Ecohydrological feedbacks in peatland development: a theoretical modelling study

Authors


Correspondence author. E-mails: pmorris@mcmaster.ca;paul.john.morris@gmail.com

Summary

1. Peatlands are complex ecohydrological systems. In a theoretical modelling study we identify three ecohydrological links – commonly omitted from existing models – as potentially important to long-term peatland development, namely those between: I oxic-zone thickness and the rates of litter addition and depth-integrated decay; II time-integrated decay and hydraulic conductivity; and III drainage and peatland lateral expansion via paludification.

2. In a simple model that includes none of these links, total peat thickness increases monotonically with annual rainfall, while oxic-zone thickness is controlled by the rates of litter addition and depth-integrated decay.

3. In an intermediate model that includes Link I, bi-stable behaviour occurs, with both ‘dry’ and ‘wet’ peatland forms possible at low rainfall, but only ‘wet’ peatland forms possible above a threshold value of rainfall. This finding agrees with those from a similar published model.

4. In a more complicated model that includes both Link I and Link II, the bi-stability of the intermediate model is lost. Increases in net rainfall lead to little change in oxic-zone thickness because the model’s feedbacks confer self-dampening (stabilizing) behaviour. Bog height after 5000 years is maximal at an intermediate anoxic decay rate, an initially counter-intuitive finding that reflects complex behaviour arising from the interacting feedbacks represented within the model.

5. In a final model that includes Links I, II and a partial representation of Link III, the mode of peatland lateral expansion (i.e. linear, logarithmic or step-wise expansion) has a strong effect on patterns and rates of peat accumulation.

6. Synthesis. Understanding long-term peatland development requires consideration of ecohydrological feedbacks; models without such feedbacks are likely to misrepresent peatland behaviour. Down-profile changes in peat properties, commonly taken to indicate external (climatic) influences in palaeoclimatic studies, may in some cases be consequences of internal peatland dynamics under a steady climate.

Introduction

As complex systems, peatlands have the potential to exhibit both self-dampening (stabilizing) and self-reinforcing (destabilizing) behaviour (Belyea & Baird 2006; Belyea 2009), which means that their structure and function may respond in a nonlinear way to external (e.g. climate) forcing (e.g. Belyea & Malmer 2004). When negative feedbacks dominate, a peatland is resistant to external forcing. Conversely, when positive feedbacks dominate, a peatland may respond disproportionately to weak external forcing (Hilbert, Roulet & Moore 2000; Belyea & Clymo 2001). Switches between self-dampening and self-reinforcing behaviour may occur, with the result that a peatland may experience long periods of stasis, dominated by negative feedbacks, punctuated by brief episodes of rapid change, dominated by positive feedbacks (Belyea 2009). Understanding how and when these switches in behaviour occur is a major challenge both to understanding peatland response to past climate change (e.g. Frolking et al. 2010), and to projecting their response to future climate change (e.g. Ise et al. 2008).

Hitherto, most models of peatland development have considered only the ecological processes of litter addition and organic matter decomposition (e.g. Clymo 1984, 1992), or have considered only short-term responses to fluctuating water levels (e.g. Frolking et al. 2002). The few models that have included ecohydrological feedbacks have shown nonlinear (e.g. Hilbert, Roulet & Moore 2000; Ise et al. 2008; Frolking et al. 2010) or self-organizing (e.g. Eppinga et al. 2008) behaviour, suggestive of switches in dominance of negative and positive feedbacks. In the theoretical models presented in this article, we focus on the links between peatland structure (height and lateral extent, thickness of the oxic zone, peat hydraulic conductivity) and key ecological and hydrological processes (rates of litter addition, depth-integrated decay, drainage and water-storage change), applied to raised bogs only. By sequentially ‘switching on’ or adding three distinct ecohydrological links, we explore how increasing the number of feedbacks and their connections affects modelled millennial-scale peatland development. Ultimately, our approach addresses the question, how do internal feedbacks regulate peatland dynamics?

Ecohydrological feedbacks in peatlands

In this section, we identify three ecohydrological links that we believe are fundamental to regulating peatland dynamics. These links are embedded within a network of feedbacks (Fig. 1), which together contribute to the nonlinear behaviour of peatlands. The links are as follows.

Figure 1.

