Ecology

Analysis of FTIRUnderstanding the functional groups involved in the biosorption of toxic metals is essential to elucidate the mechanism of this process. Groups such as carboxylic, hydroxyl and amine are among the main responsible for the absorption of metals by cellulose34 In the Fig. 1, show the FTIR of ECx.Figure 1FTIR of ECx before and after of adsorptions of Cr (VI).Full size imageAccording to13 the bandwidth at 3000–3600 cm−1 corresponds to bonds related to the -OH group. These hydrogen bonds are useful tools for cation exchange with heavy metals. This evidenced in the color spectrum (dark green) that represents an ECx sample with attached Cr (VI) after the adsorption process, where the stretching of the (OH) group lost part of its extension. The change observed in the peak from 3420 cm−1 of ECx to 3440 cm−1 in ECx-Cr indicates that these groups have a participation in the bond with the Cr (VI) ions. The variation of bands in the peak of the amines after adsorption confirms the participation of these groups in the adsorption process. This result confirmed by the ion exchange evaluation experiment discussed later in section SEM–EDX.The change in peak 3280, after Cr (VI) adsorption, indicates that EC removed Cr (VI) based on interaction with (OH), part of (OH) lost due to formation of vibrations of ascension O–Cr. Also, after Cr (VI) biosorption on ECx, the peak of the EC-S group is shifted to 590. This can be explained by surface complexation or ion exchange35.In general, comparable results reported in the literature for cellulose in the absorption of other toxic metals, as for other cellulose-derived biosorbentes in the removal of Cr (VI) ions36.One way to corroborate the information presented in the FTIR measurements is through SEM images since with these images it is possible to observe the distribution of the reagents in the ECx biomass treatment and subsequently the Cr (VI) adsorption process.SEM–EDXFigure 2 shows the micrographs obtained for the biomass before (a) the adsorption of Cr (VI), in addition to showing the distribution of the different biomass chemical modifications in (b) and in (c) it shows the distribution of chromium around all biomasses.Figure 2Biomass before (a) Cr (VI) adsorption, biomass chemical modifications in (b) and shows the distribution of chromium around the whole biomass (c).Full size imageFrom Fig. 2a, it can see that the biomass has a very irregular rough surface, with macropores and cracks. Many of these irregularities may associated with damage caused by the delignification process of E. crassipes cellulose with NaOH14. In Fig. 2b it is possible to visualize the components of the cellulose xanthogenate, coming from sodium, distributed throughout the biomass, a result like that reported in other studies35 The colored dots represent the elements in the samples, green dots represent carbon, red dots represent oxygen, and yellow dots represent the places where sodium lodged.Table 2 shows that, in addition to carbon and oxygen, the element with the greatest presence in the composition of pure waste is sodium and sulfur from the xanthogenate cellulose transformation process. Table 2 shows the physicochemical characterization of the ECx sample, through EDS.Table 2 Features of sample of ECx.Full size tableCellulose xanthogenate, is one of the cellulose transformations to improve the adsorption performance of heavy metals, this compound produced from dry and ground biomass, mixing with sodium hydroxide (NaOH) to remove lignin, creating alkaline biomass, then disulfide (CS2) added13,14. (CS2) reacts with hydratable hydroxycellulose, forming C-SNa complexes; these are responsible for the cation exchange with heavy metals. Metal ions enter the interior of E. crassipes with (CS2), exchanging with Na36,37.The SEM morphology of ECx and coupled with the high content of sulfides (7.3%) determined by the spectrum in Table 2, it further confirms that xanthate groups are successfully grafted onto the biomass of E. crassipes, and Fig. 3 represents this information based on13,36,37,38.Figure 3Prototype.Full size imageExchange biochemistry is usually identified as the main mechanism for the adsorption of metals in cellulose and its derivatives35 and through the evaluation of EDS this process could verify. Similar observations were made by36 where the adhesion of Cr (VI) in this biomass was observed. Also, in xanthogenate cellulose processes, the adhesion of Pb (II) to this type of biomass verified, concluding that this cellulose is important in the removal of heavy metals from water13.The SEM morphology of ECx with Cr (VI) coupled with the high content of sulfides determined by the spectrum in Table 3, was the determinate for the chemisorption’s of Cr (VI). The mechanism of Cr (VI) sorption by cellulose xanthate is:$$left[ {{4}left( {{text{C}}_{{6}} {text{H}}_{{{12}}} {text{O}}_{{6}} } right)} right]*{text{2CS}}_{{2}} {text{Na }} + {text{ Cr}}_{{2}} {text{O}}_{7}^{ – 2} to left{ {left[ {{4}left( {{text{C}}_{{6}} {text{H}}_{{5}} {text{O}}_{{6}} } right)} right] , *{text{2CS}}_{{2}} } right}*{mathbf{Cr}}_{{mathbf{2}}} + {text{Na}} + {text{7H}}_{{2}} {text{O}}$$where [4(C6H12O6)] *2CS2Na represents the xanthogenate biomass, and Cr2O7–2 represents the Cr (VI), that 4 parts of glucose xanthate react with the dichromate. In the Tables 3 and 4, the relationship between cellulose xanthogenate and Cr (VI), with related weights of 10.4 for Cr (VI).Table 3 Features of sample of ECx with Cr (VI).Full size tableTable 4 Researcher of process of the desorption.Full size tableMass balance in treatmentAdsorption is the phenomenon through which the removal of Cr (VI) achieved in the treatment systems; this quantified by means of the general balance equation of the treatment system as shown in Fig. 3.Adsorption is the phenomenon through which the removal of Cr (VI) achieved in treatment systems, this quantified by mass balance. Equation (1) shows the general balance of matter in the treatment system, together with the accumulation, inputs, and outputs of the system and the chemical process of adsorption.$${text{Acumulation }}upvarepsilon *frac{{partial {text{Cr}}left( {{text{VI}}} right)}}{{partial {text{t}}}} = {text{In}} frac{{partial {text{Cr}}left( {{text{VI}}} right)_{0} }}{{partial {text{t}}}} – {text{Out}}frac{{partial {text{Cr}}left( {{text{VI}}} right)}}{{partial {text{t}}}} – {text{Adsortion}},{rho b}frac{{partial {text{q}}}}{{partial {text{t}}}}$$
(1)
