Model formulation of microbial CO 2 production and efficiency can significantly influence short and long term soil C projections

Transcription

1 The ISME Journal PERSPECTIVE Model formulation of microial CO 2 production and efficiency can significantly influence short and long term soil C projections Ford allantyne IV Sharon A. illings 2 Receied: 2 August 207 / Reised: 22 Decemer 207 / Accepted: 3 Decemer 207 International Society for Microial Ecology ();,: Models of soil C dynamics continue to e rapidly deeloped in a effort to improe projections of terrestrial C fluxes under a range of climate scenarios. Recently deeloped models of soil organic C (SOC) dynamics that incorporate microes as the agents of decomposition [ 5] capture a greater numer of releant processes than their predecessors, ut it remains unclear whether the mathematical formulations and parameterization of such models are accurate [6 9]. Different model structures, parameterizations, and functions descriing responses to changing enironmental conditions exert great influence on projections of SOC stocks and CO 2 flux [, 4, 9]. Discrepancies etween projections of SOC stocks from microial models and current Earth system model (ESM) soil C modules, which can span two orders of magnitude, are highly sensitie to assumptions regarding microial respiration and efficiency [, 4, 6]. Although physical structure, chemical composition, and edaphic conditions ultimately goern the accessiility of decomposale SOC [0], physiology dictates the fate of C once it is lierated and taken up from SOC y microes, regardless of its original composition. A fundamental component of microial physiology, heterotrophic respiration, which is the conduit for C flow from SOC to the atmosphere and one of the largest potential positie feedacks to climate warming, is often incorporated in models of SOC dynamics heuristically in a term for efficiency. Alternatiely, microial respiration can e formulated more mechanistically to reflect the fact that heterotrophic * Ford allantyne IV f4@uga.edu 2 Odum School of Ecology, Uniersity of Georgia, 40 E. Green St., Athens, GA 30602, USA Department of Ecology and Eolutionary iology, Kansas iological Surey, Uniersity of Kansas, 20 Constant Ae., Lawrence, KS 66047, USA respiration in soils is largely mass specific when microes are actiely decomposing SOC []. ecause heterotrophic microial respiration influences how much of the C lierated from SOC is retained in the soil and how much is lost to the atmosphere, it is important to understand oth the short and long term dynamical consequences of formulating respiratory losses differently in models used to project future climate. Despite all the recent effort to deelop etter models, to date noody has determined if the widespread, heuristic formulation of respiratory C losses gies rise to different dynamics than the more physiologically grounded formulations of microial respiration, oer a range of time scales. Here, we address this question for the first time. Typically in terrestrial modules of ESMs, e.g., the community land model (CLM), respiratory C loss from the soil is phenomenologically defined as minus the fraction of C transferred etween C pools []. The fraction lost in these transformations is attriuted primarily to heterotrophic respiration of soil microes. Recently proposed microial modules that are candidates for replacing current CLM elowground dynamics [, 3 5, 2] incorporate respiratory C losses ia a term for efficiency, commonly referred to as C use efficiency (), that is a scalar coefficient of the uptake component of C flux into the microial iomass pool, C : d C ¼ C ½ UðS C Þ Š ðþ dt Hereafter, we refer to Eq. as the formulation. In the formulation, prescries losses from the microial iomass C pool that include allocation to enzyme production, C excretion, death or any additional generic proportional losses, and U(S C ) is the function defining C sustrate (S C ) uptake, which is usually assumed to e of Michaelis-Menten form (i.e., sustrate uptake rate is a saturating function of sustrate aailaility). The formulation is conceptually rooted in the idea of microial efficiency, which quantifies the partitioning of C flow through microial iomass. Microial efficiency is

2 F. allantyne IV, S. A. illings defined as the ratio of production or anaolism, P, to the P sum of production and respiration (cataolism), R, PþR [3], and quantifies the fraction of C lierated from SOC lost as CO 2 from the soil and the fraction remaining in microial iomass. As an integratie metric, microial efficiency reflects the context for the partitioning of C flow through microes and is typically used to characterize growth conditional on enironmental conditions. Consequently, microial efficiency should not e considered an inherent physiological trait, ut rather a deriatie of microial growth and respiratory demand. Although the fundamental P definition of efficiency, PþR, is straightforward and intuitie, making the measurements to quantify efficiency presents a