Responses of two nonlinear microbial models to warming and increased carbon input

Transcription

1 Biogeosciences, 13, , doi:1.5194/g Author(s) 216. CC Attriution 3. License. Responses of two nonlinear microial models to warming and increased caron input Y. P. Wang 1, J. Jiang 2, B. Chen-Charpentier 3, F. B. Agusto 4, A. Hastings 5, F. Hoffman 6, M. Rasmussen 7, M. J. Smith 8, K. Todd-Brown 9,11, Y. Wang 1, X. Xu 9, and Y. Q. Luo 9 1 CSIRO Ocean and Atmosphere, PMB 1, Aspendale, Victoria 3195, Australia 2 Department of Ecology and Evolutionary Biology, University of Tennessee, Knoxville, TN 37996, USA 3 Department of Mathematics, University of Texas, Arlington, TX, USA 4 Department of Mathematics and Statistics, Austin Peay State University, Clarksville TN 3744, USA 5 Department of Environmental Science and Policy, University of California, One Shields Avenue, Davis, CA 95616, USA 6 Oak Ridge National Laoratory, Computational Earth Sciences Group, P.O. Box 28, Oak Ridge, TN 37831, USA 7 Department of Mathematics, Imperial College, London, UK 8 Computational Science Laoratory, Microsoft Research, Camridge, UK 9 Department of Microiology and Plant Biology, University of Oklahoma, Norman, OK, USA 1 Department of Mathematics, University of Oklahoma, Norman, OK, USA 11 Pacific Northwest National Laoratory, Richland, WA, USA Correspondence to: Y.-P. Wang (yingping.wang@csiro.au) Received: 14 July 215 Pulished in Biogeosciences Discuss.: 7 Septemer 215 Revised: 9 January 216 Accepted: 3 January 216 Pulished: 18 Feruary 216 Astract. A numer of nonlinear microial models of soil caron decomposition have een developed. Some of them have een applied gloally ut have yet to e shown to realistically represent soil caron dynamics in the field. A thorough analysis of their key differences is needed to inform future model developments. Here we compare two nonlinear microial models of soil caron decomposition: one ased on reverse Michaelis Menten kinetics (model A) and the other on regular Michaelis Menten kinetics (model B). Using analytic approximations and numerical solutions, we find that the oscillatory responses of caron pools to a small perturation in their initial pool sizes dampen faster in model A than in model B. Soil warming always decreases caron storage in model A, ut in model B it predominantly decreases caron storage in cool regions and increases caron storage in warm regions. For oth models, the CO 2 efflux from soil caron decomposition reaches a maximum value some time after increased caron input (as in priming experiments). This maximum CO 2 efflux (F max ) decreases with an increase in soil temperature in oth models. However, the sensitivity of F max to the increased amount of caron input increases with soil temperature in model A ut decreases monotonically with an increase in soil temperature in model B. These differences in the responses to soil warming and caron input etween the two nonlinear models can e used to discern which model is more realistic when compared to results from field or laoratory experiments. These insights will contriute to an improved understanding of the significance of soil microial processes in soil caron responses to future climate change. 1 Introduction The dynamics of soil caron in most gloal iogeochemical models are modelled using first-order kinetics, which assumes that the decay rate of soil caron is proportional to the size of soil caron pool. This approach has een recently questioned on theoretical grounds (Schimel and Weintrau, 23; Fontaine and Barot, 25), and is contradicted y the oserved responses of soil caron decay to the addition of fresh organic litter (Fontaine et al., 24; Sayer et al., 211) or soil warming (Luo et al., 21; Melillo et al., 22; Bradford et al., 28). As a result, a numer of nonlinear soil microial models have een developed (Alli- Pulished y Copernicus Pulications on ehalf of the European Geosciences Union.

2 888 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input son et al., 21; Manzoni and Porporato, 27; Wutzler and Reichstein, 28) and a few of them have een applied at gloal scales (Wieder et al., 213; Sulman et al. 214). Predictions of future soil caron change y these nonlinear models can differ significantly from conventional linear models (Fontaine et al., 27; Wieder et al., 213). For example, conventional linear soil caron models predict that soil caron will decrease with increased temperature, all else eing equal (Jenkinson et al., 1991), whereas the nonlinear models predict that the soil caron can decrease or increase, depending on the temperature sensitivity of microial growth efficiency and turnover rates (Frey et al., 213; Hagerty et al., 214; Li et al., 214). However, the nonlinear models have yet to e validated against field measurements as extensively as the conventional linear soil caron models (Wieder et al., 216). They also have some undesirale features, particularly the presence of strong oscillations or ifurcations (Manzoni and Porporato, 27; Wang et al., 214) in their dynamics that are not oserved in real-world systems. Therefore it is important to improve understanding of the ehaviour of these nonlinear models efore they are used in earth system models for informing climate decisions. Nonlinear microial models can explain why the decomposition rate of recalcitrant organic soil caron varies after the addition of easily decomposale organic caron to soil, which is known as the priming effect (Kuzyakov et al., 2). This response has een oserved in the field (Fontaine et al., 24; Sayer et al., 211) ut cannot predicted y conventional linear soil caron models without modification (Fujita et al., 214). Theoretically, decomposition of soil organic caron is catalysed y extracellular enzymes that are produced y soil microes. The production rate of extracellular enzymes depends on the iomass and composition of the soil microial population and their local environment. Therefore the decomposition rate of soil organic caron should depend on oth microial iomass and sustrate concentration (Schimel and Weintrau, 23), rather than on sustrate concentration only, as assumed in conventional linear models. This