Plant and Soil

, Volume 320, Issue 1–2, pp 153–167 | Cite as

Woody plant encroachment impacts on soil carbon and microbial processes: results from a hierarchical Bayesian analysis of soil incubation data

  • Jessica M. Cable
  • Kiona Ogle
  • Anna P. Tyler
  • Mitchell A. Pavao-Zuckerman
  • Travis E. Huxman
Regular Article


Belowground processes and associated plant–microbial interactions play a critical role in how ecosystems respond to environmental change. However, the mechanisms and factors controlling processes such as soil carbon turnover can be difficult to quantify due to methodological or logistical constraints. Soil incubation experiments have the potential to greatly improve our understanding of belowground carbon dynamics, but relating results from laboratory-based incubations to processes measured in the field is challenging. This study has two goals: (1) development of a hierarchical Bayesian (HB) model for analyzing soil incubation data and complementary field data to gain a more mechanistic understanding of soil carbon turnover; (2) application of the approach to soil incubation data collected from a semi-arid riparian grassland experiencing encroachment by nitrogen-fixing shrubs (mesquite). Soil was collected from several depths beneath large-sized shrubs, medium-sized shrubs, grass, and bare ground—the four primary microsite-types found in this ecosystem. We measured respiration rates from substrate-induced incubations, which were accompanied by measurements of soil microbial biomass, soil carbon, and soil nitrogen. Soils under large shrubs had higher respiration rates and support 2.0, 1.9, and 2.6 times greater soil carbon, microbial biomass, and microbial carbon-use efficiency, respectively, compared to soils in grass microsites. The effect of large shrubs on these components is most pronounced near the soil surface where microbial carbon-use efficiency is high because of enhanced litter quality. Grass microsites were very similar to bare ground in many aspects (carbon content, microbial biomass, etc.). Encroachment of mesquite shrubs into this semi-arid grassland may enhance carbon and nutrient cycling and increase the spatial heterogeneity of soil resource pools and fluxes. The HB approach allowed us to synthesize diverse data sources to identify the potential mechanisms of soil carbon and microbial change associated with shrub encroachment.


Decomposition Respiration Soil nitrogen Sonoran desert Prosopis velutina 


Belowground processes are critical drivers of ecosystem dynamics (Ritz et al. 1994; Wardle 2002), and thus, they are important for understanding and predicting whole-ecosystem dynamics. For example, plant roots and soil microorganisms are key players in soil and ecosystem carbon cycling (e.g., Schimel et al. 1994; Williams et al. 1998; Wullschleger et al. 1994). Moreover, microbial decomposition of soil carbon is a major conduit for the loss of CO2 from the soil to the atmosphere (Raich and Potter 1995), potentially contributing to climate change (Houghton et al. 1998; Raich and Schlesinger 1992). In-turn, microbial-mediated soil carbon fluxes are being altered by climate change, either directly via the impacts of elevated temperatures and an altered hydrological cycle (Austin et al. 2004; Kirschbaum 1995; Raich and Schlesinger 1992; Saleska et al. 1999), or indirectly through changes in vegetation structure (Huxman et al. 2004; Raich and Schlesinger 1992; Waldrop and Firestone 2006). Although important, belowground processes can be difficult to directly measure, but unraveling such mechanisms is necessary for building a predictive framework of how belowground carbon cycling will respond to and feedback to global change (Hunt and Wall 2002; Saetre and Stark 2005).

Soil incubations are commonly used to unravel the mechanisms underlying soil carbon dynamics with respect to soil respiration or CO2 efflux (e.g., Dutta et al. 2006; Saetre and Stark 2005; Schuur and Trumbore 2006; Zak and Kling 2006). Generally, soil is collected in the field and is brought to the lab, treatments are applied to the soil samples, samples are “incubated” in controlled conditions, and CO2 efflux is measured. Incubation treatments may include labile carbon (e.g., sugar) substrate additions to determine active microbial biomass (Fliessbach et al. 1994), microbial diversity (Lin and Brookes 1999; Schipper et al. 2001), or variation in microbial substrate use (Hamer and Marschner 2005). Such incubation experiments have been implemented to explore microbial responses to stress or perturbations such as wet–dry cycles (Fierer et al. 2003), altered nutrients (Thirukkumaran and Parkinson 2000), and plant species shifts (Saetre and Stark 2005). Hence, incubation studies are valuable for elucidating key factors affecting soil carbon dynamics.

Integrating information from incubation experiments with complimentary field data to learn about mechanisms operating in a natural setting is challenging. For example, synthesis of incubation data is often limited to relatively simple analyses (e.g., Hook and Burke 2000; Robertson et al. 1997) that do not simultaneously consider all data sources and may not accurately reveal mechanisms necessary for understanding carbon dynamics in the field. On the other hand, the work by Saetre and Stark (2005) is unique in that they analyze substrate-induced incubation data within the context of a relatively detailed mass-balance model that couples carbon and nitrogen transformations. Another class of incubation experiments, which generally do not involve substrate additions, periodically measure CO2 efflux over several weeks or months. Generally, the goals of such experiments are to estimate the size of the initial carbon pool and the decay coefficients, but not necessarily microbial biomass or physiological activity. Functions describing carbon mineralization kinetics (such as a first-order decay model) are often fit to incubation data via non-linear regression routines (Alvarez and Alvarez 2000; Dalias et al. 2001; Grandy and Robertson 2007; Paul et al. 1999). Although these fitting approaches incorporate a semi-mechanistic model, the methods are somewhat unsatisfactory because they do not consider multiple sources of uncertainty (e.g., random and fixed effects) or multiple types of data (subsets of data are often analyzed independent of others).