 Network of ecohydrological feedbacks operating during peatland development. Links (arrows) between variables (boxes) are hollow and labelled ‘+’ if the causal influence is positive (i.e. an increase in the first variable causes an increase in the second variable), or solid black and labelled ‘−’ if the causal influence is negative (i.e. an increase in the first variable causes a decrease in the second variable). The arrow filled with grey and labelled ‘±’, indicates that that relationship can be either positive or negative, depending on the value of Z. A feedback (i.e. a sequence starting and ending at the same variable) is negative (self-dampening) if there is an odd number of negative links in the loop. Conversely, the feedback is positive (self-reinforcing) if there is an even number of negative links (or none at all). Links I, II and III (see text) are indicated. Dashed arrows represent relationships that are not fully represented in our models. Ellipses represent assumed parameters. Refer to Table 1 for variable names.

Link I: Oxic-zone thickness and litter addition.  Oxic-zone thickness is involved in two, potentially competing, feedbacks (Fig. 1). The first feedback is always negative: as thickness of the oxic zone increases, depth-integrated decay in the oxic zone increases monotonically. If the rate of water-storage change (which dictates the height of the water table) is constant, the rate of peat accumulation decreases and the oxic zone thins, counteracting the initial change. The second feedback (Link I) represents a nonlinear relationship between the rate of litter addition and oxic-zone thickness. Increases in oxic-zone thickness up to some ‘optimum’ value cause increases in the rate of litter addition; further increases in oxic-zone thickness beyond the optimum value cause the rate of litter addition to fall. This relationship has been observed in real peatlands (e.g. Wallén, Falkengren-Grerup & Malmer 1988; Belyea & Clymo 2001) and has been incorporated into a few models of peatland dynamics (Hilbert, Roulet & Moore 2000; Eppinga et al. 2009a; Frolking et al. 2010). The feedback involving Link I may be positive or negative depending on oxic-zone thickness relative to the optimum and three critical values. The three critical values represent two stable equilibria (thin and thick oxic zones) and one unstable (intermediate thickness) equilibrium. At these equilibria, litter addition minus depth-integrated decay (through both oxic and anoxic zones) is equal to water-storage change (see also Belyea & Clymo 2001; Eppinga et al. 2009a). When oxic-zone thickness is less than the optimum for litter addition but thicker than the unstable equilibrium, litter addition increases with increasing oxic-zone thickness; if the rate of storage change is constant, peat accumulation speeds up and the oxic zone thickens further. This feedback is positive. Above the optimum value but below the ‘thick’ stable equilibrium, litter addition decreases with increasing oxic-zone thickness, and the feedback is negative (self-dampening). Positive feedback dominates near the unstable equilibrium, acting either to enhance thickening of the oxic zone (e.g. hummock building) if oxic-zone thickness is above the unstable equilibrium, or to promote thinning of the oxic zone (e.g. hollow deepening) if it is below the unstable equilibrium. Negative feedback dominates near the two stable equilibria, acting to maintain a constant oxic-zone thickness that is either thick (e.g. hummock) or shallow (e.g. hollow).

Link II: Time-integrated decay and hydraulic conductivity.  The depth dependence of saturated hydraulic conductivity, K, tends to maintain the water table most of the time within a narrow range of elevations (Ivanov 1981; see also the discussion by Belyea 2009): during wet periods, the water table rises into more porous, higher-K peat and water is discharged more rapidly; conversely, during dry periods, the water table falls into denser, lower-K peat and water is discharged more slowly. This negative, purely hydrological, feedback interacts with the ecohydrological feedbacks described in Link I via the effects of decay on peat physical properties (Fig. 1). Degree of decomposition (time-integrated decay, expressed as the proportion of a peat sample’s original mass lost through decay) increases with age, and is inversely related to the structural rigidity of individual stems and of the peat matrix as a whole. With increasing age and degree of decomposition, peat is likely to be compressed by the (increasing) weight of younger, overlying peat (Clymo 1978). As peat becomes compressed, the pore spaces within it close, leading to a lower hydraulic conductivity. Once pore spaces have been mainly closed, further compression of peat is limited, as are further decreases in hydraulic conductivity. Thus it seems reasonable to expect the relationship between hydraulic conductivity and proportional remaining mass, θ (that proportion of the original mass of a peat sample that has not been lost through decay), to be nonlinear with a rapid initial decrease in hydraulic conductivity with decreasing θ and then a slowing in the rate of decrease. This link (Link II) between degree of decomposition and hydraulic conductivity opens up a wider network of ecohydrological feedbacks, via interactions with oxic-zone thickness and time-integrated decay (Fig. 1). In isolation, the hydrological feedback is always negative. However, it also interacts with the feedbacks described above (Link I) and below (Link III), in which case it may form part of a larger network in which either positive or negative feedback dominates. The influence of decomposition upon peat hydraulic conductivity has been represented in the recent Holocene Peat Model by Frolking et al. (2010), although those authors did not explicitly examine the interactive effects of this ecohydrological link with other feedbacks.