sensitivity of soil caron decomposition to the input of additional caron has important implications for the storage of caron y the iosphere in response to climate change. Soil is the largest land caron pool and therefore the direction and magnitude of the gloal caron climate feedack strongly depends on the responses of soil caron to future warming (Jones and Fallow, 29; Hargety et al., 214). A numer of nonlinear models have een developed that explicitly account for the dynamics of the soil microial community (Parnas, 1978; Smith, 1979; Schimel and Weintrau, 23; Wutzler and Reichstein, 28; Allison et al., 21; Grant, 214; Riley et al., 214; Tang and Riley, 214). Parnas (1979) explored the mechanism of priming using a nonlinear soil microial model that included oth soil caron and nitrogen dynamics. Smith (1979) developed a nonlinear model of soil caron decomposition that included the interactions among caron, nitrogen, phosphorus, and potassium. Smith s model represented multiple forms of caron, nitrogen, and phosphorus and their transformation via aiotic (such as adsorption and desorption) and iological processes y different groups of soil microes. The soil models developed y oth Parnas (1978) and Smith (1979) were ased on regular Michaelis Menten kinetics, in which the rate of caron decomposition depends linearly on the concentration of soil enzymes ut nonlinearly on sustrate concentration (Roerts, 1977). This was challenged y Schimel and Weintrau (23), who emphasized the importance of exoenzyme limitation on soil caron decomposition. Schimel and Weintrau (23) used a reverse Michaelis Menten kinetics formulation to show that the response of soil caron decomposition to caron sustrate concentration can e nonlinear regardless of caron supply. The reverse Michaelis Menten kinetics for soil caron decomposition assumes that the rate of caron decomposition depends nonlinearly on enzyme concentration ut linearly on sustrate concentration. The nonlinear soil caron models descried aove have susequently een used in a variety of studies: to explore different the fundamental mechanisms controlling soil caron decomposition (Schimel and Weintrau, 23, for example), to investigate the sensitivity of soil caron and other iogeochemical processes to warming (Grant, 214; Tang and Riley, 214), to investigate the response of soil caron to a small perturation, such as priming (Wutzler and Reichstein, 213), and to predict soil caron responses to gloal change (Wieder et al., 213; Sulman et al., 214). Some studies have explored the mathematical properties of these nonlinear models in detail (for example, Manzoni et al., 24; Manzoni and Porporato, 27; Raupach, 27; Wang et al., 214). However, to date these have een predominantly restricted to otaining insights for individual models and with a specific parameterization. In this study we use mathematical analysis to improve our understanding of the key properties of nonlinear microial models. For simplicity and analytic convenience, we choose two simple types of nonlinear microial models: one with regular Michaelis Menten kinetics and another with the reverse Michaelis Menten kinetics. These models can e considered as two special cases of the more general kinetics discussed y Tang (215). These two simple formulations are amenale to analytic approximations, whereas the formulations with more general kinetics, such as the equilirium chemistry approximations, are not. We only represent three soil caron pools with each model and ignore aiotic processes for simplicity, despite these eing potentially important under certain conditions (see Tang and Riley, 214 for an example). In comparing the two nonlinear microial models, we use the standard mathematical technique to analyse their responses to a small perturation (see Wang et al., 214), such as a step change in soil temperature or caron input, or whether two models exhiit oscillatory ehaviour under certain conditions, and how the analytic approximations to the exact model solutions differ etween the two nonlinear mod- Biogeosciences, 13, , 216

3 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input 889 els. We address the following questions. (1) How do the responses of these two models to soil warming differ, and why? (2) Can oth models simulate the response of soil caron decomposition to increased caron input as in a priming experiment and what determines the magnitude of the response in each model? 2 Methods 2.1 Model description We consider two nonlinear soil microial models: model A, which uses reverse Michaelis Menten kinetics, and model B, which uses regular Michaelis Menten kinetics (specified elow). Both models have three caron pools: litter caron, microial iomass, and soil caron. Model A is ased on the nonlinear microial model of soil caron descried Wutzler and Reichstein (213, their model A1). Their original model has four pools, modelled y dc s and dc m dc l = (1 a)f npp µ l C l C C + K, (1) = af npp + µ C µ s C s C C + K, (2) dc = εµ m C C m C m + K m µ C, (3) C C m = (µ l C l + µ s C s ) µ m C, (4) C + K C m + K m where t is time in years; C l, C s, C, and C m represent the pool sizes of litter caron, soil caron, microial iomass caron, and assimilale soil caron in g C m 2, respectively; and F npp is caron input in g C m 2 yr 1, with the fraction a going to the soil caron pool, and (1 a) to the litter caron pool. µ l, µ s, µ, and µ m are rate constants of litter caron, soil caron, microial iomass, and assimilale caron per year, respectively (see Schimel and Weintrau, 23); ε is microial growth efficiency; and K and K m are two empirical constants in g C m 2 for the dependence of the consumption of litter caron or assimilale caron y soil microes. In this study we are interested in the responses at timescales greater than 1 year. We therefore assume that C m is at steady state (dc m / = ) ecause of its relatively fast turnover (less than a few days). Therefore the dynamics of microial iomass, C, can e simplified to dc C = ε (µ l C l + µ s C s ) µ C. (5) C + K Model A as used in this paper consists of Eqs. (1), (2), and (5) unless otherwise specified. This type of formulation was also