significant challenge ecause oth P and R must e simultaneously quantified [4] with a high degree of accuracy oer a range of releant conditions. Furthermore, P and R can ary independently in response to changing edaphic conditions, changes in sustrate aailaility and composition, and microial life history. Modelers hae circumnaigated the dearth of independent empirical oserations of P and R y treating P PþR as a single parameter arying from 0 to,, that indirectly prescries respiratory C loss from heterotrophic soil microes. Incorporating respiration in this manner, with multiplying the SOC sustrate uptake function, is conenient ut can result in some logical inconsistencies. For example, respiratory losses are directly influenced y C aailaility in the enironment, not only y cellular demand, and respiratory losses remain a constant fraction of sustrate uptake. Consequently, maintenance respiration is a function of uptake rate and anishes if uptake ceases. Incorporating all respiratory losses as in Eq. means that microial iomass can persist indefinitely without respiratory costs as long as C uptake is zero, which is unrealistic []. Thus, prescriing that all respiratory C losses result from the product of and uptake is inconsistent with constitutie or aseline respiration. Furthermore, though it is oious that C enters microial iomass prior to eing respired, C lost as CO 2 in the formulation is instantaneously respired as it is lierated from SOC, and is neer depicted as haing entered microial iomass. From asic physiology, we know that C must e taken into a cell efore it is respired. Mathematically representing that respiratory C loss originates from an intracellular C pool necessitates the inclusion of a mass specific respiration rate (; for simplicity it encompasses growth and maintenance respiration). This differs from the formulation in that respiratory C losses only depend on microial iomass, ut C is lost ia the same additional pathways as in the formulation (e.g., allocation to enzyme production, C excretion prescried y in Eq., and death). Aggregating mass specific growth and maintenance respiration into a single term results in no loss of generality ecause of linearity. Such a term for maintenance respiration was recently incorporated y []. Dynamics with respiratory C losses prescried in a mass specific manner assume the following generic form: d C ¼ C ½UðS C Þ Š; ð2þ dt which we refer to as the formulation. The components of efficiency for the formulation are defined as: P = C U(S C ), and R = C ( ) U(S C ), whereas for the formulation: P = C (U(S C ) ), and R = C. Thus, the aggregate metric of microial efficiency reduces to and UðS C Þ for the and formulations, respectiely. A key difference etween the two formulations of respiratory C loss is that, for a particular set of enironmental conditions, is assumed constant for the formulation and is assumed constant for the formulation. Consequently, for a gien set of enironmental conditions, the formulation dictates that efficiency is constant irrespectie of sustrate aailaility. In contrast, the formulation portrays microial efficiency as a function of oth and sustrate aailaility, i.e., a dynamic aggregate resulting from the interaction etween microial physiology and enironmental conditions. Consequently, ariation and coariation in and in sustrate aailaility, which has een recently demonstrated in controlled experimental conditions [5], can independently influence efficiency in the formulation. The two formulations of respiratory losses can e seen as opposite ends of a continuum, with all respiratory losses proportional to uptake for a pure formulation and proportional to microial iomass C for a pure formulation. We focus on the opposite ends of the continuum to elucidate the most pronounced dynamical consequences of different mathematical formulations of microial respiratory C loss. Gross C fluxes can e made equialent for the and formulations for any leel of SOC, ut it is unclear whether the alues of and that result in equialence are mutually consistent. Efficiency can also e equialent for oth formulations if ¼ UðS C Þ, ut such equialence necessitates that the ratio of UðS C Þ is constant for all S C. This underscores the fact that the formulation treats efficiency as a pre-specified characteristic of microes, rather than an emergent feature of microial interactions with the enironment, as in the formulation. Explicitly including in models may e more physiologically realistic, ut it comes with a cost: measuring an additional parameter. The formulation is appealing ecause is an aggregate measure of efficiency, oiating the need to measure, which is