Thus, although the experimental approaches and laboratory methods associated with soil incubation experiments may be quite involved and produce a wealth of information, data analyses have lagged behind. Towards bridging this gap, we present an approach for analyzing soil incubation data obtained from substrate-induced respiration experiments that is grounded in a hierarchical Bayesian (HB) modeling framework (e.g., Clark 2005; Ogle and Barber 2008). The HB model simultaneously integrates all available and relevant laboratory and field data to yield estimates of microbial biomass, soil carbon availability, and microbial activity, facilitating inference about microbial and substrate controls on soil carbon cycling. Furthermore, the insight provided by this HB data-model integration approach may reduce the need for time-consuming and expensive laboratory analyses associated with detailed incubation studies.

Thus, the goals of this study are two-fold: (1) present a HB approach for analyzing soil incubation data and (2) apply the HB approach to an incubation experiment that explores the effects of woody-plant encroachment on soil carbon dynamics. Woody plant expansion into native grasslands is a near-global phenomenon (Archer et al. 1995; Chapin et al. 1995; Polley et al. 2003), and the resulting shifts in vegetation structure and composition can alter carbon inputs and chemical composition, affecting the decomposability, turnover, and amount of soil carbon (Hibbard et al. 2001; Jackson et al. 2000). The semi-arid riparian system that we are studying in southeastern Arizona is undergoing encroachment by a deeply rooted nitrogen fixing shrub (mesquite, Prosopis velutina) that is causing the accumulation of highly decomposable leaf litter at the soil surface and woody root material at depth. In grasslands, however, relatively decomposable root material may be deposited more uniformly to depth or concentrated near the surface (e.g., Jackson et al. 1996; Titlyanova et al. 1999). The presence of large woody plants (or shrubs) may modify other factors that affect decomposition, including microclimate (via canopy effects on soil temperature and moisture), substrate availability (via litter inputs), microbial biomass and community structure, and the efficiency of microbial carbon consumption (Saetre and Stark 2005). In semi-arid ecosystems, both carbon and water availability limit microbial activity, so vegetation characteristics that modify these resources are expected to strongly control soil carbon decomposition (Austin et al. 2004; Hibbard et al. 2001). Thus, this study aims to understand how encroachment by mesquite may affect soil carbon processes in semi-arid ecosystems.

In order to address the two goals of this study, we combine experimental data and the HB modeling approach to evaluate the following three hypotheses. We hypothesize that increased dominance of nitrogen fixing shrubs in a semi-arid riparian ecosystem will: (1) elevate soil carbon stocks throughout the soil profile due to higher productivity compared to co-occurring grasses, (2) increase microbial biomass near the soil surface due to inputs of high-quality leaf litter compared to deep soil that receives low-quality woody root litter, and (3) enhance microbial substrate-use efficiency due to inputs of more decomposable litter (high nitrogen content) compared to grasses (low nitrogen content) and open/bare spaces (low litter inputs). Evaluating these hypotheses will lend insight into the impacts of woody-plant expansion on soil carbon dynamics in this semi-arid system.

Materials and methods

Site description

This study was conducted at a medium-dense mesquite shrubland (e.g., intermediate between a grassland and closed-canopy woodland in terms of shrub density) located along the San Pedro River, southeast of Tucson, Arizona. The site is characterized by sandy-loam soils that are fairly homogenous to 50 cm (R.L. Scott, personal communication). Mean annual precipitation is 39 cm, of which approximately 60% falls in the summer (July–September) and about 22% in the winter (December–March) (average from 1971 to 2000 National Climate Data Center; NCDC 2008). Mean annual air temperature is 17°C, and the maximum occurs in June (33°C) and the minimum in January (0.6°C) (NCDC 2008). The two dominant plant species are velvet mesquite (Prosopis velutina) and giant sacaton bunchgrass (Sporobolis wrightii). Prior to the summer monsoon, the vegetation cover is about 50% mesquite, 20% sacaton, 20% other shrub species, and 10% open space (Scott et al. 2000). The field portion of this study was conducted in the dry pre-monsoon season, prior to the growth of ephemeral herbaceous plants that fill-in bare space. Seasonal activity and productivity differ between shrubs and grasses primarily because large mesquite access deep soil water and sacaton access near-surface soil water derived from recent rains (Potts et al. 2006; Scott et al. 2000).

Soil sampling and incubation experiment

In May 2006, we collected soil samples associated with four different microsites: large mesquite (>2 m tall), medium mesquite (0.5–2 m tall), sacaton (hereafter, grass), and bare ground. We examined two size classes of mesquite because large shrubs may differentially affect soil carbon compared to small shrubs due to their ability to access groundwater and their higher productivity. Pits were excavated (∼0.5 m deep) near the center of bare spaces and about 0.5 m from the base of large mesquite, medium mesquite, and grass. Bare spaces were devoid of surface litter and pits were excavated at least 1 m away from the nearest plant. Soil (∼200 g) was collected from each pit in the following increments (or depths): 0–2, 2–5, 5–10, 10–15, 15–20, 20–30, 30–40, and 40–50 cm, resulting in 96 soil samples (8 depths × 4 microsites × 3 replicates). Caliche was present at 50 cm, preventing deeper pits. Soils were transported from the field in ice-filled coolers and stored at 4°C until analysis (within 48 h). Each soil sample was sieved (2 mm) to remove roots and rocks. Gravimetric water content (mean ± 1 SD = 1.53 ± 1.01%) and water-holding capacity (WHC, 28 ± 2.96%) were measured on a subset of samples.

