structures (see: Fig. 3). Importantly, omnivorous spectra are the only ones to contain also eigenvalues with positive real part, that is, unstable eigenvalues. These food webs do therefore not converge to their equilibrium state after an invasion, but are displaying periodic or chaotic dynamics (Details: “Materials and methods”). The unstable eigenvalues are all barely larger than zero, hence hardly visible in Fig. 3. Interestingly, non-omnivorous food webs with network loops exhibit the same species richness and approximate connectivity as the omnivorous food webs, yet they do not yield unstable eigenvalues. The differences between treelike food webs and food webs with network loops discussed earlier must therefore be unrelated to the stability of the food webs, thus emphasising the difference between stability to perturbation of a given food web and its robustness to invasions. For omnivorous food webs the fraction of unstable eigenvalues increases with species richness and decreases with (beta). Intuitively, it seems reasonable that there is a relation between instability and low consumption efficiency. A species with a low consumption efficiency has to compensate by consuming more biomass, thereby putting more stress on its resources. Only for (beta =1) are there no unstable omnivorous eigenvalues.Figure 4Complex eigenvalue spectra of random matrices. Heat maps of eigenvalue spectra of random matrices, corresponding to the respective species richnesses shown in Fig. 3. Off-diagonal entries are drawn from a normal distribution with probability (p(N) = frac{N^2+21N-28}{9N(N-1)}) (Details: Sec. S5), and are otherwise set to zero. Diagonal elements are set to (-1).Full size imageWe now compare the evolved spectra (Fig. 3) to their random counterparts (Fig. 4). The diagonal entries represent self regulation of each species and are set to (d = -1). Off-diagonal entries are drawn from ({mathcal {N}}(0, 1)) with probability p(N), and are otherwise 0.$$begin{aligned} p(N) = frac{N^2+21N-28}{9N(N-1)}, ~~text {for } N >1, end{aligned}$$
(4)
where N is “species richness”, that is, the number of rows (or columns) of the matrix. This corresponds to the implemented connectivity in the simulation allowing network loops and omnivory, that is, the connectivity of omnivorous food webs given no extinctions occur (Details: Sec. S5). As predicted by spectral theory of random matrices, the spectra are centred around d on the real axis and approach a circular geometry as the size of the matrix increases. Already for (N=2) does the spectrum contain unstable eigenvalues. The fraction of unstable eigenvalues increases with N as the circle radius increases. Also for random spectra do we observe a large fraction of purely real eigenvalues. We attribute this to the small size of the matrices, being much smaller than the infinity limit for which the law was derived40.Figure 5Distribution of eigenvalues along the real axis. Normalised frequency distributions of eigenvalues along the real axis for all food web structures and random matrices for species richness (2-9). Eigenvalues representing food webs are taken from simulations using a range of values of (beta) (Table 1), since varying (beta) does not have significant effects on the real-part distributions (Details: Sec. S6). All distributions are scaled to start in (-1). Note the logarithmic vertical axis scaling.Full size imageFinally, we study the real-part frequency distributions of eigenvalues of all four types (treelike, non-omnivorous, omnivorous and random). The frequency distributions for species richness 2–9 can be seen in Fig. 5, where each distribution consists of data from various values of (beta) (see Table 1). In order to facilitate comparison of the functional form of the frequency distributions, rather than the range, the frequency distributions are scaled to be bounded by (-1) on the real axis, that is, we divide each data point by ((|min {x}|)^{-1}) where x is the data points of the distribution. Frequency distributions representing the evolved food webs follow approximately the same curve for a given species richness, and are distinctively different from the random matrices. As also seen in Fig. 3 omnivorous distributions are the only to extended to positive values for species richness greater than two.Once again, we observe quantitative differences between food webs with odd and even species richness: For odd species richness the distribution is bi-modal with a global maximum near (x=0) and a secondary maximum near the lower limit, that is (x=-1). For even species richness, the distribution is initially less strongly peaked. Yet, as species richness increases, a sharp peak emerges around (x = 0). The distribution thus becomes more similar to that of the food webs with an odd number of species.The intermediate part of the spectrum is increasingly depleted of eigenvalues at higher species richness. Comparing Fig. 5 with Fig. 3 we see that the left part of all distributions consists of purely real eigenvalues, whereas it is mostly complex eigenvalues that make up the global maximum near (x=0). This implies that perturbations can be divided into two main groups: perturbations from which the food web quickly returns to the respective steady state, and perturbations that induce oscillations from which the food web takes very long to recover. The peak consisting of purely real eigenvalues near (x=-1) does not change notably with species richness, indicating that, independent of species richness, food webs are robust to certain perturbations. In accordance with this we observe that food webs of all species richness usually return quickly to their steady states after an unsuccessful invasive species goes extinct. The main peak (near (x=0)) becomes both higher and narrower with increasing species richness, that is, the food webs become quasi-stable. In larger food webs there are more species that can be disturbed by a perturbation, which might prolong the effect of the perturbation, that is, push eigenvalues towards zero on the real axis. Overall, we thus find that the histogram of complex food webs becomes strongly bi-modal as food webs consisting of many species are approached in an evolutionary process, whereas random matrix spectra are consistently uni-modal. In Sec. S8–S9 we consider the robustness of the results in Fig. 5 by varying the parameter distributions and implementing Holling type-II response, respectively. More