Accumulation represents by Eq. (1), where ∂C(VI) is the contaminant input to the treatment system, (ε) is the porosity of the bed, which calculated as the ratio between the density of the bed of treatment and the density of the microparticle of this biomass. This parameter must be above 0.548 achieved using particle diameters less than 0.212 mm, which favors contact between the contaminant and the particle49. The contaminant input to the treatment system represents by the design speed and the amount of contaminant that the system could treat. The output in the treatment system represents by the same input speed and the amount of contaminant that comes out. With these equations, the general material balance will be complete, summarized in Eq. (2), where it can see that the accumulation is equal to the input to the system, minus the output, and minus the adsorption.$$upvarepsilon *frac{{partial {text{Cr}}left( {{text{VI}}} right)}}{{partial {text{t}}}} = frac{{partial {text{Cr}} left( {{text{VI}}} right)}}{{partial {text{t}}}} – frac{{partial {text{Cr}} left( {{text{VI}}} right)}}{{partial {text{t}}}} – frac{{text{M}}}{{text{V}}}*frac{{partial {text{q}}}}{{partial {text{t}}}}$$
(2)
where V = System volume (ml), ε = Porosity, Co = Initial concentration of Cr (VI) (mg/ml), C = Final concentration Cr (VI) in the treated solution (mg/ml), Q = design flow (ml/min), Tb = Breaking time (Min), M = amount of biomass used (g), q = Adsorption capacity of the biomass used (mg/g).$${text{V}}*upvarepsilon *{text{Co}} = {text{Q}}*{text{Tb}}*{text{Co}} – {text{Q}}*{text{Tb}}*{text{C}} – {text{M}}*{text{q}}$$
(3)
Depending on the most important parameters when building a treatment system, Eq. (3) could use to model and validate the best form of treatment, for example, the necessary amount of biomass to use to treat a certain amount of contaminant, in the present investigation it used to establish the adsorption capacity in these initial treatment conditions. The remaining Eq. (4) determines the adsorption capacity.$${text{q}} = frac{{{text{QTbCo}}}}{{text{M}}} – frac{{{text{QTbCf}}}}{{text{M}}} – frac{{upvarepsilon {text{VCo}}}}{{text{M}}}$$
(4)
Adsorption capacity is generally taken through Eq. (5) for both batch and continuous experiments20,21But unlike Eqs. (5), (4) takes into account design variables such as flow rate (Q), rupture time (Tb), particle bed porosity ε, and vessel design volume (v).$${text{q}} = frac{{{text{v}}left( {{text{Co}} – {text{C}}} right)}}{{text{m }}}$$
(5)
where m: Mass used in the treatment, V: Volume, Co: Initial concentration, C: Final Concentration, Q: adsorption capacity.However, unlike Eqs. (5),  (4) considers the design variables such as flow rate (Q), rupture time (Tb), particle bed porosity ε and vessel design volume (v).When a desorption-elution process is involved for the reuse of biomass, Eq. (4) would be:$${text{q}}_{{text{T}}} = mathop sum limits_{j = 1}^{n} left[ {frac{{{text{QTbjCo}}}}{{text{M}}} – frac{{{text{QTbjCj}}}}{{text{M}}} – frac{{upvarepsilon {text{VCo}}}}{{text{M}}}} right]$$
(6)
where Q = design flow (ml/min), Tbj = Break time of use number j (Min), Co = Initial concentration of Cr (VI) (mg/ml), C = Final concentration Cr (VI) in the treated solution (mg/ml), V = System volume (ml), ε = Porosity, M = amount of biomass used (g), q_T = Total adsorption capacity of the biomass used (mg/g).This model (6) is design to determine the adsorption capacity when different elution processes have conducted, it will used to determine the new adsorption capacity and is one of the contributions of the present investigation.Result process of adsorptionsIn Fig. 4 shows the Cr (VI) adsorption process of the system.Figure 4Percentages of Cr (VI) removal the system for ECx.Full size imageVarious researchers have extensively studied the influence of factors such as bed height, flow rate and metal inlet concentration on rupture (Tb) curves. For example, the influence and similarity of the initial contaminant concentrations should be reflected as in the case of a tannery, with initial concentrations of 600 mg/l. Figure 4 shows the progress curves obtained for the study of Cr (VI) removal by the studied biomasses, reflecting the percentage of Cr (VI) removal in contrast to the treated volume, which is a very important parameter to time to scale the process.Regarding the effect of the input concentration, it can see in Fig. 5 that the breakpoint had a better performance in all the initial concentrations in the ECx biomass. comparing it with the EC-Na biomass (see Fig. 5), always obtaining breakpoints with more treated volume.Figure 5Percentages of Cr (VI) removal the system for EC-Na.Full size imageThe difference between the rupture curves between ECx and EC-Na indicates that the cellulose xanthate modification scheme should completed, although it can also elucidate that the EC-Na biomass has high yields compared to other biomass studied. for example, in Ref.34 investigate the biomass of E. crassipes without modifying, having removals below this alkaline cellulose.Adsorption capacitiesThrough Eq. (3), the adsorption capacity of ECx, using the initial concentration of 600 mg/l, since it was the maximum concentration used.The break point was around 1200 ml according to Fig. 6 and together with the flow rate of 15 ml/min; the break time obtained in 80 min.$${text{q}} = frac{{80{*}15{*}0.6}}{40} – frac{{80{*}15{*}0.04}}{40} – frac{{0.66{*}78{*}0.6}}{40}$$q: Adsorption capacity, Co: 0.6 mg/ml, C: 0.06 mg/ml, M: 40 g, Tb: rupture time 80 min, Q: 15 Flow rate ml/min, ε: 0.6649, V: Occupied volume: 70 ml.Figure 6Adsorption capacities in the different adsorption processes in the biomass ECx.Full size imageA result of 16 mg/g obtained in this continuous study for the biomass ECx. With this same equation it gives the capacity of the biomass EC-Na, with 11 mg/g.Desorption-Elution and reuseThrough Eq. (6), the sum of the Cr (VI) adsorption capacities established, after different biomass reuses due to EDTA elution. In the second treatment process, it yielded the following results under concentrations of 6 g/l of EDTA.