Link III: Lateral expansion and hydraulic gradient.  Lateral expansion of raised bogs occurs chiefly by the process of paludification (Anderson, Foster & Motzkin 2003), in which mineral soils at the margin become saturated by water draining from the bog, providing conditions suitable for peat accumulation. Lateral expansion is linked directly to drainage (Link III) and indirectly to hydraulic gradient: as the bog expands laterally, the distance between the centre and the margin of the bog increases, causing a reduction in overall hydraulic gradient, which would be expected to cause a reduction in the loss of water through drainage. However, lateral expansion also increases the surface area of the peatland, thereby increasing the volume of rainfall received by the peatland, which will, in some circumstances, act to increase drainage rates at the margin, depending on the shape of the peatland. Whether the net effect of lateral expansion is to increase or decrease drainage rates depends on the rate of storage change and the shape of the peatland in plan (Fig. 1).

In this article, we used four 1-D models of peatland development to explore formally and quantitatively the effects on peatland behaviour of sequentially adding the links described above. The advantage of a modelling approach is that it removes extraneous or confounding variables from consideration so that the effect of the different feedbacks on system behaviour can be clearly seen. The limitation of the approach is that it is purely theoretical, and care needs to be taken when applying the results to real peatlands.

Models and methods

In all of our models, we simulated the 1-D (vertical) growth of a raised bog and the development of a groundwater mound within the peat soil. Peat accumulates in the models when time-integrated addition of litter at the peatland surface exceeds time-integrated decay. The groundwater mound rises or falls within the model peatland in response to rainfall and drainage to the model’s boundary. The models calculate two state variables: the height of the peatland surface and the thickness of the oxic zone (i.e. depth to water table from the peatland surface).

To ensure that we properly identified the effects of the ecohydrological links on model behaviour, we used a consistent set of default parameters across the four models (see Table 1). The models we used were as follows: (i) A simple model (henceforth referred to as ‘Model 1’), in which the three links described above are inactive. If our modelling is thought of as a numerical experiment, then Model 1 can be thought of as a type of experimental control; (ii) An intermediate model (‘Model 2’) in which Link I is included; (iii) a more complicated model (‘Model 3’), which includes both Link I and Link II; and (iv) an extension of the complicated model (‘Model 4’), which includes Links I and II and some of Link III, but not the tentative links between drainage and lateral expansion, or between lateral expansion and volume of rainfall received by the peatland. We used Model 4 to explore, in a simplified way, how lateral expansion might affect long-term peatland development (see Model 4 (Links I, II and III), below).

Table 1.   Glossary of symbols used and parameter default values for the four models (default values assumed at all times unless otherwise stated)
SymbolNameDimensionsModel 1Model 2Model 3Model 4
ACross-sectional area of flowL2
aEqn 10 coefficientL T−10.001 cm s−10.001 cm s−1
αanAnoxic specific decay rateT−10.0001 year−10.0001 year−10.0001 year−10.0001 year−1
αoxOxic specific decay rateT−10.015 year−10.015 year−10.015 year−10.015 year−1
BBog heightLState variableState variableState variableState variable
B*Steady-state peatland heightLsee eqn 6
bEqn 10 exponent88
HWater-table heightLAuxiliary variableAuxiliary variableAuxiliary variableAuxiliary variable
H*Steady-state water-table heightLSee eqn 2
KHydraulic conductivityL T−1200 000 cm year−1200 000 cm year−1Variable (eqn 10)Variable (eqn 10)
LLateral extentL50 000 cm50 000 cm50 000 cmVariable (Fig. 2)
MCumulative mass per unit areaM L−2Auxiliary variableAuxiliary variableAuxiliary variableAuxiliary variable
pLitter addition rateM L−2 T−10.0864 g cm−2 year−1Variable (eqn 8)Variable (eqn 8)Variable (eqn 8)
QDischargeL3 T−1
rPore radiusL
ρPeat dry bulk densityM L−30.1 g cm−30.1 g cm−30.1 g cm−30.1 g cm−3
UNet rainfall rateL T−130 cm year−130 cm year−130 cm year−130 cm year−1
sPeat drainable porosity0.30.30.30.3
tModel timeT
θProportional remaining massAuxiliary variableAuxiliary variableAuxiliary variableAuxiliary variable
ZOxic-zone thicknessLState variableState variableState variableState variable
Z*Steady-state oxic-zone thicknessLsee eqn 7