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Study sites, experimental design, and field samplingThe Tibetan alpine permafrost region, the largest area of permafrost in the middle and low latitudes of the Northern Hemisphere64, stores substantial soil C (15.3–46.2 Pg C within 3 m depth)65,66,67. With continuous climate warming, permafrost thaw has triggered the formation of widespread thermokarst landscapes across this permafrost area13,68. To explore the impacts of thermokarst formation and development on soil C dynamics, we collected topsoil samples (0–15 cm) from a thaw sequence in 2014 and from five additional sites spread across the region in 2020. The thermokarst landscape was characterized as thermo-erosion gullies (Supplementary Table 2). The elevation of these six sites is between 3515 and 4707 m. The mean annual temperature across this area ranges from −3.1 to 2.6 °C, and the average annual precipitation varies from 353 to 436 mm. The vegetation type across these sites is swamp meadow, with the dominant species being Kobresia tibetica, Kobresia royleana and Carex atrofuscoides. Although the dominant species did not change after permafrost collapse, the forb coverage increased along the thaw sequence and across the five additional thermokarst-impacted sites. The main soil type is Cryosols on the basis of the World Reference Base for Soil Resources69, with soil pH ranging from 5.6 to 7.3 (Supplementary Fig. 1e). The active layer thickness varies between 0.7 and 1.1 m across the six study sites and the underlying soil parent material is either siliciclastic sedimentary or unconsolidated sediments (Supplementary Table 2).To evaluate the dynamics of soil C fractions after permafrost collapse, we collected soil samples across the Tibetan alpine permafrost region based on the following two steps (Supplementary Fig. 5). In the first step, we established six collapsed plots (~15 × 10 m) along a thaw sequence (located in Shaliuhe close to Qinghai Lake, Qinghai Province, China), which had been collapsed for 1, 3, 7, 10, 13, and 16 years before the sampling year of 201413. The collapse time of each plot was estimated by dividing the distance between the collapsed plot and the gully head by the retreat rate (~8.0 m year−1; the rate of the head-wall retreat was determined by Google Earth satellite images and in situ monitoring)13. Then, we set up six paired control (non-collapsed) plots adjacent to these collapsed plots. To limit experimental costs, we selected three paired control and collapsed plots (collapsed for 1, 10, and 16 years, representing the early, middle, and late stages of collapse) to examine the responses of POC, MAOC and OC-Fe to permafrost collapse (Supplementary Fig. 5). Within each collapsed plot, we collected topsoil (0–15 cm) samples from all vegetated patches (Supplementary Fig. 6), and then evenly selected 10 vegetated patches for this study considering the heavy workload and high cost. In each selected vegetated patch, 5–8 soil cores were sampled and completely mixed as one replicate. Within each control plot, topsoil samples were randomly collected from five quadrats at the center and four corners of the plot. In each quadrat, 15–20 soil cores were sampled and mixed as one replicate. Thereby, ten replicates were acquired in each collapsed plot (n = 10), and five replicates were obtained in each control plot (n = 5). In total, we acquired 45 soil samples, including 30 samples from the three collapsed plots and 15 samples from the non-collapsed control for subsequent analysis.In the second step, to further verify the universality of collapse effects on SOC fractions, we collected topsoil (0–15 cm) samples from an additional five similar sites located near the towns of Ebo, Mole, Huashixia, and Huanghe across a 550 km permafrost transect in August 2020 (Fig. 1). Specifically, paired collapsed and control plots (15 × 10 m) were established at the end of a gully and in adjacent non-collapsed areas in each site (Supplementary Fig. 5). In the collapsed plot, we set five 5 × 3 m quadrats at the center and four corners of the plot, and then collected topsoil samples within all the vegetated patches in these quadrats. In each quadrat, all the collected soil cores (15–20 cores) were completely mixed as one replicate, and finally, five replicates were acquired in each collapsed plot (n = 5). Similarly, five replicates were obtained from the five quadrats in each control plot (n = 5). In total, we collected 50 topsoil samples across these five thermokarst-impacted sites. After transportation to the laboratory, all the soil samples were handpicked to remove surface vegetation, roots and gravels, and sieved (2 mm) for subsequent analysis.It should be noted that the space for time approach was only used for the permafrost thaw sequence, not for the other five sites over the regional scale. Across these five sites, we focused on the impact of permafrost collapse on POC, MAOC as well as OC-Fe by comparing