Each soil sample was divided into a pair of 50 g sub-samples that were placed in 0.5 pint glass jars. The soil in paired jars was brought to 50% WHC (±2%): one with pure deionized (DI) water and the second with a dextrose–DI water solution (4.2 mg dextrose/g soil, 30 mg dextrose/mL water) to determine substrate-induced respiration (SIR) (West and Sparling 1986). The sealed jars were incubated at a constant temperature of 25°C in the dark. Respiration (CO2 efflux, μmol CO2 g−1 soil s−1) was measured with a closed-loop gas-exchange system that pumped air from the jars through a Li-820 (LiCor Inc., Lincoln, NE) and a flow meter (0.5 L/min). Respiration was measured prior to substrate addition (dry soil) and at 24 and 48 h after water and substrate addition. Jar lids were removed for 120 s to vent the high CO2 concentrations, and respiration was subsequently measured for 60 to 120 s.

Soil not used for incubations was processed for microbial biomass carbon (n = 25) and organic carbon, inorganic carbon, and nitrogen content (n = 91). Microbial biomass was determined on subsamples selected to give a representative coverage of the soil system; subsamples were distributed among the microsites and depths as follows: bare (0–2, 2–5, 5–10, 15–20, 40–50 cm), grass (0–2, 2–5, 5–10, 20–30, 30–40 cm), medium mesquite (0–2, 2–5, 10–15, 40–50 cm), and big mesquite (0–2, 5–10, 15–20, 20–30, 30–40 cm). Field moist soil samples (5 g) were treated via chloroform fumigation–extraction using K2SO4 in a soil:extractant ratio of 1:4 (Vance et al. 1987). Microbial carbon in the extracts was determined with a total carbon analyzer (Shimadzu-5000 Kyoto, Japan) and calculated using an extraction efficiency of 0.38 (Vance et al. 1987). Additional soil samples were ground to a fine powder and inorganic carbon was measured with a modified pressure-calcimeter method (Sherrod et al. 2002); total carbon and nitrogen were measured by dry combustion (NC2100 soil analyzer, CE Elantech, Lakewood, NJ). Organic carbon was calculated as the difference between total and inorganic carbon.

Hierarchical Bayesian analysis of incubation data

Here we describe a hierarchical Bayesian (HB) modeling approach (e.g., Berliner 1996; Clark 2005; Ogle and Barber 2008; Wikle 2003) for analyzing the different types of incubation data that we obtained in this study. The HB method provides a fully consistent statistical framework for analyzing the diverse data within the context of a Michaelis–Menten type process model for microbial respiration. One of the two goals of this study is to describe the general HB approach and apply it to data to address the second goal of this study, which is to elucidate the effects of different microsites on microbial activity and soil respiration. The HB model has three components: (1) the data model that defines the likelihood of the observed data, (2) the probabilistic process model of microbial respiration, and (3) the parameter model that defines the prior distributions for the process model parameters and variance terms. See Table 1 for descriptions of all the symbols used in the following sections.
Table 1

Description of symbols used in the HB model, including units and the type of data or node denoted by each symbol





t, m, d, q, s



Indices for time (t), microsite (m), depth (d), soil pit (q), and treatment addition type (s)



Log(μmol m−2 s−1 layer−1)

Log respiration rates



Log(μmol m−2 s−1 layer−1)

Mean or predicted log-respiration value

τLr, τμLr



Precision terms for Lr and μLr

τC, τB



Precision terms for CLayer and BLayer



g C m−2 layer−1

Observed carbon content



g M m−2 layer−1

Observed microbial biomass carbon



g C m−2 layer−1

Mean or latent amount of carbon



g M m−2 layer−1

Mean or latent amount of carbon




Relative amount of carbon in each soil layer (depth) and microsite



G C m−2

Total amount of carbon in a column of soil in each microsite




Relative amount of microbial biomass in each soil layer and microsite



G M m−2

Total amount of microbial biomass in a column of soil in each microsite

μ × μLr


log(μmol m−2 s−1)

Latent or mean log respiration rate



μmol CO2 g C−1 s−1

Substrate-use efficiency of microbes



μmol CO2 g M−1 s−1

Base-line microbial metabolic activity



% cm−1

Relative amount of microbial biomass per cm of soil



% cm−1

Relative amount of carbon per cm of soil




Limitation index: relative importance of microbial activity vs. substrate availability to respiration



μmol CO2 g soil−1 s−1

Reduced model: respiration rate when substrate is limiting (microbial biomass unlimiting)



μmol CO2 g soil−1 s−1

Reduced model: respiration rate when microbes are limiting (amount of substrate unlimiting)

SD stochastic data, SP stochastic parameter or quantity, LN logical node or described by deterministic function

ag M is grams of microbial biomass carbon (dry weight) and g C is grams of soil carbon