Analytical analysis

Before describing each 1-D model, we consider how Ingram’s (1982) ground-water mound hypothesis (GMH) (see also Childs & Youngs 1961) can be used to help understand the controls on peatland height or thickness in the absence of ecohydrological feedbacks. We use the GMH to calculate change in water-table height, H [dimensions of L], in the centre of an idealized raised bog that meets the following assumptions: (i) the bog is hemi-elliptical in cross section, and is constrained laterally by two parallel streams to which it drains; (ii) the bog possesses uniform hydraulic conductivity; and (iii) the bog is underlain by a flat, impermeable substrate. Under such conditions,

image(eqn 1)

where H is water-table height in the centre of the bog, t is time [T], U is net rainfall (i.e. precipitation minus evapotranspiration) [L T−1], K is hydraulic conductivity [L T−1] and L is peatland lateral extent [L], equal to half the distance between the parallel streams. When set to zero, eqn 1 may be rearranged to give steady-state water-table height, H* [L]:

image(eqn 2)

The central height of the peatland surface, B [L], above the mineral substrate (i.e. the thickness of the entire peat deposit, including both oxic and anoxic zones) at any point in model time may be expressed as the quotient of cumulative peat mass per unit area, M [M L−2] (again, representing both oxic and anoxic zones), and peat dry bulk density, ρ [M L−3] (ρ is assumed to be constant in time and space; see Table 1):

image(eqn 3)

Furthermore, the depth Z [L] of the water table below the peatland surface (i.e. oxic-zone thickness) may be described in terms of B and H:

image(eqn 4)

We adopt a convention, defined in eqn 4, whereby positive values of Z represent water-table positions below the peatland surface, while negative values represent water ponded above the peatland surface. We assumed that all ponded water above the peatland surface was lost as runoff, i.e. we prevented Z from becoming negative. This would appear to be a reasonable assumption in the case of a non-patterned bog (cf. Swanson 2007). We also assumed that conditions below the water table are fully saturated and anoxic (although see, e.g. Tokida et al. 2004) and that conditions above the water table are unsaturated and oxic (i.e. like in Frolking et al. 2001, our models do not represent a capillary fringe).

Peat decay throughout the peat profile or column representing the centre of the bog occurs at one of two rates: a faster, oxic rate, αox [T−1], above the water table, and a slower, anoxic rate, αan [T−1], below the water table. Change in peatland height may therefore be described in terms of the rate, p [M L−2 T−1], of addition of fresh litter at the peatland surface; water-table depth, Z; and the two specific decay rates:

image(eqn 5)

The form of eqn 5 is almost identical to one used by Hilbert, Roulet & Moore (2000; eqn 2, page 232).

Steady-state peatland height, B* [L], can be obtained by setting eqn 5 to zero and substituting eqn 2 for H and eqn 4 for Z:

image(eqn 6)

Steady-state water-table depth, Z* [L], may be similarly written as follows:

image(eqn 7)

The steady-state solutions demonstrate two important relationships. First, peatland height (eqn 6) is controlled mainly by hydrological parameters (L, U and K) and to a lesser extent by processing of material in the oxic zone (p/αox). The magnitude of the term p/ραox for realistic parameter values is small compared to L(U/K)1/2, meaning that processing of material in the oxic zone exerts a second-order control over steady-state peatland height, B* (eqn 6), compared to hydrological parameters. Steady-state peatland height is only weakly controlled by anoxic decay rate, due to its low value relative to oxic decay (for most reasonable parameter values, the term αanox is very small). Secondly, steady-state oxic-zone thickness, Z* (eqn 7), is controlled mainly by processing of material in the oxic zone (i.e. p/αox), because the small value of αanox reduces the model’s sensitivity to the hydrological parameters. However, large changes in the quotient αanox will have a greater relative effect on Z than on B, due to the reduced influence of hydrological parameters in eqn 7 compared to eqn 6. Unlike Clymo’s (1984) bog growth model, which suggests that bog height is limited ultimately by depth-integrated anoxic decay, our analysis of eqns 6 and 7 suggests that, in the absence of ecohydrological feedbacks, net rainfall (U) may be a more important control on peatland height than anoxic decay.