$${text{q}}left( {text{T}} right) = frac{{60{*}15{*}0.6}}{40} – frac{{50{*}15{*}0.06}}{40} – frac{{0.66{*}68{*}0.6}}{40}$$Co: 0.6 mg/ml, C: 0.06 mg/ml, M: 45 g Biomass eluted with EDTA, Tb: rupture time: 60 min, Q: 15 Flow ml/min, ε: 0.6649, V: Occupied volume: 68 ml, q: 10 mg/g.Five Cr (VI) adsorption cycles performed using ECx and EC-Na cellulose in a continuous system to evaluate the regeneration and reuse potential. Between each biosorption cycle, a desorption cycle performed using three different concentrations of EDTA eluent.According to Figs. 6 and 7, although the adsorption capacity gradually decreases from the first adsorption process, it could consider that it is a satisfactory biomass recycling process and a design parameter for later stages of this treatment system.Figure 7Adsorption capacities in the different adsorption processes in the biomass EC-Na.Full size imageIn the experiments with concentrations of 6 g/l, five reuse processes obtained, obtaining a final sum of 52 mg/g. In concentrations of 3 g/l of EDTA, final capacities of 51 mg/g obtained lower than concentrations of 6 g/l but with half of this reagent. With concentrations of 1 g/l, final capacities of 33 mg/g obtained.The desorption processes of the EC-Na biomass with initial capacities of 11 mg/g were also evaluated and through desorption processes with EDTA of 3 g/l this biomass recycled on 5 occasions, reaching 32 mg/l in capacities of adsorption and like the EC-Na biomass, the ideal concentration in the process for desorption processes is 3 g/l, due to the considerable increase in reuse processes and low concentration compared to 6 g/l, which, although higher, does not this value is significant in the absorption capacity.Through Eq. (6) and with different bibliographic references, representative data obtained to feed this equation, determining the capacities of each of these biomasses together with the new capacities determining the desorption power of the different eluents shown and summarized in Table 4.For the EDTA eluent and with Eq. (6), satisfactory results evidenced by removing Al (II), reaching almost 150% of its adsorption capacity, corroborating what presented in the present investigation, also the EDTA reagent obtained interesting yields to recycle the cassava biomass increasing up to 40 mg/g. In Ref.39 used the biomass of Phanera vahlii to remove Cr (VI) obtaining results of 30 mg/g and with NaOH they reached capacities in the reuse process of this biomass up to 62 mg/g, reaching almost double of its total capacity41, also used NaOH for desorption processes with green synthesized nanocrystalline chlorapatite biomass, achieving results of 75% more. The eluent HCl is also a good chemical agent to use in desorption processes since it reached more than 100% in the reuse of biochar alginate for Cr (VI) but not so significant with biomass A. barbadensis Miller to remove Ni (II) and in40 significant results were also obtained to remove Pb (II) with pine cone Shell biomass. With the chemical agent HNO3, interesting contaminant recycling processes obtained, since more than 100% of the adsorption capacity of the biomasses used in this process used1,45.Mathematical models of adsorptionIn general, the models presented R2 greater than 0.95 for the adjustment of all the advance curves, which indicates a good adherence to the data, the model that best describes the behavior of the ECx system was the phenomenological model Thomas, which presented all the R2 values above 0.99.This model could use for the extension of the Cr (VI) ion biosorption system using cellulose xanthogenate, in the literature it is possible to observe that this model often tends to better adapt to the experimental data of the adsorption systems that use cellulose for the absorption of toxic metals28,30,31.With qt values remarkably close to the experimental values of Eq. (4) designed and presented in this investigation, indicating the validity of this equation where it reflects the maximum capacity obtained. Table 5 shows the adsorption constant of the Thomas model (Kt), which corresponds to the adsorption rate of Cr (VI) in the biomass49 This value was 0.048 (ml/mg*min) reflecting the speed with which Cr (VI) is chemisorbed in the biomass of ECx, in the EC-Na cellulose there was a Thomas model speed of 0.039 (ml/ mg*min) evidencing a lower adsorption rate than ECx. In the adsorption of Cr (VI) by rice biomass, the Thomas constant is 0.1 (ml/mg*min)47,50 also in the adsorption of Cr (VI) by biomass. Nanocrystalline chlorapatite biomass obtained at the Thomas constant 0.013 (ml/mg*min)49.Table 5 Summary of the experiments obtained with material ECx.Full size tableIn the Table 6, it presents summary of the experiments obtained with material EC-Na.Table 6 Summary of the experiments obtained with material EC-Na.Full size tableThe Cr (VI) adsorption process in the EC-Na biomass represented through the Bohart equation, since the sorption rate is proportional to the biomass capacity, obtaining an adsorption rate of 0.85(ml/mg*min). Having an alkalized biomass represents this model due to the homogeneity of this adsorbent.Mathematical models in desorption processesThe continuous desorption process with its fit to the Thomas model for biomass ECx always shows the fit of this model with significance, because this type of model fits representatively to desorption processes with good performance32,51 It can also verify that with values of qt it is close to the experimental values of Eq. (6) designed and presented in this research, indicating the validity of this equation again, where it reflects the maximum capacity obtained.In the Table 7. Show Summary of the experiments obtained with material ECx in process of desorption’s.Table 7 Summary of the experiments obtained with material ECx in process of desorption’s.Full size tableIn the Table 8 the EC-Na biomass had a different behavior and in its second and third cycle it adjusted to the Yoon model and later to the Bohart model.Table 8 Summary of the experiments obtained with material EC-Na in process of desorption’s.Full size tableThis behavior is due to the alkalinization of the biomass and this process makes the biomass a little more unstable. The values of qt, although a resemblance evidenced, were not so representative due to the little adjustment that there was with respect to the Thomas model. More