3 Model formulation of microial CO 2 production and efficiency can significantly... Fig. Discrepancy in steadystate SOC pool sizes etween the and formulations for four alues of /. Along the thick lack line, ¼ =þ ^S C and thus ¼. The contours depict the deiation in the steady state SOC pool sizes. The shaded region corresponds to the range of and / alues from the literature. aries from h [2] to h [], and aries from to h [5, 7] =^S C = = = 64 = difficult in practice [4], and ecause it heuristically captures partitioning etween anaolism and cataolism. The question of whether the more conenient, heuristic and the more physiologically-ased formulations of respiratory C losses exhiit similar short term and long term dynamics has not een addressed, and needs to e answered efore we continue moing forward with -ased microial dynamics in soil iogeochemical modules of ESMs. Answering this question will tell us whether or not the more conenient formulation faithfully portrays dynamics of a more physiologically realistic formulations of respiratory C losses from soil microes. Our goal here is to delineate the scenarios for which the and formulations of respiratory C loss result in significantly different soil C dynamics and when they agree. The different means of parsing C lierated from SOC into iomass and respired CO 2 represented y Eqs. and 2 can result in significantly different long term microial iomass C and SOC pool sizes, and in significantly different dynamical responses of microial iomass C, and associated soil respiration. To see this, we must explicitly couple SOC sustrate (S C ) dynamics to microial iomass C dynamics. In the most astract sense, the SOC decomposition process can e formalized with two general equations, one for SOC and one for microial iomass C: d C dt ds C dt ¼ I C UðS C Þ S C λ S þ C ε >< C ½ UðS C Þ Š; formulation ¼ or >: C ½UðS C Þ Š; formulation ð3þ The SOC pool increases as a function of external inputs, litter, (I) and C lost from microial iomass ( C ) and decreases as a result of microial uptake (U) and leaching (note we consider microial iomass distinct from SOC throughout). Microial uptake, U(S C ), microial respiration rate,, and other losses,, are all mass specific rates, and ε is the fraction of C lost from microes recycled ack to the sustrate pool. We only consider a single SOC pool to highlight dynamical differences etween different formulations of respiratory C loss. Such differences will e independent of sustrate particulars such as quality or recalcitrance ecause our focus is on the fate of C fate after it is taken up. For the comparison of interest here, we do not need to address arying propensities of acquisition for

4 F. allantyne IV, S. A. illings S = 0.0 S = 0. S = 0.5 I = 0. I = I = 0 Fig. 2 Regions of parameter space consistent with iomass C to SOC ratios etween 0 and The λ S / ratio reflects the relatie loss rates from the SOC pool to non-respiratory loss rate from the microial iomass pool, and the x-axis reflects the relatie rates of maximum SOC uptake and non-repiratory losses. Aoe the enelopes for and elow the enelopes for ; ^ C =^S C > As the colors darken across enelopes, the relatie magnitude of input rate to the SOC pool I increases relatie to non-respiratory loss rate from microial iomass different sustrates. Differences in SOC composition will certainly influences rates of acquisition and microial growth rate, ut ecause our focus is on how C that is already lierated is lost as CO 2, SOC chemistry will not influence the dynamical consequences of different mathematical formulations of microial respiration. Although a soil enzyme C pool could hae een explicitly included, we omitted enzymes for clarity. Howeer, ε effectiely prescries that some C lost ia increases the size of the sustrate pool in a functionally similar manner to soil enzyme actiity. The explicit inclusion of soil enzymes would shift C from either or oth the SOC and iomass pools into an enzyme pool, ut ecause soil enzyme C is usually assumed to e a small fraction of total soil C, the effect would e minimal. The same would hold for additional intermediate C pools, e.g., DOC []. The different formulations of respiratory C loss can result in three types of discrepancies: in equilirium pool sizes for the same C pool, in the relatie size of oth C pools (iomass C and SOC), and in the response to perturation, i.e., transient dynamics. For the first two, we examine equilirium expressions for sustrate C and iomass C: U λ >< ; formulation ^S C ¼ or >: U ð þ Þ; formulation I U λ ð Þλ S >< ð ^ C ¼ εþ ; formulation or >: I U ðþ Þλ S þð εþ ; formulation ð4þ It is clear from Eq. (4) that discrepancies etween equilirium pool sizes will occur if λ is sustantially different from +. The standard assumption is that the uptake S function U(S C ) is of Michaelis Menten form, C ðkþs C Þ with equaling maximum sustrate uptake rate and K the halfsaturation constant. Explicit and straightforward steadystate expressions for S C in terms of the Michaelis-Menten