The data model

The data model combines the data likelihoods for observed respiration rates, organic carbon contents, and microbial biomass. First, we work with measured respiration rates (r) that have units of μmol CO2 g−1 soil s−1. Note that r is positive-valued, its variance tended to increase with its mean, and Lr = log(r), the natural logarithm of r, is approximately normally distributed. Thus, for microsite m (four types), soil depth d (eight layers), soil pit q (three reps), substrate-addition type s (water or dextrose), and time period t (24 or 48 h), we assume:
$${\text{Lr}}_{\left\{ {m{\text{,}}d,q,s,t} \right\}} \sim {\text{Normal}}\left( {\mu {\text{Lr}}_{\left\{ {m,d,q,s} \right\}} ,\tau _{{\text{Lr}}} } \right),$$
\(\mu {\text{Lr}}_{\left\{ {m,d,q,s} \right\}} \) is the mean or latent log-flux value and τLr is the precision (1/variance) that describes observation error. Within each substrate-addition type, there were no systematic differences between the 24 and 48 h time periods, so time period is treated as a replicate and used to estimate τLr.
Respiration is the response variable of interest, and organic carbon and microbial biomass are covariates in the Michaelis–Menten model (see Eq. 6). Let \(C_{{\text{Layer}}\left\{ {m,d,q} \right\}} \) and \(B_{{\text{Layer}}\left\{ {m,d,q} \right\}} \) denote the observed amounts of organic carbon (g C·m−2 per layer) and microbial biomass (g dw·m−2 per layer) from the fumigation–extractions, respectively; we assume:
$$C_{{\text{Layer}}\left\{ {m,d,q} \right\}} \sim {\text{Normal}}\left( {\mu C_{{\text{Layer}}\left\{ {m,d} \right\}} ,\tau _C } \right),$$
$$B_{{\text{Layer}}\left\{ {m,d,q} \right\}} \sim {\text{Normal}}\left( {\mu B_{{\text{Layer}}\left\{ {m,d} \right\}} ,\tau _B } \right),$$
\(\mu C_{{\text{Layer}}\left\{ {m,d} \right\}} \) and \(\mu B_{{\text{Layer}}\left\{ {m,d} \right\}} \) are the mean or latent amounts of organic carbon and microbial biomass, respectively, and the precision parameters τC and τB describe variability introduced by soil pit random effects. We further define \(\mu C_{{\text{Layer}}\left\{ {m,d} \right\}} \) and \(\mu B_{{\text{Layer}}\left\{ {m,d} \right\}} \) as follows:
$$\mu C_{{\text{Layer}}\left\{ {m,d} \right\}} = c_{\left\{ {m,d} \right\}} \times C_{\left\{ m \right\}}^ * ,$$
$$\mu B_{{\text{Layer}}\left\{ {m,d} \right\}} = b_{\left\{ {m,d} \right\}} \times B_{\left\{ m \right\}}^ * ,$$
c{m,d} and b{m,d} (both are unitless) are the relative amounts of carbon or biomass in a given microsite and layer such that \(\sum\limits_d {c_{\left\{ {m,d} \right\}} } = \sum\limits_d {b_{\left\{ {m,d} \right\}} } = 1\), and \(C_{\left\{ m \right\}}^ * \) and \(B_{\left\{ m \right\}}^ * \) are the total amounts of carbon (g C/m2) and biomass (g dw/m2) in an entire 0–50 cm column of soil. The depth-dependent distribution of carbon and microbes (c, b) and their total amounts (C*, B*) are quantities that we wish to estimate.

The process model

Latent respiration, microbial biomass, and organic carbon are unobservable quantities that are informed by the data. Latent respiration is described by a process model that includes process uncertainty, whereas we implemented an errors-in-variables-type model (Dellaportas and Stephens 1995) for biomass and carbon (Eqs. 3a and 3b). We define the process model for latent respiration \(\mu {\text{Lr}}_{\left\{ {m,d,r,s} \right\}} \)as:
$$\mu {\text{Lr}}_{\left\{ {m,d,r,s} \right\}} \sim {\text{Normal}}\left( {\mu \times \mu {\text{Lr}}_{\left\{ {m,d,s} \right\}} ,\tau _{\mu {\text{Lr}}} } \right),$$
the mean \(\mu \times \mu {\text{Lr}}_{\left\{ {m,d,s} \right\}} \) is described by a Michaelis–Menten (MM)-type equation, and τμLr is the precision, which describes process error due to, for example, soil pit random effects that cannot be captured by the relatively simple MM model (i.e., Eq. 5). Such random effects could reflect differences in microbial community structure, soil texture, or other soil or microbial properties that were not measured.
We chose a MM-type model partly because this fairly simple model lends insight into key parameters that describe microbial substrate use efficiency and “inherent” microbial activity. When coupled with the incubation data within the HB framework, estimates of these parameters facility inferences about microbial activity. The general form of the MM model for respiration (R) as a function of the amount of carbon substrate (C) and microbial biomass (B) is:
$$R = \frac{{A_C \times A_B \times C \times B}}{{A_C \times C + A_B \times B}} \cdot $$
AC (μmol CO2·g organic C−1·s−1) describes microbial carbon substrate-use efficiency, and AB (μmol CO2·g microbial C−1·s−1) is an index of the inherent microbial activity or metabolism in the absence of competition for carbon substrate. Note that when the substrate is saturating (C→ ∞), Eq. 5 reduces to R = AB·B such that respiration is limited by and proportional to microbial biomass. We assume that sugar addition results in substrate saturation, and applying Eq. 5 gives:
$$ \mu \times \mu {\text{Lr}}_{{{\left\{ {m,d,s} \right\}}}} = \left\{ {\log \begin{array}{*{20}c} {{{\left( {\frac{{A_{{C{\left\{ {m,d} \right\}}}} \times A_{B} \times \mu C_{{{\text{Layer}}{\left\{ {m,d} \right\}}}} \times \mu B_{{{\text{Layer}}{\left\{ {m,d} \right\}}}} }} {{A_{{C{\left\{ {m,d} \right\}}}} \times \mu C_{{{\text{Layer}}{\left\{ {m,d} \right\}}}} + A_{B} \times \mu B_{{{\text{Layer}}{\left\{ {m,d} \right\}}}} }}} \right)}}} & {{{\text{if}}\;s = {\text{water}}}} \\ {{\log {\left( {A_{B} \times \mu B_{{{\text{Layer}}{\left\{ {m,d} \right\}}}} } \right)}}} & {{{\text{if}}\;s = {\text{sugar}}}} \\ \end{array} } \right..$$
Note that \(\mu C_{{\text{Layer}}\left\{ {m,d} \right\}} \) and \(\mu B_{{\text{Layer}}\left\{ {m,d} \right\}} \) are given in Eq. 3, and that Eq. 6 explicitly links the different data sources via their associated latent processes.
One might expect AB to be affected by the composition of the microbial community, but because these data were unavailable and since all samples were collected in close proximity, we assume that AB is independent of microsite or soil depth. We tested this assumption with the model and found no differences in AB across microsite and depth. On the other hand, we expect that AC will depend on the quality and chemical composition of the available substrate, and we assume that AC is related to the nitrogen content (%N) of the organic matter such that:
$$A_{C\left\{ {m,d} \right\}} = \alpha + \beta \times \left( {{\text{N}}_{\left\{ {m,d} \right\}} - {\text{aveN}}} \right),$$
where α is the value of AC when nitrogen content is equal to aveN (the average %N measured in this study; aveN = 0.053%), and β describes the sensitivity of AC to changes in %N.