used y Schimel and Weintrau (23) and Drake et al. (213). Model B, ased on the model used y Allison et al. (21) and Wieder et al. (213) with one additional assumption that oth enzyme and dissolved organic caron pools are at steady states, is given y dc s and dc dc l = (1 a)f npp C V l C l C l + K l, (6) = af npp + µ C C V s C s C s + K s, (7) = ε C ( Vl C l C l + K l + V s C s C s + K s ) µ C, (8) where K l and K s are Michaelis Menten constants in g C m 2, and V l and V s are maximum rates of sustrate caron (litter or soil) assimilation rate per unit microial iomass per year. This type of kinetics was used y Riley et al. (214), Wieder et al. (214) and Wang et al. (214). These two models make different assumptions aout the rate-limiting step in caron decomposition. Both models assume that microes have similar access to litter and soil caron. In model A, caron decomposition is assumed to depend nonlinearly on the numer of inding sites or the amount of sustrate and linearly on enzymes or microial iomass (Schimel and Weintrau, 23). In model B, caron decomposition is assumed to depend nonlinearly on enzymes or microial iomass and linearly on the numer of inding sites or the amount of sustrate (Allison et al., 21). When caron input, F npp, is equal to zero, the steady-state solution is zero for litter and soil caron pools for oth models (a trivial solution). When F npp >, the steady-state solutions to model A are Cl = (1 a)f npp + (ε 1 1)(1 a)µ K, µ l µ l (9) C = F npp (ε 1, 1)µ (1) and C s = (a + ) 1 Fnpp ( ( )) ε a ε 1 µ K 1. (11) 1 µ s µ s The steady-state solutions to model B are C l = C = and C s = K l εv l (1 ε)(1 a)µ 1, (12) F npp ( µ ε 1 ), (13) 1 K s (14) V s ε µ ε+a(1 ε) 1. Biogeosciences, 13, , 216

4 89 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input CO 2 efflux from the decomposition of soil organic caron (F s ) is calculated as C F s = (1 ε)µ s C s (15) C + K for model A and V s C s F s = (1 ε)c (16) C s + K s for model B. 2.2 Parameter values We allow all model parameters to vary with soil temperature (T s ) with the exception of parameter a. Based on the work of Allison et al. (21) and Hagerty et al. (214), we model the temperature dependence of parameters as ε = ε R x (T s T R ) (17) and µ = µ R exp( (T s T R )) (18) for oth models, where T R is reference soil temperature in C (i.e. 15 C), ε R and µ R are the values of ε and µ at T s = T R, respectively, and x and are two empirical constants (see Tale 1 for their default values). Previously there has een deate aout the temperature sensitivities of ε and µ (see Frey et al., 213; Hargety et al., 214). The microial models as developed y Allison et al. (21) and used y Wieder et al. (213), and Wang et al. (214) assumed that ε was temperature-sensitive and µ was temperature-insensitive (or = ). This assumption was recently challenged y Hargety et al. (214), who found that µ was temperature-sensitive and ε was not, ased on a laoratory soil-warming experiment. Here we will explore the consequence of different assumptions aout the temperature sensitivities of ε and µ on the simulated response of soil caron to warming y the two models (see Sect. 3.2). We also assume that three additional model parameters in model A, K, µ l, and µ s depend on soil temperature exponentially, with K = K R exp(α k (T s T R )), (19) µ l = µ lr exp(α l (T s T R )), (2) and µ s = µ sr exp(α s (T s T R )), (21) where K R, µ lr, and µ sr are the values of K, µ l and µ s when soil temperature (T s ) is equal to the reference temperature, T R (15 C in this study), and α k, α l, and α s are three empirical constants with their default values listed in Tale 1. For model B, we assume that K l, K s, V l, and V s increase with soil temperature exponentially: K l = K lr exp(β kl (T s T R )), (22) K s = K sr exp(β ks (T s T R )), (23) and V l = V lr exp(β vl (T s T R )), (24) V s = V sr exp(β vs (T s T R )), (25) where K lr, K sr, V lr, and V sr are the values of K l, K s, V l, and V s at the reference soil temperature (T R ), respectively, and β kl, β ks, β vl and β vs are four empirical constants for model B (see Tale 1). As found y Wang et al. (214), the microial iomass as simulated y model B using the parameter values of Wieder et al. (213) was low (< 1 % of total soil caron). We therefore reduced the turnover rate of microial iomass to 1.1 yr 1 y assuming that 2 % of total soil organic caron is microial iomass caron at a soil temperature of 15 C. Some parameter values in model A at the reference temperature were otained y calirating the equilirium litter and soil caron pool sizes against those from model B for a soil temperature of 15 C and caron input of 4 g C m 2 yr 1, as used in Wang et al. (214). 2.3 Analytic solutions and numerical simulations We derived and used analytic solutions whenever possile for comparing the two models. Specifically, we mathematically analysed the temperature dependence of steady-state soil caron pool size, and derived an analytic approximation of soil temperature at which equilirium soil caron is at a minimum (e.g. Eq. B4 for model B). We also derived an approximate solution for the maximum CO 2 loss from soil caron decomposition after the increased caron input for each model (e.g. Eq. C12 for model A and Eq. C15 for model B). When an analytic solution was not possile or too cumersome, we used numerical simulations to show the differences etween the two models in their responses of caron pools to a small perturation in litter or microial caron pool sizes, and the response of CO 2 efflux from soil caron decomposition to litter addition at a tropical forest site (Sayer et al., 211). 3 Results Before comparing the responses of our models to soil warming and increased caron input, we first analyse some key properties of their responses to a small perturation, i.e. whether oth models oscillate in response to a small change in their initial pool sizes and what determines the period and amplitude of the oscillation. As a step change in soil temperature or caron input can e considered to e a perturation, Biogeosciences, 13, , 216