Model 1 (no ecohydrological links)

We used finite-difference solutions (Euler-type integration) to eqns 1 and 5 to simulate, for a 1-D column at the centre of a bog, the effect of varying values of U, p, αox and αan on the state variables, B and Z. All other terms were assumed constant (see Table 1). The structure of eqn 1 means that increasing the net rainfall rate has the same directional effect upon peatland height and oxic-zone thickness as decreasing hydraulic conductivity or increasing lateral extent (see also eqn 2). Hence, we only manipulated net rainfall. All simulations were run for 5000 years of model time, with timesteps of 0.3–1 year (see below). We found that 5000 years may not necessarily be long enough to achieve a genuine steady state (eqns 6 and 7), but that in most cases model results were very close to steady-state values after this time (see Results, below). Additionally, radiocarbon estimates of peatland basal ages suggest that approximately a third to a half of all northern peatlands are 5000 years or younger (MacDonald et al. 2006; Yu, Beilman & Jones 2009; Korhola et al. 2010). While we could have analysed Model 1 using the analytical steady-state solutions presented above (eqns 6 and 7), our more complicated models are not readily analytically tractable. Therefore, we ran all of our models using numerical solutions for 5000 years of simulated time. Separate solution tests not reported here confirmed that our yearly timesteps provide accurate (and convergent) solutions for Model 1 and Model 2, in which numerical diffusion errors are small (i.e. do not materially affect the results shown later). For Models 3 and 4, solution testing indicated that timesteps should be reduced to 0.3 years in order to ensure a stable, convergent solution, with small diffusion errors. The smaller timesteps were needed because of higher values of near-surface K (see Model 3 (Links I and II), below).

One advantage of using the finite-difference approach is that it is possible to simulate explicitly the down-profile properties of the peat. In effect, the finite-difference model is a peat cohort model. Each timestep, a cohort of new litter is added to the top of the model peat column, and the cohorts from previous timesteps undergo decay and other changes (see Models 3 and 4).

Finally, it is important to appreciate that our model is a 1-D, vertical model representing the centre of a bog. Although eqn 1 represents the lateral flow of water from a bog to its margin, it considers only flow from the centre and not the total loss of water from the entire peatland. Therefore, it is appropriate to apply only to a central 1-D column of peat.

Model 2 (Link I)

Model 2 is the same as Model 1 except that it includes Link I, in which the rate of addition of plant litter is a function of oxic-zone thickness. We altered Model 1 so as to predict p as a function of Z for all water-table depths between zero (as stated above, we assumed that all ponded water is lost as runoff, so that Z is never negative) and 66.8 cm below the surface, according to the function provided by Belyea & Clymo (2001), and assuming that Z may reasonably be used as a proxy for surface soil moisture (Hayward & Clymo 1982). Formally:

image(eqn 8)

The relationship described by eqn 8 is similar to that implemented in the peat accumulation model of Hilbert, Roulet & Moore (2000) and the surface patterning models of Eppinga et al. (2009a,b).

Model 3 (Links I and II)

We altered Model 2 so as to include Link II, in which K is a function of the degree of peat decomposition; in this model K varies both through time and with depth in the peat profile. Our finite-difference model represents the peat column as many vertically stacked, slab-like cohorts or layers of peat, with layer age increasing downwards through the column. The model keeps track of the remaining peat mass in each layer as a proportion, θ, of that layer’s original mass. In Model 3, we used this value of θ for each layer to calculate K. Link II applies to both the oxic and anoxic zones in our model, meaning that the saturated hydraulic conductivity of any layer continues to fall in response to decay, even after that layer is submerged beneath the water table. However, the K value of any layer is for saturated hydraulic conductivity, and so only affects drainage when that layer is below the water table.