The parameter model

Ultimately, we want to estimate the unobserved quantities c, b, C*, B*, AB, and AC (i.e., α and β) and the precision terms τLr, τC, τB, and τμLr, and we specify prior distributions for these quantities. (The HB model will also estimate missing data—for example, we were unable to measure microbial biomass for all depth by microsite combinations, and the associated missing data model is given by Eq. 2b.) We assign independent, non-informative (diffuse) normal priors to log(AB), α, β, \(\log \left( {C_{\left\{ m \right\}}^ * } \right)\), and \(\log \left( {B_{\left\{ m \right\}}^ * } \right)\). We assume Dirichlet priors for the depth-dependent distributions of carbon substrate, c{m,.}, and microbial biomass, b{m,.}, where the {m,.} refers to all depths such that c{m,.} and b{m,.} are vectors that vary by microsite. The Dirichlet prior constrains the proportions (i.e., c{m,d} and b{m,d}) to be between 0 and 1 and, within a microsite, the proportions sum to one across all depths. We assume a non-informative Dirichlet prior that is equivalent to a uniform prior in the one-dimensional case. Finally, we assume independent, diffuse gamma priors for the precision parameters (i.e., τLr, τC, τB, and τμLr).

We also estimated several quantities that are deterministic functions of the above parameters. We divided c{m,d} and b{m,d} by soil layer thickness (dL, cm) to obtain the relative amounts of soil carbon (ccm, %/cm) and microbial biomass (bcm, %/cm), allowing for direct comparisons of ccm and bcm between layers and microsites. We also computed the average microbial carbon-use efficiency associate with each microsite as a weighted average of AC{m,d} with weights given by c{m,d}:
$${\text{AveA}}_{C\left\{ m \right\}} = \sum\limits_{d = 1}^{Nd} {c_{\left\{ {m,d} \right\}} } \times A_{C\left\{ {m,d} \right\}} .$$
To explore the relative importance of microbial activity and carbon substrate availability to heterotrophic soil respiration, we calculated a “limitation” index (λ):
$$\lambda _{\left\{ m \right\}} = \left( {\frac{{A_B \times B_{\left\{ m \right\}}^ * }}{{{\text{AveA}}_{C\left\{ m \right\}} \times C_{\left\{ m \right\}}^ * }}} \right)$$
If λ = 1, microbial activity and carbon substrate availability are equally limiting (or controlling) respiration; if λ > 1, substrate is most limiting and if λ < 1, microbial activity is most limiting.

Combining the data model (Eqs. 1, 2a, and 2b), the process model (Eqs. 4, 5, 6, and 7), and the parameter model (or priors) results in joint and marginal posterior distributions for the unknown or latent quantities of interest (e.g., Berliner 1996; Ogle and Barber 2008). The HB model that is specified by the above equations was programmed in WinBUGs (Lunn et al. 2000). We ran two parallel MCMC (Markov chain Monte Carlo) chains, and we used the BGR diagnostic tool to evaluate convergence of the chains to the posterior distribution (Brooks and Gelman 1998; Gelman et al. 2004). Chains converge once the values for parameters “settled” on (or varied around) the “best estimates” (e.g., the posterior means). The simulations yielded an independent sample of 3,000 values for each parameter from the joint posterior distribution.

Re-parameterizing the model to accommodate less data

To determine the sensitivity of this modeling approach to available data, we compared the results given by full model (as described above) with a reduced version that only uses respiration data (i.e., no soil carbon or microbial biomass data). Note that if one were to fit Eq. 5, as modified in Eq. 6, to incubation data that only included respiration rates, then B*, C*, AC, and AB are nonidentifiable (i.e., there is no unique solution for these parameters). Without additional carbon content and microbial biomass data, we can only estimate the relative amount of microbes (b), the relative amount of carbon (c), and indices of microbial activity. The following re-parameterized model can be used in such cases, and we re-write Eq. 5 as:
$$R = \frac{{{\text{ac}} \times {\text{ab}} \times c \times b}}{{{\text{ac}} \times c + {\text{ab}} \times b}}$$
where ac = AC × C* and ab = AB × B* (both have units of μmol·m−2·s−1), and recall that C = C* × c and B = B* × b. Since AC, C*, and M* are expected to vary between microsites, ac and ab also vary between microsites. Note that Eq. 10 describes the water addition data and R = ab × b corresponds to the sugar addition data (substrate saturated). We can estimate ac, ab, b, and c by specifying the complete HB model that includes Eq. 1 (data model), Eq. 4 (process model), Eq. 6 (with process mean modified according to Eq. 10), and the prior models previously defined for c, b, τLr, and τμLr. Additional prior distributions are needed for ac and bc, and we assumed non-informative normal priors for log(ac) and log(bc). Note that the ratio ac/ab is equal to the limitation index (λ, Eq. 9), and thus λ can still be estimated.