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Now consider a generalized model in which the species interactions are heterogeneous. A natural way of introducing heterogeneity in the system is by having a species diversify into subpopulations with different interaction strengths12,13,14,15. This way of modeling heterogeneity is useful as it can describe different kinds of heterogeneity. For example, the subpopulations could represent polymorphic traits that are genetically determined or result from plastic response to heterogeneous environments. A population could also be divided into local subpopulations in different spatial patches, which can migrate between patches and may face different local predators. We can also model different behavioral modes as subpopulations that, for instance, spend more time foraging for food or hiding from predators. We study several kinds of heterogeneity after we introduce a common mathematical framework. By studying these different scenarios using variants of the model, we show that our main results are not sensitive to the details of the model.We focus on the simple case where only the prey species splits into two types, (C_1) and (C_2), as illustrated in Fig. 1b. The situation is interesting when predator A consumes (C_1) more readily than predator B and B consumes (C_2) more readily than A (i.e., (a_1 / a_0 > b_1 / b_0) and (b_2 / b_0 > a_2 / a_0), which is equivalent to the condition that the nullclines of A and B cross, see section “Resources competition and nullcline analysis”). The arrows between (C_1) and (C_2) in Fig. 1b represent the exchange of individuals between the two subpopulations, which can happen by various mechanisms considered below. Such exchange as well as intraspecific competition between (C_1) and (C_2) result from the fact that the two prey types remain a single species.The system is now described by an enlarged Lotka-Volterra system with four variables, A, B, (C_1), and (C_2): $$begin{aligned} dot{A}&= varepsilon _A ,alpha _{A1} , A , C_1 + alpha _{A2} , A , C_2 – beta _A , A end{aligned}$$
(3a)
$$begin{aligned} dot{B}&= varepsilon _B , alpha _{B1} , B , C_1 + alpha _{B2} , B , C_2 – beta _B , B end{aligned}$$
(3b)
$$begin{aligned} dot{C_1}&= C_1 , (beta _C – alpha _{CC} , C)-alpha _{A1} , C_1 A-alpha _{B1} , C_1 B – sigma _1 , C_1 + sigma _2 , C_2 end{aligned}$$
(3c)
$$begin{aligned} dot{C_2}&= C_2 , (beta _C – alpha _{CC} , C) -alpha _{A2} , C_2 A -alpha _{B2} , C_2 B + sigma _1 , C_1 – sigma _2 , C_2 end{aligned}$$
(3d)
The parameters in these equations and their meanings are listed in Table 1. Here we assume that the prey types (C_1) and (C_2) have the same birth rate and intraspecific competition strength, but different interaction strengths with A and B. Note that (C_1) and (C_2) are connected by the (sigma _i) terms, which represent the exchange of individuals between these subpopulations through mechanisms studied below; these terms indicate a major difference between our model of a prey with intraspecific heterogeneity and other models of two prey species. For the convenience of analysis, we transform the variables (C_1) and (C_2) to another pair of variables C and (lambda), where (C equiv C_1 + C_2) is the total population of C as before, and (lambda equiv C_2 / (C_1 + C_2)) represents the composition of the population (Fig. 1c). After this transformation and rescaling of variables (described in “Methods”), the new dynamical system can be written as: $$begin{aligned} dot{A}&= A , big ( C , (a_1 (1-lambda ) + a_2 lambda ) – a_0 big ) end{aligned}$$
(4a)
$$begin{aligned} dot{B}&= B , big ( C , (b_1 (1-lambda ) + b_2 lambda ) – b_0 big ) end{aligned}$$
(4b)
$$begin{aligned} dot{C}&= C , big ( 1 – C – A (a_1 (1-lambda ) + a_2 lambda ) – B (b_1 (1-lambda ) + b_2 lambda ) big ) end{aligned}$$
(4c)
$$begin{aligned} dot{lambda }&= lambda (1-lambda ) , big ( A (a_1 – a_2) + B (b_1 – b_2) big ) + eta _1 (1-lambda ) – eta _2 lambda end{aligned}$$
(4d)
Here, (a_i) and (b_i) are the (rescaled) feeding rates of the predators on the prey type (C_i); (a_0) and (b_0) are the death rates of the predators as before; (eta _1) and (eta _2) are the exchange rates of the prey types (Table 1). The latter can be functions of other variables, representing different kinds of heterogeneous interactions that we study below. Notice that Eqs. (4a–4c) are equivalent to the homogeneous Eqs. (2a–2c) but with effective interaction strengths (a_text {eff} = (1-lambda ) , a_1 + lambda , a_2) and (b_text {eff} = (1-lambda ) , b_1 + lambda , b_2) that both depend on the prey composition (lambda) (Fig. 1c).Table 1 Model parameters (before/after rescaling) and their meanings.Full size tableThe variable (lambda) can be considered an internal degree of freedom within the C population. In all of the models we study below, (lambda) dynamically stabilizes to a special value (lambda ^*) (a bifurcation point), as shown in Fig. 3a. Accordingly, a new equilibrium point (P_N) appears (on the line (mathscr {L}) in Fig. 2), at which all three species coexist. For comparison, Fig. 3b shows the equilibrium points if (lambda) is held fixed at any other values, which all result in the exclusion of one of the predators. Thus, heterogeneous interactions give rise to a new coexistence phase (see Fig. 4 below) by bringing the prey composition (lambda) to the value (lambda ^*), instead of having to fine-tune the interaction strengths. The exact conditions for this new equilibrium to be stable are detailed in “Methods”.Figure 3(a) Time series of (lambda) for systems with each kind of heterogeneity. All three systems stabilize at the same (lambda ^*) value, which is the bifurcation point in panel (b). (b) Equilibrium population of each species (X = A), B, or C, with (lambda) held fixed at different values. Solid curves represent stable equilibria and dashed curves represent unstable equilibria (see Eq. (9) in “Methods”). The vertical dashed line is where (lambda = lambda ^*), which is also the bifurcation point. Notice that the equilibrium population of C is maximized at this point (for (a_1 > a_2) and (b_2 > b_1)). Parameters used here are ((a_0, a_1, a_2, b_0, b_1, b_2, rho , eta _1, eta _2, kappa ) = (0.25, 0.5, 0.2, 0.4, 0.2, 0.6, 0.5, 0.05, 0.05, 50)).Full size imageInherent heterogeneityWe first consider a scenario where individuals of the prey species are born as one of two types with a fixed ratio, such that a fraction (rho) of the newborns are (C_2) and ((1-rho )) are (C_1). This could describe dimorphic traits, such as the winged and wingless morphs in aphids12 or the horned and hornless morphs in beetles13. We call this “inherent” heterogeneity, because individuals are born with a certain type and cannot change in later stages of life. The prey type given at birth determines the individual’s interaction strength with the predators. This kind of heterogeneity can be described by Eq. (4d) with (eta _1 = rho (1-C)) and (eta _2 = (1-rho ) (1-C)) (see “Methods”).Figure 4Phase diagrams showing regions of parameter space identified by the stable equilibrium points. Yellow region represents (P_C) (predators A, B both extinct), red represents (P_A) (A excludes B), blue represents (P_B) (B excludes A), and green represents (P_N) (A, B coexist). The middle point (black dot) is where the preferences of the two predators are identical, (a_2/a_0=b_2/b_0) and (b_1/b_0=a_1/a_0). The coexistence phase appears in all three kinds of heterogeneity modeled here. (a–d) Inherent heterogeneity: Individuals of the prey population are born in two types with a fixed composition (rho). In the extreme cases of (rho = 0) and 1, the prey is homogeneous and there is no coexistence of the predators. (e–h) Reversible heterogeneity: Individual prey can switch types with fixed switching rates (eta _1) and (eta _2). As the switching rates increase, the coexistence region shrinks because the prey population becomes effectively homogeneous (the occasional green spots are numerical artifacts because the time to reach the equilibrium becomes long in this limit). (i–l) Adaptive heterogeneity: The switching rates (eta _i) dynamically adapt to the predator densities, so as to maximize the growth rate of the prey. As the sharpness (kappa) of the sigmoidal decision function is increased, the prey adapts more optimally and the region of coexistence expands. Parameters used here are ((a_0, a_1, b_0, b_2) = (0.3, 0.5, 0.4, 0.6)).Full size imageThe stable equilibrium of the system can be represented by phase diagrams that show the identities of the species at equilibrium. We plot these phase diagrams by varying the parameters (a_2) and (b_1) while keeping (a_1) and (b_2) constant. As shown in Fig. 4a–d, the equilibrium state depends on the parameter (rho). In the limit (rho = 0) or 1, we recover the homogeneous case because only one type of C is produced. The corresponding phase diagrams (Fig. 4a, d) contain only two phases where either of the predators is excluded, illustrating the competitive exclusion principle. For intermediate values of (rho), however, there is a new phase of coexistence that separates the two exclusion phases (Fig. 4b, c). There are two such regions of coexistence, which touch at a middle point and open toward the bottom left and upper right, respectively. The middle point is at ((a_2/a_0 = b_2/b_0, b_1/b_0 = a_1/a_0)), where the feeding preferences of the two predators are identical (hence their niches fully overlap). Towards the origin and the far upper right, the predators consume one type of C each (hence their niches separate). The coexistence region in the bottom left is where the feeding rates of the predators are the lowest overall. There can be a region (yellow) where both predators go extinct, if their primary prey type alone is not enough to sustain each predator. Increasing the productivity of the system by increasing the birth rate ((beta _C)) of the prey eliminates this extinction region, whereas lowering productivity causes the extinction region to take over the lower coexistence region. Because the existence and identity of the phases is determined by the configuration of the equilibrium points (Fig. 2, see also section “Mathematical methods”), the qualitative shape of the phase diagram is not sensitive to changes of parameter values.The new equilibrium is not only where the predators A and B can coexist, but also where the prey species C grows to a larger density than what is possible for a homogeneous population. This is illustrated in Fig. 3b, which shows the equilibrium population of C if we hold (lambda) fixed at different values. The point (lambda = lambda ^*) is where the system with a dynamic (lambda) is stable, and also where the population of C is maximized (when A and B prefer different prey types). That means the population automatically stabilizes at the optimal composition of prey types. Moreover, the value of (C^*) at this coexistence point can even be larger than the equilibrium population of C when there is only one predator A or B. This is discussed further in section “Multiple-predator effects and emergent promotion of prey”. These results suggest that heterogeneity in interaction strengths can potentially be a strategy for the prey population to leverage the effects of multiple predators against each other to improve survival.Reversible heterogeneityWe next consider a scenario where individual prey can switch their types. This kind of heterogeneity can model reversible changes of phenotypes, i.e., trait changes that affect the prey’s interaction with predators but are not permanent. For example, changes in coat color or camouflage14,16,17, physiological changes such as defense15, and biomass allocation among tissues18,19. One could also think of the prey types as subpopulations within different spatial patches, if each predator hunts at a preferred patch and the prey migrate between the patches20,21. With some generalization, one could even consider heterogeneity in resources, such as nutrients located in different places, that can be reached by primary consumers, such as swimming phytoplankton22. We can model this “reversible” kind of heterogeneity by introducing switching rates from one prey type to the other. In Eq. (4d), (eta _1) and (eta _2) now represent the switching rates per capita from (C_1) to (C_2) and from (C_2) to (C_1), respectively. Here we study the simplest case where both rates are fixed.In the absence of the predators, the natural composition of the prey species given by the switching rates would be (rho equiv eta _1 / (eta _1 + eta _2)), and the rate at which (lambda) relaxes to this natural composition is (gamma equiv eta _1 + eta _2). Compared to the previous scenario where we had only one parameter (rho), here we have an additional parameter (gamma) that modifies the behavior of the system. Fig. 4e–h shows phase diagrams for the system as (rho) is fixed and (gamma) varies. We again find the new equilibrium (P_N) where all three species coexist. When (gamma) is small, the system has a large region of coexistence. As (gamma) is increased, this region is squeezed into a border between the two regions of exclusion, where the slope of the border is given by (eta _1/eta _2) as determined by the parameter (rho). However, this is different from the exclusion we see in the case of inherent heterogeneity, which happens only for (rho rightarrow 0) or 1, where