5 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input 891 Tale 1. Default values of model parameters and their temperature sensitivities ( C 1 ). Four parameters were tuned: 1 tuned using the microial iomass data measured from a tropical forest site (see Sayer et al., 211), and 2 tuned against the soil caron pool size simulated y model B y Wang et al. (214). Default value Source Temperature Source sensitivity ε R =.39 Allison et al. (21) x =.16 Allison et al. (21) µ R = 1.1 yr 1 This study 1 =.63 Hagerty et al. (214) µ lr =.84 yr 1 This study 2 α l =.63 Hagerty et al. (214) µ sr =.28 yr 1 This study 2 α s =.63 Hagerty et al. (214) K R = 1 g C m 2 This study 1 α k =.7 Allison et al. (21) K lr = g C m 2 Wang et al. (214) β kl =.7 Allison et al. (21) K sr = g C m 2 Wang et al. (214) β kss =.7 Allison et al. (21) V lr = 172 yr 1 Wang et al. (214) β vl =.63 Allison et al. (21) V sr = 32 yr 1 Wang et al. (214) β vs =.63 Allison et al. (21) identifying differences in those key properties will help us understand the differences in the responses of the two models to soil warming and increased caron input. The response of model B to perturation has already een analysed y Wang et al. (214), and will not e elaorated here, ut the results from that analysis will e used to compare the period and amplitude of the response to perturation to that of model A. 3.1 Comparison of the perturation responses of oth models Perturation analysis is a standard mathematical technique for analysing the ehaviour of a dynamic system near its equilirium state (see Drazin, 1992, for further details). There are two kinds of perturation responses: stale or unstale. The system states, or caron pool sizes in this study, will always approach their equilirium states for a stale response, or otherwise for an unstale response. For oth stale and unstale responses, the transient change in a caron pool size over time can e oscillatory or monotonic. As shown in Appendix A, the response of a caron pool to a small perturation is always stale, and oscillatory only if F npp < 4 (1 ε)2 ε µ l µ 2 K (µ µ l ) 2, or monotonic otherwise for model A. This region of oscillation in the two-dimensional space of caron input and soil temperature is shown in lack in Fig. 1. The response of model A to a small perturation is oscillatory under most conditions; the conditions with low soil temperature and high caron input are uncommon in terrestrial ecosystems. The results of a singular perturation analysis are strictly applicale only when the perturation is small. However, our simulations show that the predictions from the perturation analysis approximate well the responses of our two models to any realistic perturation (see Appendix A of this paper and Appendix B in Wang et al., 214). Therefore we can predict how soil caron or other caron pools change over time in Parameter values (relative to 15 o C) ε V l K Caron input (g C m -2 year -1 ) Non-Oscillate Oscillate Figure 1. Variation in microial growth efficiency (ε), V l, and K with soil temperature (left panel) or the region in which model A has oscillatory or non-oscillatory response to a small perturation (right panel) at different caron input and soil temperature. response to a change in caron inputs or soil warming (i.e. a perturation of the external environment) and explain why the responses of the caron pools are different etween the two models. To illustrate how the responses of caron pools to a small perturation differ etween the two models, we numerically simulated the recovery of all three caron pools in each model after a 1 % reduction at time t = in oth litter and microial caron from their respective steady-state values, while no perturation was applied to soil caron at t = (see Fig. 2). The amplitude of the initial oscillation is aout 7 g C m 2 for the litter pool (see Fig. 2) and 7 g C m 2 for the microial caron pool (see Fig. 2d) in model B, compared to aout 25 g C m 2 (see Fig. 2a) for the litter pool and 4 g C m 2 for the microial pool (see Fig. 2c) in model A. After 2 years, oth the litter and microial caron pools are very close to their respective steady-state values in model A, ut continue to oscillate in model B. The oscillatory response can e mathematically characterized y its half-life (t.5 ) and period (p). For a stale oscillatory response, the amplitude of the oscillation decays exwww.iogeosciences.net/13/887/216/ Biogeosciences, 13, , 216

6 892 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input Litter caron Microial caron Soil caron (a) (c) (e) Time () (d) Time (f) Figure 2. Dynamics of litter caron (a, ), microial caron (c, d), and soil caron (e, f) for model A (a, c, e) and model B (, d, f) after a 1 % reduction of initial pool size in litter and microial caron. The unit is g C m 2 for caron pool on the y axis and year for time. All initial pools are steady-state values for a caron input of 2 g C m 2 yr 1 at a soil temperature is 25 C. ponentially. The time for the amplitude to reach 5 % of its initial value is defined as the half-life (t.5 ). The smaller t.5 is, the faster the oscillation dampens. As explained in Appendix A, values of t.5 and p for model A are much smaller than model B for any given soil temperature and perturation. This explains why the oscillatory response of model A dampens much faster than model B. There are significant differences in the response of soil caron etween the two models. While there is no response of soil caron to a small perturation in litter caron and microial iomass in model B, soil caron in model A decreases initially to a minimum value at 5 years after the perturation, then gradually increases to its steady-state value. These differences in the response of soil caron etween the two models can e explained y the differences in the structure of eigenvectors for litter caron and microial iomass etween the two models (see Appendix A for further details). 