soil C fractions inside and outside the gully in each site rather than among the study sites. Given the low coefficient of variation of parameters (i.e., edaphic variables and soil minerals) in the control plot of each site (Supplementary Table 3), the pristine soils in each site could also be regarded as homogeneous70, and the differences in parameters inside and outside the gully could be attributed to the effects of permafrost collapse. Along the permafrost thaw sequence, to verify whether the plots with different collapse times (1, 10, and 16 years) were comparable, we analyzed a series of parameters (i.e., vegetation biomass, edaphic variables, and soil minerals) for the three control plots which were located outside the gully but adjacent to three collapsed plots within the gully (Supplementary Fig. 5). By comparing aboveground biomass, belowground biomass, SOC, soil moisture, pH, bulk density, soil texture, and soil minerals (see below for details of the analytical method), we observed that the above parameters were not significantly different among the three control plots along the thaw sequence (all P  > 0.05; Supplementary Fig. 7). These comparisons demonstrated that the study area was homogeneous before permafrost thaw and thus it was reasonable to adopt the space for time approach along the permafrost thaw sequence.It should also be noted that the collected topsoil samples used in this study were less affected by physical mixing and translocation due to thaw phenomena at the thermokarst-impacted sites. Specifically, to examine changes in soil properties upon permafrost thaw, we chose to collect topsoil within the vegetated patches rather than from the exposed soil areas in the collapsed plots (Supplementary Fig. 6). These vegetated patches (40–60 cm thickness) are formed during the landscape fragmentation after permafrost collapse13. Although permafrost collapse inevitably led to soil translocation, these vegetated patches maintained their original shapes, especially for the topsoil because it is protected by mattic epipedon in this swamp meadow ecosystem on the Tibetan Plateau (which has an intensive root network protecting soils against interference)71,72. Moreover, we collected 0–15 cm of topsoil within the vegetated patches, in which soil cores were at least 10 cm away from the edge of the patch. Due to these two points, topsoil should not be mixed with the subsoil in our case. To test this deduction, we compared the non-collapsed (control) plot with the collapsed plot occurring for 1 year (the early stage of the permafrost thaw sequence), and observed no significant differences in soil properties such as bulk density, SOC, pH, soil texture and soil minerals (all P  > 0.05; Supplementary Fig. 8). These comparisons illustrated that permafrost collapse did not cause soil physical mixing for the topsoil samples involved in this study, and soil layers were comparable between the collapsed and control plots.SOC fractionationWe separated POC and MAOC from bulk soils using a fractionation method based on a combination of density and particle size18 using the following three steps. First, 10 g of soil was put into a 100 mL centrifuge tube, and added with 50 mL of 1.6 g cm−3 NaI. After being completely mixed, the mixture was sonicated and then centrifuged at 1800 × g. The floating particulate organic matter, together with the supernatant, was poured into a GF/C filter membrane for filtration, completely washed with deionized water, and then dried at 60 °C to constant weight. Then, the C content of the particulate organic matter was determined as POC. Second, deionized water were added to the remaining soils in the tube to wash out any residual NaI. The washed soils were then separated with a 53-μm sieve. The residues on the sieve ( >53 μm) were dried and determined as heavy POC. Third, the organic matter that passed through the sieve ( More

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RESEARCH HIGHLIGHT
29 August 2022

North America’s largest land mammal can double the diversity of native grasses through its grazing.

Home on the range: the American bison’s taste for prairie grasses helps to boost diversity of native flora (pictured, stiff goldenrod, Solidago rigida). Credit: Jill Haukos/Kansas State University

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Grazing animals can shape the grasslands they dine on by preferentially eating certain species, allowing other species to find a foothold. To quantify this effect, Zak Ratajczak at Kansas State University in Manhattan and his colleagues analysed 29 years’ worth of data from plots in an unploughed native tallgrass prairie in eastern Kansas1. Since 1992, the plots have been managed in one of three ways: year-round grazing by bison (Bison bison); seasonal grazing by cattle; or no grazing at all.

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doi: https://doi.org/10.1038/d41586-022-02332-4

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