the borders are horizontal or vertical (Fig. 4a,d). Here the predators exclude each other despite having a mixture of prey types in the population.This special limit can be understood as follows. For a large (gamma), (lambda) is effectively set to a constant value equal to (rho), because it has a very fast relaxation rate. In other words, the prey types exchange so often that the population always maintains a constant composition. In this limit, the system behaves as if it were a homogeneous system with effective interaction strengths (a_text {eff} = (1-rho ) , a_1 + rho , a_2) and (b_text {eff} = (1-rho ) , b_1 + rho , b_2). As in a homogeneous system, there is competitive exclusion between the predators instead of coexistence. This demonstrates that having a constant level of heterogeneity is not sufficient to cause coexistence. The overall composition of the population must be able to change dynamically as a result of population growth and consumption by predators.An interesting behavior is seen when we examine a point inside the shrinking coexistence region as (gamma) is increased. Typical trajectories of the system for such parameter values are shown in Fig. 5. As (gamma) increases, the system relaxes to the line (mathscr {L}) quickly, then slowly crawls along it towards the final equilibrium point (P_N). This is because increasing (gamma) increases the speed that (lambda) relaxes to (lambda ^*), and when (lambda rightarrow lambda ^*), (mathscr {L}) becomes marginally stable. Therefore, the attraction to (mathscr {L}) in the perpendicular direction is strong, but the attraction towards the equilibrium point along (mathscr {L}) is weak. This leads to a long transient behavior that makes the system appear to reach no equilibrium in a limited time23,24. It is especially true when there is noise in the dynamics, which causes the system to diffuse along (mathscr {L}) with only a weak drift towards the final equilibrium (Fig. 5). Thus, the introduction of a fast timescale (quick relaxation of (lambda) due to a large (gamma)) actually results in a long transient.Figure 5Trajectories of the system projected in the A-B plane, with parameters inside the coexistence region (by holding the position of (P_N) fixed). As (gamma) increases, the system tends to approach the line (mathscr {L}) quickly and then crawl along it. The grey trajectory is with independent Gaussian white noise ((sim mathscr {N}(0,0.5))) added to each variable’s dynamics. Noise causes the system to diffuse along (mathscr {L}) for a long transient period before coming to the equilibrium point (P_N). Parameters used here are ((a_0, a_1, a_2, b_0, b_1, b_2) = (0.2, 0.8, 0.5, 0.2, 0.6, 0.9)), chosen to place (P_N) away from the middle of (mathscr {L}) to show the trajectory drifting toward the equilibrium.Full size imageAdaptive heterogeneityA third kind of heterogeneity we consider is the change of interactions in time. By this we mean an individual can actively change its interaction strength with others in response to certain conditions. This kind of response is often invoked in models of adaptive foraging behavior, where individuals choose appropriate actions to maximize some form of fitness25,26. For example, we may consider two behaviors, resting and foraging, as our prey types. Different predators may prefer to strike when the prey is doing different things. In response, the prey may choose to do one thing or the other depending on the current abundances of different predators. Such behavioral modulation is seen, for example, in systems of predatory spiders and grasshoppers27. Phenotypic plasticity is also seen in plant tissues in response to consumers28,29,30.This kind of “adaptive” heterogeneity can be modeled by having switching rates (eta _1) and (eta _2) that are time-dependent. Let us assume that the prey species tries to maximize its population growth rate by switching to the more favorable type. From Eq. (4c), we see that the growth rate of C depends linearly on the composition (lambda) with a coefficient (u(A,B) equiv (a_1 – a_2) A + (b_1 – b_2) B). Therefore, when this coefficient is positive, it is favorable for C to increase (lambda) by switching to type (C_2). This can be achieved by having a positive switching rate (eta _2) whenever (u(A,B) > 0). Similarly, whenever (u(A,B) < 0), it is favorable for C to switch to type (C_1) by having a positive (eta _1). In this way, the heterogeneity of the prey population constantly adapts to the predator densities. We model such adaptive switching by making (eta _1) and (eta _2) functions of the coefficient u(A, B), e.g., (eta _1(u) = 1/(1+mathrm {e}^{kappa u})) and (eta _2(u) = 1/(1+mathrm {e}^{-kappa u})). The sigmoidal form of the functions means that the switching rate in the favorable direction for C is turned on quickly, while the other direction is turned off. The parameter (kappa) controls the sharpness of this transition.Phase diagrams for the system with different values of (kappa) are shown in Fig. 4i–l. A larger (kappa) means the prey adapts its composition faster and more optimally, which causes the coexistence region to expand. In the extreme limit, the system changes its dynamics instantaneously whenever it crosses the boundary where (u(A,B) = 0), like in a hybrid system31. Such a system can still reach a stable equilibrium that lies on the boundary, if the flow on each side of the boundary points towards the other side32. This is what happens in our system and, interestingly, the equilibrium is the same three-species coexistence point (P_N) as in the previous scenarios. The region of coexistence turns out to be largest in this limit (Fig. 4l).Our results suggest that the coexistence of the predators can be viewed as a by-product of the prey’s strategy to maximize its own benefit. The time-dependent case studied here represents a strategy that involves the prey evaluating the risk posed by different predators. This is in contrast to the scenarios studied above, where the prey population passively creates phenotypic heterogeneity regardless of the presence of the predators. These two types of behavior are analogous to the two strategies studied for adaptation in varying environments, i.e., sensing and bet-hedging33,34. The former requires accessing information about the current environment to make optimal decisions, whereas the latter relies on maintaining a diverse population to reduce detrimental effects caused by environmental changes. Here the varying abundances of the predators play a similar role as the varying environment. From this point of view, the heterogeneous interactions studied here can be a strategy of the prey species that is evolutionarily favorable. More