3.2 Response of soil caron to warming Here we explore how soil caron responds to a step increase in soil temperature, as in many soil-warming experiments (Luo et al., 21; Mellilo et al., 22), and ignore the response of caron input to warming. As explained in Appendix A, the response of soil caron to warming is always stale in oth models and is likely to e weakly oscillatory in model A and monotonic in model B. The transient change in soil caron after warming can e predicted using the generalized solution for soil caron for each model (see Eq. B1 of Wang et al., 214). Therefore the directional change in soil caron in response to warming, i.e. increasing or decreasing only, depends on the sensitivity of the equilirium soil caron pool to soil temperature in oth models. As shown in Appendix B, the equilirium pool size of soil caron of model A always decreases with soil warming if caron input does not increase with warming. For model B, the equilirium pool size of soil caron can increase or decrease in response to warming, depending on soil temperature and model parameter values. In Appendix B, we show that a soil temperature (T x ) may exist at which the equilirium soil caron is at a minimum for model B. Identifying T x is important for predicting the directional change in soil caron y model B in a warmer world, ecause soil caron will decrease if the warmed soil temperature is elow T x, and will increase otherwise. The value of T x for model B depends on three parameters: the fraction of caron input directly into the soil pool (a), microial iomass turnover rate (µ or its temperature sensitivity ), and microial growth efficiency (ε or its temperature sensitivity x). Figure 3a shows that T x for model B decreases with an increase in a or x. Over the ranges of values of x and a, T x can vary across the range of air temperature experienced y most terrestrial ecosystems. For example, T x is > 4 C when x <.5 C 1 and a <.5; therefore the equilirium soil caron predicted y model B decreases with warming when the warmed soil temperature is elow 4 C. When a >.4 and x>.2 C 1, T x is < C (the lack region on the top left corner of Fig. 3a); therefore the simulated equilirium soil caron y model B increases with warming if the warmed soil temperature is aove C. Figure 3 shows that T x for model B decreases with an increase in or x. When the turnover rate of microial iomass is not sensitive to soil temperature ( = ) and x =.16 C 1 as the default value for model B, T x is aout 35 C. For =.63, as estimated y Hagerty et al. (214), T x < C; therefore the equilirium soil caron pool size as simulated y model B always increases with soil warming for most terrestrial ecosystems, irrespective of the value of x. Therefore the simulated responses of the soil caron pool to warming y the two models can e quite different: the equilirium soil caron pool size always decreases with soil warming in model A, ut can increase or decrease in model B, depending on the temperature sensitivities of microial growth efficiency and microial turnover rate and the fraction of caron input entering soil caron pool directly. 3.3 The response of soil caron to an increased litter input We compare the simulated responses of soil caron to litter addition y the two models with field measurements from an experiment descried y Sayer et al. (211). The experiment Biogeosciences, 13, , 216

7 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input 893 Parameter a (fraction) Parameter ( o C -1 ) Parameter x ( o C -1 ) (a) Parameter x ( o C -1 ) 1 () Figure 3. (a) Variation in T x, or the soil temperature at which the equilirium soil caron pool is minimum, with the temperature sensitivity of microial growth efficiency (x) and the fraction of caron input directly into soil caron pool (a). µ was fixed at 1.1 yr 1 (or = ) for this plot. () Variation in T x with x and. Parameter a was fixed at.5 for plot (). The unit is C for all the numers along the contour lines in oth (a) and (). The lack region in () represents T x < C. used three treatments: litter removal (L ), with aoveground litter eing removed regularly; increased litter input (L + ) with the added litter from the litter removal treatment; and a control (C). Measurements of CO 2 efflux from soil were made and the contriution of root rhizosphere respiration to soil respiration was estimated using a δ 13 C technique. Sayer et al. (211) found that the CO 2 efflux from the decomposition of soil organic caron in the L + treatment was 46 % higher than in the control. Therefore, increased litter addition accelerated the decomposition of soil organic caron. Here we assess whether the oserved response of soil caron Litter input (g C m -2 day -1 ) Year Control (C) L+ Figure 4. Mean monthly total (aove- and elowground) litter caron input to the control or litter addition treatment. decomposition to increased litter input can e reproduced y running oth models for L + and C treatments. Inputs to each model, including the monthly data of soil temperature and litter input from 22 to 28 for two treatments (C and L + ) at the site, were compiled from Sayer and Tanner (21a, ; see Fig. 4 for monthly litter input as an example). We also assumed that the contriution of fine-root respiration to total soil respiration (root respiration plus heterotrophic respiration) was 35 % for the control treatment and 21 % for the litter addition treatment, ased on the estimates y Sayer et al. (211). The initial sizes of all pools were otained y running each model with the monthly inputs for the first 2 years repeated until all pools reached steady state (i.e. the change in pool size etween two successive cycles is less than.1 %). Using the initial pool sizes for each model and the monthly input from 22 28, we numerically integrated oth models and calculated the average contriutions to total soil CO 2 efflux from the decomposition of litter and soil organic caron for the last 2 years (27 28) and compared the simulated results with the estimates from field measurements y Sayer et al. (211). By tuning values of two model parameters (µ R and K R ) (see Tale 1), we otained an initial microial iomass caron 24 g C m 2 for oth models, very close to the measured microial iomass caron of 219 g C m 2 y Sayer et al. (27). The simulated initial soil caron is 6715 g C m 2 for model A and 6945 g C m 2 for model B, which is higher than the estimated soil caron of 511 g C m 2 in the top 25 cm (Cavelier et al., 1992) and lower than the estimated soil caron of 9272 g C m 2 in the top 5 cm soil (Grimm, 27). The estimated total soil CO 2 efflux from the control treatment y Sayer et al. (211) was 18 g C m 2 yr 1 from 27 to 28, which was closely simulated y oth models (14 g C m 2 yr 1 y model A and 18 g C m 2 yr 1 y model B). However, oth models overestimated the total soil CO 2 efflux from the litter addition treatment. The estiwww.iogeosciences.net/13/887/216/ Biogeosciences, 13, , 216