5 Model formulation of microial CO 2 production and efficiency can... uptake kinetics can easily e otained and are: K ; formulation λ >< ^S C ¼ or K ; formulation >: þ ð5þ Steady state pool sizes (^S C in Eqs. 4 and 5) and microial efficiency will e equal for oth formulations if ¼, ut if the steady state SOC pools λ þ λ þ for the two formulations will differ in size. Using the empirically osered and often assumed ariation in for soil microes [, 2, 4, 6], and recently quantified ariation in for a common soil microe [5, 7], the potential for a discrepancy in steady-state SOC pool size etween the two formulations can e assessed. In Fig., the suset of parameter space corresponding to the assumed and empirically quantified ranges of and, shown as a shaded ox, is superimposed on contour plots in which the contours denote the ratios of steady state SOC pool sizes for the formulation relatie to the formulation. The lowest assumed alue for found in the literature, h [], results in a discrepancy in SOC pool sizes of up to three orders of magnitude etween the two formulations (the right edge of the shaded ox in Fig. ), whereas the highest assumed alue, 0.05 d [2], results in a less pronounced discrepancy (the left edge of the shaded ox in Fig. ) when comined with the lower portion of the empirical range for, h [5, 7]. Smaller SOC pools for the formulation relatie to the formulation are possile (lower left corner of each panel in Fig., which corresponds to ery low and low relatie to λ ), ut the parameter space for this situation is restricted and is not supported y empirical eidence []. Thus, projections of microial iomass C are likely to e significantly larger and projections of SOC are likely to e smaller using the formulation with alues in the current literature. The two formulations also differ in how the steady state ratio of iomass C to SOC (^ C =^S C ) aries as function of microial physiology (Fig. 2). For ratios of / aoe 50, lower alues of ( 0.2) than typically assumed, inferred from, or measured y experiments [, 2, 5 7] are required for ^ C =^S C to e less than 0.05, the empirically osered upper limit [9]. This is especially true in enironments characterized y large inputs to the SOC pool relatie to non-respiratory losses from microial iomass (darker enelopes in Fig. 2 corresponding to large alues of I/ ). Howeer, if / is smaller in magnitude, assumed alues are consistent with ^ C =^S C 0:05. In contrast, for large ratios of maximum uptake rate to non-respiratory loss rate (high alues of / ), the formulation is more consistent with low ratios of microial iomass C to SOC. The / ratio will likely e large in soils though, ecause realized uptake that depends on sustrate aailaility must compensate for oth respiratory and non-respiratory C losses from microes. The consistency across columns in Fig. 2 reflects the fact that as one component of microial physiology changes, e.g., /, another, efficiency in this case, must simultaneously change to constrain microial iomass within reasonale limits. Although incorporating an additional SOC pool would expand the parameter range consistent with ^ C =^S C 0:05 for either model (sensu [4]), the qualitatie differences etween respiratory formulations, namely that the formulation tends to generate greater microial iomass C relatie to SOC than the formulation, will persist. The discrepancies in steady state pool sizes for the and formulations are primarily the result of specific choices of parameter alues. If the assumed alues or the assumed ranges of alues for,, non-respiratory loss rate ( ), and maximum sustrate uptake rate () are incorrect, then discrepancies may e less distinct. Although the lowest assumed alue for in the soil literature is 0.3, aquatic microes [3] and litterag decomposition rates [2] are often characterized y 0.2. Assuming a alue 0.2 for would indeed decrease the discrepancy in steady state soil C pool sizes, ecause lower alues of yield results more consistent with the formulation and empirically osered ratios of ^ C =^S C. This fact is interesting ecause [4] hae argued that alues reported in the soil literature are likely to oerestimate true efficiency. Additionally, ecause the ratio of iomass C to SOC considered here and in recent microial models [ 5, 2] is est thought of as the ratio of microial iomass C to SOC accessile for microial reakdown and assimilation, the upper limit of 0.05 that is accurate for total SOC may e an underestimate when considering the aailale SOC pool [9]. If 0.2 and the upper limit for the ratio of microial iomass C to accessile SOC is higher than 0.05, the model will produce steady state results more in line with empirical oserations. Howeer, this will not affect the comparison of the and formulations if oth incorporate a single SOC pool. Transient dynamics, i.e., responses to perturations or arying enironmental conditions such as temperature, also differ etween the two formulations of respiratory C loss. For a dynamical system, e.g., Eq. 3 here, transient dynamics are typically characterized y linear staility analysis, which inoles computing the eigenalues of the the linearized system. The signs of the eigenalues prescrie qualitatie staility and the magnitudes of the eigenalues prescrie the rate at which the system returns to steady-state. oth the and systems hae eigenalues with negatie real parts and are thus stale in the dynamical sense