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Data were analyzed from samples collected, processed, and published previously [21, 25, 29] and have been summarized here. The present analysis, which consisted of sequence data processing, the calculation of taxon-specific isotopic signatures, and subsequent analyses, reflects original work.Sample collection and isotope incubationTo generate experimental data, three replicate soil samples were collected from the top 10 cm of plant-free patches in four ecosystems along the C. Hart Merriam elevation gradient in Northern Arizona. From low to high elevation, these sites are located in the following environments: desert grassland (GL; 1760 m), piñon-pine juniper woodland (PJ; 2020 m), ponderosa pine forest (PP; 2344 m), and mixed conifer forest (MC; 2620 m). Soil samples were air-dried for 24 h at room temperature, homogenized, and passed through a 2 mm sieve before being stored at 4 °C for another 24 h. This produced three distinct but homogenous soil samples from each of the four ecosystems that were subject to experimental treatments. Three treatments were applied to bring soils to 70% water-holding capacity: water alone (control), water with glucose (C treatment; 1000 µg C g−1 dry soil), or water with glucose and a nitrogen source (CN treatment; [NH4]2SO4 at 100 µg N g−1 dry soil). To track growth through isotope assimilation, both 18O-enriched water (97 atom %) and 13C-enriched glucose (99 atom %) were used. In all treatments isotopically heavy samples were paired with matching “light” samples that received water with a natural abundance isotope signatures. For 18O incubations, this design resulted in three soil samples per ecosystem per treatment (across four ecosystems and three treatments, n = 36) while 13C incubations were limited to only C and CN treatments (n = 24). Previous analyses suggest that three replicates is sufficient to detect growth of 10 atom % 18O in microbial DNA with a power of 0.6 and a growth of 5 atom % 18O with a power of 0.3 (12 and 6 atom % respectively for 13C) [30]. All soils were incubated in the dark for one week. Following incubation, soils were frozen at −80 °C for one week prior to DNA extraction.Quantitative stable isotope probingThe procedure of qSIP (quantitative stable isotope probing) is described here but has been applied to these samples as previously published [17, 21, 25]. DNA extraction was performed on soils using a DNeasy PowerSoil HTP 96 Kit (MoBio Laboratories, Carlsbad, CA, USA) and following manufacturer’s protocol. Briefly, 0.25 g of soils from each sample were carefully added to deep, 96-well plates containing zirconium dioxide beads and a cell lysis solution with sodium dodecyl sulfate (SDS) and shaken for 20 min. Following cell lysis, supernatant was collected and centrifuged three times in fresh 96-well plates with reagents separating DNA from non-DNA organic and inorganic materials. Lastly, DNA samples were collected on silica filter plates, rinsed with ethanol and eluted into 100 µL of a 10 mM Tris buffer in clean 96-well plates. To quantify the degree of 18O or 13C isotope incorporation into bacterial DNA (excess atom fraction or EAF), the qSIP protocol [31] was used, though modified slightly as reported previously [21, 24, 32]. Briefly, microbial growth was quantified as the change in DNA buoyant density due to incorporation of the 18O or 13C isotopes through the method of density fractionation by adding 1 µg of DNA to 2.6 mL of saturated CsCl solution in combination with a gradient buffer (200 mM Tris, 200 mM KCL, 2 mM EDTA) in a 3.3 mL OptiSeal ultracentrifuge tube (Beckman Coulter, Fullerton, CA, USA). The solution was centrifuged to produce a gradient of increasingly labeled (heavier) DNA in an Optima Max bench top ultracentrifuge (Beckman Coulter, Brea, CA, USA) with a Beckman TLN-100 rotor (127,000 × g for 72 h) at 18 °C. Each post-incubation sample was thus converted from a continuous gradient into approximately 20 fractions (150 µL) using a modified fraction recovery system (Beckman Coulter). The density of each fraction was measured with a Reichart AR200 digital refractometer (Reichert Analytical Instruments, Depew, NY, USA). Fractions with densities between 1.640 and 1.735 g cm−3 were retained as densities outside this range generally did not contain DNA. In all retained fractions, DNA was cleaned and purified using isopropanol precipitation and the abundance of bacterial 16S rRNA gene copies was quantified with qPCR using primers specific to bacterial 16S rRNA genes (Eub 515F: AAT GAT ACG GCG ACC ACC GAG TGC CAG CMG CCG CGG TAA, 806R: CAA GCA GAA GAC GGC ATA CGA GGA CTA CVS GGG TAT CTA AT). Triplicate reactions were 8 µL consisting of 0.2 mM of each primer, 0.01 U µL−1 Phusion HotStart II Polymerase (Thermo Fisher Scientific, Waltham, MA), 1× Phusion HF buffer (Thermo Fisher Scientific), 3.0 mM MgCl2, 6% glycerol, and 200 µL of dNTPs. Reactions were performed on a CFX384 Touch Real-Time PCR Detection System (Bio-Rad, Hercules, CA, USA) under the following cycling conditions: 95 °C at 1 min and 44 cycles at 95 °C (30 s), 64.5 °C (30 s), and 72 °C (1 min). Separate from qPCR, retained sample-fractions were subject to a similar amplification step of the 16S rRNA gene V4 region (515F: GTG YCA GCM GCC GCG GTA A, 806R: GGA CTA CNV GGG TWT CTA AT) in preparation for sequencing with the same reaction mix but differing cycle conditions – 95 °C for 2 min followed by 15 cycles at 95 °C (30 s), 55 °C (30 s), and 60 °C (4 min). The resulting 16S rRNA gene V4 amplicons were sequenced on a MiSeq sequencing platform (Illumina, Inc., San Diego, CA, USA). DNA sequence data and sample metadata have been deposited in the NCBI Sequence Read Archive under the project ID PRJNA521534.Sequence processing and qSIP analysisIndependently from previous publications, we processed raw sequence data of forward and reverse reads (FASTQ) within the QIIME2 environment [33] (release 2018.6) and denoised sequences within QIIME2 using the DADA2 pipeline [34]. We clustered the remaining sequences into amplicon sequence variants (ASVs, at 100% sequence identity) against the SILVA 138 database [35] using a pre-trained open-reference Naïve Bayes feature classifier [36]. We removed samples with less than 3000 sequence reads, non-bacterial lineages, and global singletons and doubletons. We converted ASV sequencing abundances in each fraction to the number of 16S rRNA gene copies per gram dry soil based on qPCR abundances and the known amount of dry soil equivalent added to the initial extraction. This allowed us to express absolute population densities, rather than relative abundances. Across all replicates, we identified 114 543 unique bacterial ASVs.We calculated the 18O and 13C excess atom fraction (EAF) for each bacterial ASV using R version 4.0.3 [37] and data.table [38] with custom scripts available at https://www.github.com/bramstone/. Negative enrichment values were corrected using previously published methods [17]. ASVs that appeared in less than two of the three replicates of an ecosystem-treatment combination (n = 3) and less than three density fractions within those two replicates were removed to avoid assigning spurious estimates of isotope enrichment to infrequent taxa. Any ASVs filtered out of one ecosystem-treatment group were allowed to be present in another if they met the frequency threshold. Applying these filtering criteria, we limited our analysis towards 3759 unique bacterial ASVs which accounted for a small proportion of the total diversity but represented 68.0% of all sequence reads, and encompassed most major bacterial groups (Supplementary Fig. 1).Analysis of life history strategies and nutrient responseAll statistical tests were conducted in R version 4.0.3 [37]. We assessed the ability of phylum-level assignment of life history strategy to predict growth in response to C and N addition, as proxied by the incorporation of heavy isotope during DNA replication [39, 40]. Phylum-level assignments (Table 1) were based on the most frequently observed behavior of lineages with a representative phylum (or subphylum) as compiled previously [23]. We averaged 18O EAF values of bacterial taxa for each treatment and ecosystem and then subtracted the values in control soils from values in C-amended soils to determine C response (∆18O EAFC) and from the 18O EAF of bacteria in CN-amended soils to determine C and N response (Δ18O EAFCN). Because an ASV must have a measurable EAF in both the control and treatment for a valid Δ18O EAF to be calculated, we were only able to resolve the nutrient response for 2044 bacterial ASVs – 1906 in response to C addition and 1427 in response to CN addition.We used Gaussian finite mixture modeling, as implemented by the mclust R package [41], to demarcate plausible multi-isotopic signatures for oligotrophs and copiotrophs. For each treatment, we calculated average per-taxon 13C and 18O EAF values. To compare both isotopes directly, we divided 18O EAF values by 0.6 based on the estimate that this value (designated as µ) represents the fraction of oxygen atoms in DNA derived from the 18O-water, rather than from 16O within available C sources [42]. Two mixture components, corresponding to oligotrophic and copiotrophic growth modes, were defined using the Mclust function using ellipsoids of equal volume and shape. We observed several microorganisms with high 18O enrichment but comparatively low 13C enrichment, potentially indicating growth following the depletion of the added glucose, and that were reasonably clustered as oligotrophs in our mixture model.We tested how frequently mixture model clustering of each microorganism’s growth (based on average 18O–13C EAF in a treatment) could predict its growth across replicates (n = 12 in each treatment—although individual). We applied the treatment-level mixture models defined above to the per-taxon isotope values in each replicate, recording when a microorganism’s life history strategy in a replicate agreed with the treatment-level cluster, and when it didn’t. We used exact binomial tests to test whether the number of “successes” (defined as a microorganism being grouped in the same life history category as its treatment-level cluster) was statistically significant. To account for type I error across all individual tests (one per ASV per treatment), we adjusted P values in each treatment using the false-discovery rate (FDR) method [43].To determine the extent that life history categorizations may be appropriately applied at finer levels of taxonomic resolution, we constructed several hierarchical linear models using the lmer function in the nlme package version 3.1-149 [44]. To condense growth information from both isotopes into a single analysis, 18O and 13C EAF values were combined into a single variable using principal components analysis separately for each treatment. Across the C and CN treatments, the first principal component (PC1) was able to explain – respectively – 86% and 91% of joint variation of 18O and 13C EAF values. In all cases, we applied PC1 as the response variable and treated taxonomy and ecosystem as random model terms to limit the potential of pseudo-replication to bias significance values. We used likelihood ratio analysis and Akaike information criterion (AIC) values to compare models where life history strategy was determined based on observed nutrient responses at different taxonomic levels (Eq. 1) against a model with the same random terms but without any life history strategy data (Eq. 2). Separate models were applied to each treatment. To reduce model overfitting, we removed families represented by fewer than three bacterial ASVs as well as phyla represented by only one order. In addition, we removed bacterial ASVs with unknown taxonomic assignments (following Morrissey et al. [21]). This limited our analysis to 1 049 ASVs in the C amendment and 984 in the CN amendment.$${{{{{rm{PC}}}}}}{1}_{{18{{{{{rm{O}}}}}} – 13{{{{{rm{C}}}}}}}}sim {{{{{rm{strategy}}}}}} + 1|{{{{{rm{phylum}}}}}}/{{{{{rm{class}}}}}}/{{{{{rm{order}}}}}}/{{{{{rm{family}}}}}}/{{{{{rm{genus}}}}}}/{{{{{rm{eco}}}}}}$$
(1)
$${{{{{rm{PC}}}}}}{1}_{{18{{{{{rm{O}}}}}} – 13{{{{{rm{C}}}}}}}}sim 1 + 1|{{{{{rm{phylum}}}}}}/{{{{{rm{class}}}}}}/{{{{{rm{order}}}}}}/{{{{{rm{family}}}}}}/{{{{{rm{genus}}}}}}/{{{{{rm{eco}}}}}}$$
(2)
Here, life history strategy was defined at each taxonomic level using the mixture models above and based on the mean 18O and 13C EAF values of each bacterial lineage (Supplemental Fig. 2). We compared these models with the no-strategy model (Eq. 2) directly using likelihood ratio testing. More

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