8 894 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input Soil CO2 efflux (g C m -2 year -1 ) (a) Control L+ Treatment Soil CO2 efflux (g C m -2 year -1 ) () Control L+ Treatment Figure 5. Simulated response of soil CO 2 efflux in control and litter addition (L+) experiments as descried y Sayer et al. (214) using model A (a) or B (). The dark-grey ar and lack ars represent CO 2 effluxes from litter and soil organic caron decomposition, respectively. The light-grey ar for the litter addition treatment represents the additional CO 2 efflux from soil organic caron decomposition due to additional litter input. mated efflux y Sayer et al. (211) was 138 g C m 2 yr 1, as compared with the simulated flux of 1425 g C m 2 yr 1 y model A and 152 g C m 2 yr 1 y model B (see Fig. 5). The additional CO 2 efflux from the decomposition of soil caron in the litter addition treatment was estimated to e 18 ± 5 g C m 2 year 1 y Sayer et al. (211), which was quite well simulated y model B (15 g C m 2 yr 1 ) (see Fig. 5) ut was underestimated y model A (29 g C m 2 yr 1 ) (see Fig. 5a). The difference in the simulated response of soil organic caron decomposition to increased litter input y the two models can e explained y differences in their sustrate kinetics. The rate of caron loss from the decomposition of soil caron depends on oth soil caron and microial iomass in oth models. Because soil caron is unlikely to change significantly within a few years, the rate of CO 2 emission from soil caron decomposition will largely depend on microial iomass, and that dependence is nonlinear following the reverse Michaelis Menten equation in model A (see Eq. 2), ut is linear in model B (see Eq. 7). Therefore the simulated response of soil organic caron decomposition to increased litter input y model B is more sensitive to microial iomass than model A. 3.4 Response to priming: maximum CO 2 efflux from soil caron decomposition Results from the aove comparison of the responses of two models to the increased litter input are likely dependent on soil temperature, caron input, and model parameter values. To understand the differences in the responses of our two models to litter addition at different rates and soil temperatures for any parameter value, we use the analytic approximations to maximum CO 2 efflux from the priming treatment for each model to identify key differences in their response to priming. Priming is defined as the change in organic caron decomposition rate after the addition of an easily decomposale organic sustance to soil (Kuzyakov et al., 2). In la priming experiments, a given amount of isotopically laelled C sustrate is added to the primed treatment only at the eginning of the experiment (t = ) and no sustrate is added to the control. CO 2 effluxes from soil caron decomposition are estimated from measurements for the following weeks or longer (Cheng et al., 214). The effect of priming, p, is calculated as (R p R c )/R c, where R c and R p are the CO 2 efflux from the decomposition of soil organic caron in the control and primed treatments, respectively. Maximum values of p are usually reported in most priming studies (see Cheng et al., 214). However, analytic approximations to p for oth models are quite cumersome for analysing their differences in the responses to priming. Another way to quantify the priming effect is y measuring the maximum CO 2 efflux from soil organic caron decomposition after caron addition at time t = (Jenkinson et al., 1985; Kuzyakov et al., 2). This quantity can e easily measured in the laoratory or field. In oth models, the equilirium soil microial iomass is proportional to caron input (see Eqs. 11 and 13). In the primed treatment, the amount of caron added at t = usually is well aove the rate of the caron input under natural conditions, and no further caron is added. Therefore the microial iomass will increase until reaching a maximum value, then decreases with time after t =. As shown in Appendix C, the maximum CO 2 efflux from soil caron decomposition in the primed treatment, F max, depends on the maximum microial iomass and microial growth efficiency for oth models, as well as on soil caron turnover rate for model A (see Eq. C12 for F A ) and the microial turnover rate for model B (see Eq. C15 for F B ). Figure 6 shows that F max (or F A for model A, F B for model B) increases with caron input, and decreases with an increase in soil temperature for oth models. However, the sensitivity of F max to caron input at different soil temperatures is different etween the two models. For model A, the sensitivity of F max to caron input is greatest around 25 C, and is quite small at < 5 C. For model B, the sensitivity of F max to caron input decreases with an increase in soil temperature (see Fig. 6). The sensitivity of F max to soil temperatures in oth models can e explained y the analytic approximations (Eq. C12 for model A and C15 for model B). Maximum CO 2 efflux is proportional to soil caron in model A, and to the maximum microial iomass in model B. Both soil caron and maximum microial iomass in oth models decrease with an increase in soil temperature for the parameter values we used (see Fig. 6c); therefore F max also decreases with an increase in soil temperature. Differences in the sensitivity of F max to caron input at different soil temperatures in the two models can also e explained y their respective analytic approximations, par- Biogeosciences, 13, , 216