6 F. allantyne IV, S. A. illings Fig. 3 Ratio of the largest eigenalue for the system to the largest eigenalue for the system. The region of microe existence differs ased on alues of I K, /, λ S /, and ϵ. For this particular figure, I K =, λ S / = 0., and ϵ = 0.2. The shaded rectangle corresponds to the same parameter ranges as in Fig.. The thick lack line corresponds to the ratio of eigenalues for the case that steady state pools are equal for oth formulations = = = 64 = (see Appendix for details). As with the discrepancy etween steady-state SOC pool sizes, the difference in the eigenalues for the two formulations depends on the relationship etween and. As aoe, if ¼ λ þ λ þ, pool sizes will e identical for oth formulations, ut the formulation will almost always hae a larger magnitude dominant eigenalue, and as a result will e more responsie to perturation than the formulation (see Appendix). Thus, een if steady state pools sizes are identical for oth formulations, transient dynamics, i.e., the responses to punctuated or continued perturations, will differ etween the two formulations of respiratory C loss. In Fig. 3, the suset of parameter space corresponding to the assumed and empirically quantified ranges of and, shown y a shaded ox as in Fig., is superimposed on contour plots in which the contours denote ratios of dominant eigenalues for the formulation relatie to the formulation. For the same ranges of and delineated in Fig., the dominant eigenalue of the formulation is approximately an order of magnitude larger than the dominant eigenalue for the formulation. In general, the formulation is characterized y larger magnitude negatie eigenalues (the shaded ox in Fig. 3 oerlaps with ratios sustantially smaller than ), meaning that formulations will oth return to steadystate more quickly after a perturation (a faster approach to steady state in Fig. 4a, c) and will track fluctuations in the enironment more closely (smaller magnitude oscillations in Fig. 4, d) gien the parameter alues widely used in the literaure. In contrast, the formulation results in a slower and less damped approach to steady state (red lines s. lue lines in Fig. 4a, c) ecause of relatiely smaller magnitude negatie eigenalues, and less of a tendency to respond to high frequency enironmental ariation (red lines s. lue lines in Fig. 4), depicted here for illustratie purposes in response to ariation in temperature (see Appendix for details). For larger alues of, the sensitiity of the formulation decreases (the magnitude of the real part of the dominant eigenalue decreases in asolute alue) due to increased respiratory loss, and can lead to much longer transient dynamics (compare the red lines in Fig. 4a, to the red lines in to Fig. 4c, d). Howeer, for smaller alues of, the formulation can more

7 Model formulation of microial CO 2 production and efficiency can... Fig. 4 Simulated SOC and microial iomass C dynamics using dimensionless ersions of equations ( formulation) and 2 ( formuiation) presented in the Appendix. Simulations in panels a and c start at the same aritrary initial conditions, and simulations in panels and d start at the same aritrary initial conditions. Inputs to the SOC pool, uptake, and loss are identical for all I panels, K ¼, ¼ ¼ 6, λ S ¼ λs ¼ 0:, which are the same alues for Fig. 3. In panels a and, = 0.2 and = 0, and in panels c and d, = 0.5 and = 4. In panels and d, enironmental ariation is implicitly incorporated y arying T sinusoidally oer time, assuming Arrhenius dependence of on T, with amplitude and frequency identical for oth formulations. The units for time and pool size are dimensionless. See Methods for details Pool Size (dimensionless) Pool Size (dimensionless) a c S S d Time (dimensionless) Time (dimensionless) closely track high frequency enironmental ariation (red lines in Fig. 4d) and will exhiit transient ehaior similar to that of the formulation. In general, prescriing respiratory losses solely as mass specific slows down the flow of C from SOC into the atmosphere relatie to a formulation ecause C must enter the iomass C pool and leae as CO 2 after a period time dictated y the alue of. Oscillatory ehaior is known to e a feature of nonlinear models of SOC dynamics [20], ut whether long period oscillations exist in nature is unknown. Perhaps more importantly though, the difference in responsieness can result in different steady-state pool sizes for the formulation in the presence of enironmental ariation (the difference etween in the red dashed lines in panels a and and etween panels c and d in Fig. 4). As a consequence, the formulation will therefore not generate the potentially significant changes in SOC pool size in response to enironmental ariation that result from the formulation. The way that microial respiration is formulated in models of SOC dynamics influences projections of terrestrial C fluxes. Different formulations of microial respiration can generate discrepancies in steady-state SOC and microial iomass C pool sizes of more than two orders of magnitude, depending on assumed