Summaries of observed respiration rates associated with the incubation experiment are shown in Fig. 1. Compared to pre-treatment dry soil, sugar water addition increased respiration 23 to 140 times, and pure water addition increased respiration by about 1.2 to 51 times. The relative increase in respiration varied with soil depth such that surface soil was most affected and deeper soil was least affected by the addition treatments. For both addition types, respiration rates were highest from soils near the surface (0–2 and 2–5 cm layers) and lowest for soils from deeper (15–50 cm) layers. Respiration also differed between microsites whereby respiration rates were generally greater for soil collected beneath big and medium mesquite compared to grass and bare microsites (Fig. 1). Likewise, soil nitrogen was highest under big mesquite (mean ± 1 SE = 0.088 ± 0.030%; 0–55 cm), followed by medium mesquite, grass, and bare microsites (0.040 ± 0.004, 0.046 ± 0.004, and 0.033 ± 0.001%, respectively). Soil nitrogen was highest in the surface layer (0.275 ± 0.041%; across all microsites) compared to deeper layers.
Fig. 1

Summary (sample means and standard errors) of observed respiration rates following substrate additions. For both the pure water or sugar water additions, respiration rates were similar when measured at 24 and 48 h, so we show the data pooled across time periods. Respiration rates following (1a) pure water addition (measures potential carbon mineralization) and (2a) sugar water addition (substrate induced respiration). The insets (1b and 2b) display values on the log-transformed scale. Respiration rates are shown for each microsite (bare triangles, grass circles, medium mesquite squares, and big mesquite diamonds) and for eight depths (0–2, 2–5, 5–10, 10–15, 15–20, 20–30, and >40 cm)

The HB model successfully captured the observed patterns in soil respiration, soil organic carbon, and microbial biomass. For example, regressions of observed vs. predicted log respiration (i.e., Lr vs. μ × μLr, Eq. 6), organic carbon (i.e., CLayer vs. μCLayer, Eq. 2a), and microbial biomass (i.e., BLayer vs. μBLayer, Eq. 2b) indicate that the full model successfully fit the observed data (Fig. 2). The full HB model accounted for 94%, 96%, and 61% of the variation in observed (log) soil respiration, microbial biomass, and organic carbon (Fig. 2).
Fig. 2

Evaluation of model goodness-of-fit by comparing log of observed (measured) and model-predicted (a) respiration rates, (b) microbial biomass, and (c) organic carbon. For a, solid circles are respiration rates associated with pure water addition and open circles are rates associated with sugar–water addition. The dotted lines are the 1:1 lines, and solid lines are the least squares regression fits with the following coefficients: a water addition: \({\text{Log}}\left( y \right) = - 0.09 + 0.99 \times {\text{Log}}\left( x \right),{\text{ }}R^2 = 0.94\), sugar–water addition: \({\text{Log}}\left( y \right) = - 0.37 + 0.84 \times {\text{Log}}\left( x \right),{\text{ }}R^2 = 0.96\), b\({\text{Log}}\left( y \right) = - 1.2 + 1.3 \times {\text{Log}}\left( x \right),{\text{ }}R^2 = 0.48\), c\({\text{Log}}\left( y \right) = - 1.6 + 0.68 \times {\text{Log}}\left( x \right),{\text{ }}R^2 = 0.61\)

To evaluate the full versus reduced model, we compare parameters shared by both models (c, b and λ). We consider two quantities significantly different if the 95% credible interval (CI)—the interval defined by the 2.5th and 97.5th percentiles—for each quantity does not contain the other quantity’s posterior mean. For example, the full model and reduced model yielded similar estimates for λ, c, and b as the 95%CIs obtained from the full model contained the associated parameter’s posterior mean for the reduced model (Fig. 3a–c). The full model resulted in slightly more precise estimates for these parameters, which is reflected in the narrower CIs; the mean widths of the 95%CIs for the full vs. reduced models were: 0.14 vs. 0.14 (c), 0.11 vs. 0.15 (b), and 6.1 vs. 6.9 (λ).
Fig. 3

For bare, grass, medium mesquite, and big mesquite microsites, full and reduced model comparisons of a posterior means and 95% credible intervals for λ (grey error bars for the reduced model), and posterior means for the relative amounts of b microbial biomass (b) and c organic carbon (c) in each soil layer or depth (each point is a depth for each microsite). The dotted line in b and c is the 1:1 line

Now we focus on parameters unique to the full model, and posterior results for a subset of parameters are given in Table 2. The posterior distributions of ccm (% C/cm) and bcm (% dw/cm) show that both organic carbon content and microbial biomass were most concentrated near the surface and declined with depth (Fig. 4a,b). The strongest depth-dependent decline occurred under big mesquite; conversely, microbes and carbon were more uniformly distributed with depth in bare soil. Differences in the posterior estimates of total carbon content and total microbial biomass were greatest between non-mesquite microsites (grass, bare; low C* and B*) and mesquite microsites (big and medium shrubs; high C* and B*) (Table 2). In general, soils in bare areas have the lowest C* and B* while soils under big mesquite had the highest C* and B* (Table 2). Despite strong microsite differences in C* and B*, microbial biomass per unit of carbon (i.e., B*/C*) was similar across microsites (Fig. 4c).
Fig. 4

Posterior means and 95% credible intervals (CIs) for the relative density of a microbes (bcm, %/cm) and b carbon substrate (ccm, %/cm) across the eight soil depths within each of the four microsites; and c the ratio of microbes to carbon substrate, i.e., B*/C* (g dw/g C) across the four microsites. CIs that do not contain posterior means for other microsites or depths and/or different letters indicate that means are significantly different

Table 2

Posterior means and 95% credible intervals (2.5th and 97.5th percentiles) and units for a subset of the HB model parameters. Mesq. refers to mesquite



Mean (95%CI)



9.52 (7.66, 11.5)



4.33 (3.09, 5.91)

τC × 105


2.73 (1.81, 38.9)



6.18 (1.87, 16.0)


μmol CO2 g soil−1 s−1

150 (103, 238)


μmol CO2 g C−1 s−1

0.032 (0.025, 0.040)


μmol CO2 g C−1 s−1%N−1

0.712 (0.371, 1.08)


μmol CO2 g C−1 s−1




0.017 (0.014, 0.022)a



0.026 (0.021, 0.031)b

 Med. Mesq.