9 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input 895 Maximum CO 2 efflux Microial iomass ( g C m -2 ) (g C m -2 year -1 ) (a) (c) Litter caron ( g C m -2 ) () (d) Figure 6. Dependence of maximum rate of CO 2 efflux from the decomposition of soil caron in the primed treatment (F max ) as a function of soil temperature and caron addition at time t = for model A (a) and B (). At each soil temperature, the caron input was varied from 1 to 1 g C m 2, and F max increases with an increase in caron input as shown y the arrow in each plot. (c) Variation in equilirium soil microial iomass with soil temperature and caron input at 2 (solid lack line), 6 (long-dashed line), and 1 (short-dashed line) g C m 2 yr 1 for model A. (d) Variation in equilirium litter caron with soil temperature in model B. ticularly the dependence of maximum microial iomass on oth caron input and initial microial iomass in model A (see Eq. C11) and on equilirium litter caron pool size in model B (see Eq. C14), ecause F max depends on the maximum microial iomass in oth models. In model A, F A nonlinearly varies with maximum microial iomass (see Eq. C12), which increases linearly with caron addition at t = ( C l ) and varies nonlinearly with the initial pool size of microial iomass (C ) (see Eq. C11). Because C increases with a decrease in soil temperature or an increase in C l (see Fig. 6c), F A increases with an increase in C l (either directly (Eq. C11) or via the effect on C ), and with a decrease in soil temperature (via the temperature dependence of C ). In model B, the sensitivity of F B to caron input is determined y the maximum microial iomass (C max, B ), which varies with equilirium litter pool size (Cl ) following the regular Michaelis Menten equation (C max, B M l in Eq. C14) for a given amount of caron input ( C l ). The equilirium litter caron pool size increases with soil temperature, and is independent of caron input ased on Eq. (12) (see Fig. 6d). When soil temperature is low, Cl is low, and therefore sensitivity of F B to caron input is high. When soil temperature is high, C l is high and the sensitivity of F B in model B to caron input is low ecause of saturating response in the regular Michaelis Menten equation. 4 Discussion Here we analysed the responses of different caron pools to perturation, soil warming, and increased caron input in two nonlinear microial soil caron models. Tale 2 lists the key differences in those responses. Some of the differences etween the two models also depend on the chosen parameter values for each model. For example, there has een deate aout the temperature sensitivities of microial iomass turnover rate and microial growth efficiency (Frey et al., 213; Hargety et al., 214), and the simulated sensitivity of soil caron to warming (Hagerty et al., 214). Regardless of the temperature sensitivity of microial growth efficiency, model A always simulates a decrease in the equilirium soil caron under warming, whereas model B can simulate an increase or a decrease in the equilirium soil caron under warming, depending on the temperature sensitivities of microial growth efficiency and turnover rate. If microial growth efficiency is sensitive to soil temperature and microial turnover rate is not, as found y Frey et al. (213), the simulated responses of equilirium soil caron to warming y the two nonlinear models are quite similar in the direction of response over temperate and oreal regions, ut different in the tropical regions. This is ecause the minimum soil caron temperature, T x, for model B is aout 25 C for x =.15 K 1 and a =.5, the values used y Allison et al. (21) and German et al. (212) (see Fig. 3a). In that case the equilirium soil caron, as simulated y model B, will decrease over most temperate and oreal regions, for which the mean soil temperature within the rooting zone is elow 25 C for most of the growing season, and will increase in tropical regions, for which the mean soil temperature in the top 1 cm of soil is close to 25 C for most of the year. However, if microial turnover rate is sensitive to soil temperature and microial growth efficiency is not, as found y Hargety et al. (214), then T x is < C at α s >.55 ( C) 1 for model B, causing equilirium soil caron to increase in model B with warming, ut decrease in model A with warming. Therefore, the predicted responses of soil caron to warming y the two nonlinear models differ significantly across all major gloal iomes where mean rooting zone soil temperature over the growing season is aove C. Some of the key differences in the responses of the two nonlinear models can e used to discern which model is more applicale to the real world. For example, the oscillatory response of model A generally is quite small (<1%), which is quite consistent with the results from litter removal experiments (Sayer et al., 27, for example). The relatively large and more persistent oscillation in model B has not een oserved in the field, and the insensitivity of soil caron to Biogeosciences, 13, , 216

10 896 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input Tale 2. Key differences etween the two nonlinear soil microial models. Response to Model A Model B Pool size perturation More frequent and faster oscillations in litter and microial caron pools Soil caron pool may oscillate Less frequent and slower oscillations in litter and microial caron pools Soil caron pool does not oscillate Warming Soil caron pool always decreases Soil caron may increase or decrease Caron input Sensitivity of maximum CO 2 efflux increases with soil temperature Sensitivity of maximum CO 2 efflux decreases with soil temperature a perturation in the litter or soil microial caron pool in model B also needs to e assessed against long-term field experiments such as the DIRT experiment (Nadelhoffer et al., 24). Model B in its present form may not e applicale under field conditions. It has een argued that the influences of microial community structure and their activities on mineral soil caron decomposition at field scale may e much smaller than at the rhizosphere scale (Schimel and Schaeffer, 212), ecause sustrate concentration rather microial activity is the rate-limiting step for the decomposition of soil organic matter in mineral soils. A recent study y Sulman et al. (214) clearly showed the importance of physical protection of microial y-products in forming stale soil organic matter, and its implications