Water flow through a unit cross section of a cylindrical pore (under a unit energy gradient) is a quadratic function of pore diameter (Poiseuille’s law; cf. Pfitzner 1976). If it is assumed that a similar relationship applies to pores in a peat soil, and an energy gradient of unity is also assumed, we may write:

image(eqn 9)

where Q is flow [L3 T−1], A is pore cross-sectional area [L2] and r is pore radius [L]. Therefore, if pore sizes decreased linearly with θ, we would expect a quadratic decrease in K with θ. However, as noted above, pore diameter will not show a linear decrease with θ. If we assume that the relationship between pore radius and θ is exponential, then the relationship between K and θ is also exponential, thus:

image(eqn 10)

where a [L T−1] and b [dimensionless] are parameters, and = 2β.

Hydraulic conductivity, K, is generally held to decrease with increasing degree of decomposition, and there is evidence that the relationship is nonlinear along the lines suggested by eqn 10 (e.g. Boelter 1969, 1972; Ivanov 1981). However, it is also clear that other factors, such as botanical composition and depth (peat compression) affect K, and these will complicate any relationship between K and θ (Päivänen 1973; Ivanov 1981). Until more work is done on the shape of K(θ) for a range of peats, we use eqn 10 as a first approximation.

Now that K is no longer a constant, eqn 1 must be rewritten to reflect the fact that depth-averaged K is a function of time and water-table position:

image(eqn 11)

We assumed that = 0.001 cm s−1 and = 8 (see eqn 10), leading to near-surface K values (fresh litter; high values of θ) that are arguably lower than would usually be expected, and deep K values (highly decomposed peat; low values of θ) that are higher than might be expected for highly decomposed peat (Fraser, Roulet & Laffleur 2001; Clymo 2004). Therefore, the values chosen for a and b probably represent a conservative decrease in K with decreasing θ, so reducing the effect of Link II on model behaviour.

Model 4 (Links I, II and III)

We extended Model 3 to include peatland lateral expansion, by representing its negative relationship with hydraulic gradient (Link III). However, we did not represent the positive relationship between L and drainage (via increased volume of rainfall received by a peatland as it expands) or an explicit, process-based description of the positive effects of drainage upon L. Rather, we simply prescribed increases in L through time. By allowing L to increase through model time, the knock-on effects of lateral expansion on drainage and water storage, via a reduced hydraulic gradient, are readily apparent (see eqn 11). We know of no mechanistic description of peatland expansion as an explicit function of drainage, although a few authors have put forward conceptual (Anderson, Foster & Motzkin 2003; Belyea & Baird 2006) and empirical (Korhola et al. 1996) models. We decided to omit the tentative link between drainage and lateral expansion (see Fig. 1) because empirical evidence is lacking and because theoretical foundations are shaky. Instead, we imposed increases in L as simple linear, logarithmic and step-wise functions of time (Fig. 2) and compared the results to those of Model 3 (constant L). For all three modes of expansion, we assumed that L increases from an initial value of 10 000 to 50 000 cm after 5000 simulated years. Because we prescribed changes in L rather than allowing it to respond to drainage and the peatland surface area that receives rainfall, Model 4 shows the unidirectional effects of lateral expansion rather than the more complex effects of expanding the ecohydrological feedback network to include the two other distinct loops shown in Fig. 1.

Figure 2.

 Constant peatland lateral extent, L, in Model 3, and assumed increases in L as linear, logarithmic and step-wise functions of time, in Model 4.