0.021 (0.017, 0.026)ab

 Big. Mesq.


0.067 (0.044, 0.095)c


g M m−2




2.81 (1.64, 4.07)a



3.37 (1.88, 5.11)a

 Med. Mesq.


6.19 (3.76, 8.70)b

 Big. Mesq.


6.51 (3.72, 9.12)b


g C m−2




2,122 (1,596, 2,650)a



2,561 (2,043, 3,112)a

 Med. Mesq.


3,272 (2,720, 3,850)b

 Big. Mesq.


4,176 (3,552, 4,803)c

Statistically significant differences are denoted by different letters (i.e., posterior means are not contained within another’s 95%CI)

Depth averaged microbial carbon substrate-use efficiency (AveAC) was lowest for bare soil, intermediate for grass and medium mesquite microsites, and highest for big mesquite microsites (Fig. 5a, Table 2). Depth-varying microbial carbon-use efficiency (AC) was positively correlated with soil N content (β > 0, Table 2, Fig. 5b). Over the range of N contents measured in this study, the posterior mean for AC increased from ca. 0.01 (N = 0.02%) to 0.19 μmol·g C−1·s−1 (N = 0.28%) (Fig. 5b). Pronounced depth-dependent variation in soil N under big mesquite led to a strong depth-dependent decline in AC (Fig. 5c). In this system, carbon substrate availability rather than microbial activity appears to be the dominant controller of respiration (λ > 1) (Fig. 6). Substrate availability is most limiting in medium mesquite and bare microsites \(\left( {\lambda \cong 10.6} \right)\) and least limiting under big mesquite \(\left( {\lambda \cong 3.5} \right)\) (Fig. 6).
Fig. 5

Posterior estimates of microbial carbon-use efficiency (AC), where a shows posterior means and 95% credible intervals for substrate-use efficiency averaged across depth for each microsite (i.e., AveAC) (statistical differences denoted by different letters); b shows the predicted relationship between AC and bulk soil nitrogen content (%), plotted for soil N values that span the N contents measured in this study, where the middle line is the posterior mean, the upper and lower curves define the 95%CI, and the symbols (circles) indicate the predicted AC values associated with the measured N values; and c shows the posterior means and 95%CIs for AC by soil depth and microsite

Fig. 6

Posterior means and 95% credible intervals for the limitation index (λ, see Eq. 9), which describes the primary controller of respiration (microbes or carbon). Statistical differences in the means are denoted by different letters


Woody plant encroachment and soil carbon turnover

We used a hierarchical Bayesian (HB) modeling approach to bridge lab and field studies within a unified statistical framework, with the goal of gaining mechanistic insight into soil carbon–microbial interactions. This framework was used with soil incubation data to explore how encroachment of a nitrogen-fixing shrub (mesquite, Prosopis velutina) into a semi-arid riparian grassland may be affecting soil carbon cycling, and we evaluated three hypotheses related to this question. In support of Hypothesis #1, total organic carbon under mesquite (big and medium shrubs) was 1.8 and 1.5 times greater than total carbon under bare and grass microsites, respectively (C*, Table 2). Big mesquite shrubs also resulted in 2.2 times greater accumulation of organic carbon in surface layers (0–2 cm) compared to bare, grass, and medium mesquite (Fig. 4b). In support of Hypothesis #2, microbial biomass was 1.9 and 2.3 times greater under mesquite (big and medium shrubs) compared to grass and bare microsites, respectively (B*, Table 2). Similar to the soil carbon patterns, big mesquite appear to enhance the relative amount of microbes by nearly threefold in surface layers compared to grass and bare microsites (Fig. 4a). In support of Hypothesis #3, microbes associated with big mesquite had approximately threefold greater carbon substrate-use efficiency than the other three microsites (Fig. 5a).

Woody plant encroachment into other arid and semi-arid grasslands has led to changes in soil carbon stocks (Jackson et al. 2002) that are consistent with our observations. While Jackson, Banner, Jobbagy et al. (2002) did not observe a significant or consistent effect of shrub encroachment on the depth distribution of carbon in the top 100 cm, we found that big mesquite facilitate accumulation of carbon and microbial biomass in the near-surface layers (Fig. 4a,b). The large shrubs in this ecosystem access deep water (Scott et al. 2000), resulting in potentially more aboveground production and carbon accumulation in the soil surface compared to the shrubs studied by Jackson et al. (2002). Despite the proliferation of microbes beneath big mesquite, the amount of microbial biomass relative to carbon content (B*/C*) was similar across microsites (Fig. 4c). This suggests that the amount of carbon, regardless of the quality of this carbon, constrains the amount of microbial biomass across microsites; however, the carbon-use efficiency of these microbes does depend on carbon (or substrate) quality.

In our system, microbial respiration was controlled by carbon availability to a lesser extent under big mesquite compared to the other microsites (Fig. 6). This was expected because big mesquite microsites have an extensive litter layer (>7.6 cm deep, data not shown), high surface soil carbon content, and nitrogen-rich litter compared to bare soil, grass microsites, and the lesser-developed medium mesquite microsites. Because Prosopis velutina is a nitrogen-fixer, soil beneath the big shrubs supported microbes with high carbon substrate-use efficiency (Fig. 5a), particularly near the surface (Fig. 5c). This suggests that soil carbon was of higher quality (e.g., higher N) compared to bare, grass, and medium mesquite microsites (low N = low decomposability, Ball 1997; Fierer et al. 2006). In fact, carbon substrate-use efficiency was positively correlated with substrate quality as described by the nitrogen content of the bulk soil (Fig. 5b). Thus, high respiration rates in the soil surface beneath big mesquite are attributed to large, labile carbon stocks and high nitrogen contents that facilitate the relatively rapid decomposition of high quality litter.