for the response of gloal soil caron to caron inputs. This mechanism has een recently incorporated into a nonlinear soil microial caron model (Wieder et al., 214). Whether the large oscillatory responses of model B will e significantly dampened y the addition of such physical protection mechanism is yet to e studied. The two models also have quite different sensitivities to soil warming (see Tale 2), particularly in warm regions. Results from a decade-long soil-warming experiment showed that warming did not reduce soil caron, ecause plant caron production increased as a result of the increased availaility of soil mineral nitrogen in a nitrogen-limited forest (Melillo et al., 22). However, this is quite a different mechanism ecause model B in our study includes neither a nitrogen cycle nor the response of caron input to warming. Overall oth models can simulate the priming response to a change in caron inputs, although model A simulates a weaker response than model B and the sensitivities to caron input at different soil temperature are different etween the two models, particularly under cool climate conditions (see Tale 2). So far, results from litter manipulation experiments in the field have not een analysed for their sensitivity to soil temperature. The differences in the responses of soil caron decomposition to an increased caron input we identified etween the two models can also e used to assess which model is more applicale in the field using experiments with different caron input under cool (mean annual air temperature < 1 C) and warm (mean annual temperature > 2 C) conditions. If the sensitivity of soil caron decomposition to an increased caron input under cool conditions is greater than that under warm conditions, then model B is more appropriate than model A. This has yet to e tested. Our analysis here does not include some other key processes, such as the transformations of different forms of organic caron sustrates y different microial communities as included in some models (see Grant, 214; Riley et al., 214, for example). Therefore the conclusions from this study aout the two nonlinear models should e interpreted with some caution. As shown y Tang and Riley (214), interactions among soil mineral sorption, caron sustrate, and microial processes can generate transient changes in the apparent sensitivity of soil caron decomposition to soil temperature; therefore the static dependence of microial processes on soil temperature as used in our study may not e applicale. Our simplification of the soil microial community and soil caron fractions is necessary for analytic tractaility, ut may also limit the applicaility of our results to field experiments. For example, Allison (212) showed that the apparent kinetics of soil caron decomposition can vary with the spatial scale: the regular Michaelis Menten kinetics at microsites coupled with an explicit representation of different strategies for facilitation and competition among different microial taxa generated litter caron decomposition kinetics similar to the reverse Michaelis Menten equation. Therefore, the identified differences etween the two models should vary with spatial scale. The regular and reverse Michaelis Menten kinetics can e considered as two special cases of a more general kinetics, as discussed y Tang (215). Both models use different mass alance constraints (see Tang, 215), which are unlikely to hold across a wide range of conditions. In the real world, the kinetics and parameter values of caron decomposition likely depend on a numer of other factors, such as soil physical properties, sustrate quality, and soil nutrient availaility (Manzoni and Porporato, 29). Future studies of soil caron decomposition kinetics need to include those factors and the role of root growth dynamics and photosynthetic activities in rhizosphere priming (see Kuzyakov, 22). Finally, oth models have a numer of parameters, and their values are largely ased on laoratory studies (Allison Biogeosciences, 13, , 216

11 Y.-P. Wang et al.: Responses of two nonlinear microial models to warming and increased caron input 897 et al., 21). The values of those parameters may e quite different under field conditions. Evaluation of their applicaility under a wide range of field conditions will require an integrated approach, such as applications of model data fusion using a range of field experiments (Wieder et al., 216). This will eventually lead to a etter understanding of the significance of microial activity on soil caron decomposition and more accurate predictions of caron climate interactions. 5 Conclusions This study analysed the mathematical properties of two nonlinear microial soil caron models and their responses to soil warming and caron input. We found that the model using the reverse Michaelis Menten kinetics (model A) has shorter and more frequent oscillations than the model using regular Michaelis Menten kinetics (model B) in response to a small perturation. The responses of soil caron to warming can e quite different etween the two models. Under gloal warming, model A always simulates a decrease in soil caron, ut model B will likely simulate a decrease in soil caron in temperate and oreal regions, and an increase in soil caron in tropical regions, depending on the sensitivities of microial growth efficiency and microial iomass turnover rate. The response to caron input varies with soil temperature in oth models. The simulated maximum response to priming y model A generally is smaller than that y model B. The maximum rate of CO 2 efflux from SOC decomposition (F max ) to caron input in the primed treatment decreases with an increase in soil temperature in oth models, and the sensitivity of F max to the amount of caron input increases with soil temperature in model A ut decreases monotonically with an increase in soil temperature in model B. Based on those differences etween the two models, we can design laoratory or field experiments to assess which model is more applicale in the real world and, therefore, advance our understanding of the importance of microial processes at regional to gloal scales. Biogeosciences, 13, , 216