relatie rates of respiratory and non-respiratory losses for microes reported in the literature. Contrasting formulations of microial respiration can also drie diergent transient responses to enironmental perturations, with formulations exhiiting a reduced propensity for transient SOC and microial iomass C dynamics than formulations ased on mass specific respiratory losses. Thus, the distinction etween efficiency ased respiratory losses and mass specific respiratory losses is not merely syntactic or semantic. Howeer, ecause there is significant uncertainty regarding parameters that influence discrepancies etween formulations (e.g., and ), it is currently impossile to quantify the exact magnitude of these discrepancies in oth short and long-term dynamics. Regardless, the results presented here highlight the critical dependence of modeled SOC stock dynamics on respiratory and non-respiratory C losses from microes, on the way these fluxes are represented mathematically, and the need to more accurately quantify oth fluxes. As a consequence, the growing community of soil modelers must reassess how microial soil respiration is formally incorporated in models, and the parameter alues that prescrie microial efficiency. Acknowledgements We dedicate this paper to Dr. Henry Gholz, who was an inspiration and a supporter of our work on soil C dynamics. We

8 F. allantyne IV, S. A. illings wish to thank Chao Song and Stefano Manzoni for conersations and comments that improed the manuscript. We also thank o Sinsaaugh, Josh Schimel, Will Wieder, and an anonymous reiewer for comments on preious ersions of the manuscript. This work was supported y a grant from the US National Science Foundation (DE ). Compliance with ethical standards Conflict of interest The authors declare that they hae no conflict of interest. Mathematical appendix To simplify analysis, we non-dimensionalized Eq. 3 to arrie at ds C S C dτ ¼ I C ðþs C Þ S C λ S þ C ε h i C S C >< ðþs d C dτ ¼ C Þ ; formulation or h i >: S C C ðþs C Þ ; formulation in which S C ¼ S C K, C ¼ C K, τ ¼ t, I ¼ I K, ¼ λ S ¼ λ S, and ¼. The equiliria are gien y ð Þ ; formulation >< ^S C ¼ or >: ; formulation I λ S ð εþ ; formulation >< ^ C ¼ or I λ S >: þ þ ε ; formulation In the asence of life,, ð6þ ^S C ¼ I λ ¼ I ; ð7þ λ S S and for iota to persist, > and λ S < I ( ) for the formulation, and > + and λ S <I þ for the formulation. The ratio of equilirium SOC for the formulation to equilirium SOC for the formulation is gien y ^S C ¼ ð : Þ ^S C þ ðþ The ratio will equal if ¼, denoted y the þ old line in the figure panels. To address the discrepancy in the response to perturation, we analyze the characteristic equation resulting from the determinants of the Jacoians for the two formulations. The Jacoian ealuated at the equilirium for oth systems assumes the general form 2 ð ^J X ¼ λ ^S Þ 2 C S þ ϵ þ^s C 7 ^ C X 5 ð9þ 0 ð Þ 2 ^ C þ^s C þ^s C in which X = or and ^S ¼ þ^s or + for the and formulations, respectiely. The eigenalues for the associated characteristic equation are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ λ S Þ ± ða þ λ S Þ2 þ 4AXðϵ Þ ð0þ 2 in which A ¼ ^ C for oth formulations and ¼ ðþ^s CÞ 2 or + for the and formulations, respectiely. In the asence of life, we simply recoer λ S as the only eigenalue. Thus, the addition of liing microes to the soil system increases the asolute magnitude of the dominant eigenalue, therey decreasing return time to equilirium and increasing system staility. The real parts of all eigenalues for oth formulations are negatie, reflecting the staility of the steady-state. To most directly compare staility etween formulations, we assume that the equiliria are the same for oth, i.e ¼ þ. The discrepancy in staility etween the two formulations is most easily seen for the case of no recycling, ε = 0, ecause the only difference in the eigenalues arises in the AX term. For the formulation, AX simply reduces to A ecause X = and ¼, which also means that the response to perturation is independent of in the asence of recycling. For the formulation, AX = A ( + ) ecause X = and = +. A is the same y assumption, which means that in the case of no recycling the return to identical equilirium will e faster for the formulation. In general, increased rates of recycling, ε > 0, promote a faster return to equilirium and shorter duration transient dynamics. >, which is sustantiated in other work [], is a sufficient condition to guarantee the same damped response to perturation and longer transient dynamics of the formulation as for the case of no recycling. The influence of on staility is more seerely constrained ecause oth ε and are ounded y 0 and, whereas can, and will often e significantly greater than. Equations 3 were simulated with I =, = 6, = [0.2,0.5], = [4,0], and λ S = 0. to generate temporal trajectories of S C and C. To simulate the response to temperature ariation, we made the standard assumption that exhiits an Arrhenius temperature dependence [],

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