Conversely, bare and grass microsites are devoid of surface litter, thereby limiting carbon inputs to surface layers. Low decomposability of soil carbon (low N) in these microsites resulted in low microbial abundance and reduced carbon substrate-use efficiency. Although grass and bare microsites are functionally similar in terms of litter inputs and microbial activity, at the ecosystem level grasses may not be equivalent to bare ground because they support a root system and dense canopies with high production rates (Potts et al. 2006). Medium mesquite were functionally similar to big mesquite in that they had similar amounts of soil microbial biomass and soil carbon (Table 2), but they were functionally similar to grass and bare soil in terms of their microbial carbon-use efficiencies (Fig. 5a). The latter was somewhat surprising since the chemical composition of soil organic matter formed from the litter from both medium and large shrubs was expected to differ from that of grass litter. As shrubs grow larger, however, the soil beneath their canopies changes whereby all of the following tend to increase: soil carbon stocks, microbial biomass, substrate-use efficiency, and soil respiration. However, it appears that microbial biomass responds most rapidly to shrub encroachment, followed by soil carbon, then microbial substrate-use efficiency. Thus, our results suggest that there appears to be a lag in soil carbon processes associated with the conversion of grassland to mesquite shrubland.

The hierarchical Bayesian modeling approach

The HB modeling approach that we employed allowed us to synthesize experimental and observational data related to soil carbon–microbe interactions. Although we applied this approach to short-term, substrate-induced incubation data, it could be applied to long-term incubation studies for understanding mineralization kinetics (Alvarez and Alvarez 2000; Dalias et al. 2001; Grandy and Robertson 2007; Paul et al. 1999); more generally, it can be used to synthesize data from microcosm studies. Although we worked with a specific dataset and particular process model, the HB approach is highly flexible and can accommodate different types of data, experimental designs, sampling protocols, and process models (Ogle 2008). Traditional methods for analyzing incubation data generally do not use the data to their fullest potential and tend to misrepresent uncertainty, which will impact subsequent inferences. Further, multiple datasets are often analyzed in a piece-wise fashion rather than in a single analysis that explicitly accounts for multiple sources of uncertainty. The HB approach presented here overcomes these issues by employing a probabilistic framework that links semi-mechanistic process models with diverse sources of data that inform processes of interest (e.g., in this study, microbial decomposition of soil carbon).

Although our study produced a fairly rich dataset, other studies may be limited in the types of data available. For example, measurements of microbial biomass can be time consuming and expensive, prohibiting the collection of such data. We note, however, that the reduced model (e.g., Eq. 10 and associated text) is appropriate for incubation experiments that only measure respiration rates. Compared to the full model, the reduced model produced similar estimates of the relative amounts of microbial biomass and organic carbon (b and c), soil respiration, and the importance of microbial activity vs. substrate availability to soil respiration (λ). This suggests that carbon and microbial biomass are not critical to measure when the goal is to estimate b, c, and λ. However, if the goal is to estimate microbial metabolic parameters (AC and AB) and/or the total amount of carbon or microbes (C* and B*), then organic carbon and microbial biomass data must be collected and used with the full model described herein.

The full model was very successful at predicting respiration, but it was comparatively less successful at predicting microbial biomass and organic carbon (Fig. 2b,c). There are at least two potential explanations for this result. First, latent respiration rates were mostly informed by two data sources (water and sugar–water incubation flux data), but estimated total carbon and total biomass were primarily informed by one source each (measured organic carbon and microbial biomass carbon, respectively). Further, the biomass dataset was small compared to the carbon and respiration data, and data for some depths were missing across all microsites. Second, the respiration data and Michaelis–Menten model inform us about relatively fast processes related to highly mineralizable soil carbon and metabolically active micro-organisms. Conversely, the soil carbon data integrate over disparate temporal and spatial scales associated with, for example, labile and recalcitrant carbon pools, and the biomass data may not accurately describe micro-organisms that are active at the time of incubation. However, the data-sensitivity analysis that compares the reduced and full models suggests that including soil carbon data improves parameter estimates and reduces uncertainty in predicted soil carbon. In summary, the HB approach can easily accommodate different types and amounts of data that can be analyzed within the context of a process model, facilitating inferences about key parameters and processes related to, for example, soil carbon cycling.



We thank Dr. David Williams and Dr. Russell Scott for the access to field sites and intellectual contributions; Greg Barron-Gafford, Ben Collins, Kevin “the Red” Gilliam, and Amelia Hazard for the field assistance; and Mary Kay Amistadi and Jon Chorover, School of Natural Resources, University of Arizona for the TOC analysis of microbial biomass samples. We acknowledge funding from SAHRA (Sustainability of Semi-Arid Hydrology and Riparian Areas) under the STC program of NSF, and NSF awards to TEH, Jake F. Weltzin, and David G. Williams. The experiments herein comply with the current laws of the USA. The statistical analysis was partly supported by a DOE NICCR grant (K.O., T.H.).


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Jessica M. Cable
    • 1
  • Kiona Ogle
    • 1
    • 2
  • Anna P. Tyler
    • 3
  • Mitchell A. Pavao-Zuckerman
    • 3
  • Travis E. Huxman
    • 3
    • 4
  1. 1.Department of BotanyUniversity of WyomingLaramieUSA
  2. 2.Department of StatisticsUniversity of WyomingLaramieUSA
  3. 3.Department of Ecology and Evolutionary BiologyUniversity of ArizonaTucsonUSA
  4. 4.Biosphere 2, B2 EarthscienceUniversity of ArizonaTucsonUSA

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