Parameter values and model simulations

For all models, we varied assumed net precipitation, U, between values of 5 and 100 cm year−1, representing the ‘climate space’ of the majority of northern peatlands (cf. Yu, Beilman & Jones 2009). We varied anoxic decay rates, αan, between 10−5 and 10−3 year−1. Estimates of anoxic decay rates vary greatly. Litterbag studies such as those by Johnson & Damman (1991) might be taken to suggest rates as high as 10−2 year−1, although initially high experimental decay rates often decrease rapidly during the study period. Johnson & Damman (1991) recognized that their results were unlikely to be representative of genuine, long-term anoxic decay rates. Using an analysis of peat-core data, Yu et al. (2001) estimated anoxic decay rates of the order of 10−4 to 10−5 year−1 for their study site in western Canada, while Clymo (1984) obtained estimates of the order of 10−6 year−1 for various bogs by back-calculating from his own model. Our assumed rates fall within the bounds of these various estimates. We assumed oxic specific decay rates, αox, of between 0.05 and 0.005 year−1, consistent with estimates from litterbag studies, which appear to provide more reliable estimates of oxic decay rates than of anoxic rates (e.g. Johnson & Damman 1991; Hogg 1993), and from peat-core data (e.g. Yu et al. 2001). In Models 1 and 2, we assumed a constant hydraulic conductivity, K, of 2 × 105 cm year−1 (approximately equal to 0.006 cm s−1), within the normal range of field and laboratory estimates for deep peat (e.g. Clymo 2004; Baird, Eades & Surridge 2008). In Models 1, 2 and 3, we assumed a constant default lateral extent, L, of 50 000 cm (500 m). In Model 1, we assumed values of p of 0.0864 g cm−2 year−1, equivalent to the maximum rate of litter addition observed by Belyea & Clymo (2001) at their study site in SW Scotland, as well as 90% and 75% of this value. These p values represent those predicted by eqn 8 for water tables between Z = 16.9 and 43.6 cm, equivalent to c. 40% of the range of water-table values considered by eqn 8. Finally, in all simulations we held drainable porosity, s, constant at 0.3. While reasonable for shallow peat, this value is higher than might reasonably be expected for deep, highly decomposed peat. Vorob’ev (1963) found that deep peat samples from what he described as ‘Sphagnum-shrub high-lying swamps’ had s values of < 0.1, although he also found surface values approaching 0.5.

Assumptions

We have endeavoured to keep our models as simple as possible while including what we consider to be key ecohydrological feedbacks. Wherever possible, we replicate the simplifying assumptions of other similar studies (e.g. Hilbert, Roulet & Moore 2000) so as to facilitate comparison between our results and theirs. For example, we assume that specific decay rates are unaffected by geochemical and botanical composition of the litter (cf. Johnson & Damman 1991; Turetsky et al. 2008), physicochemical conditions such as moisture and soil temperature (cf. Hogg, Lieffers & Wein 1992; Scanlon & Moore 2000) or end-product build-up (cf. Beer & Blodau 2007). Unlike Frolking et al. (2010), we assumed that peat dry bulk density does not change with increasing decomposition, despite our assumption that changes in K are due to compression (see Model 3 (Links I and II), above). We also assumed that net rainfall, U, is constant throughout any simulation, and that evapotranspiration, E, is independent of water-table depth Z (Lafleur et al. 2005; although see Ingram 1983). It might be argued, therefore, that our models omit a number of potentially important feedbacks, although their simplicity allows us to explore the interactive effects of specific ecohydrological feedbacks. Particularly, the effects of decay upon K have rarely been considered in models of the long-term development of peatlands (although see Frolking et al. 2010). As already stated, our models are 1-D and represent only the centre of a raised bog. Ingram’s (1982) assumption of a hemi-elliptical groundwater mound (eqn 2) accounts for the role of lateral extent in determining areal rate of delivery of rainfall to, and drainage from, the model peatland. Therefore, in Model 4, we did not explicitly represent increases in drainage as a result of increased lateral extent. The only effect of increased L in Model 4 is to decrease the overall hydraulic gradient.

Results

Model 1 (no ecohydrological links)

The finite-difference solutions after 5000 years of model time showed the general patterns expected from the steady-state solutions: B increases nonlinearly with increasing U (Fig. 3a), increases weakly with decreasing αox (Fig. 3a), and is almost unresponsive to changes in p and αan (Fig. 3b). Water-table depth, Z, is largely unresponsive to U (Fig. 3c) but increases strongly with decreasing αox (Fig. 3c) and αan (Fig. 3d), and with increasing p (Fig. 3d). The strong control of αan over Z reflects the large range of values assumed for αan. For the parameter values chosen, B is controlled mainly by hydrological parameters, whereas Z is controlled primarily by the rate of litter addition and specific oxic and anoxic decay rates (Fig. 3).

Figure 3.

 Monotonic response of Model 1 to assumed net rainfall rate, U, anoxic specific decay rate, αan, oxic specific decay rate, αox, and the rate of litter addition, p.

We evaluated eqns 6 and 7 for all of the parameter combinations used in simulations with Model 1, in order to assess how close the model’s outputs are to the steady-state solutions after 5000 simulated years. The greatest deviation of any model output from the steady-state solutions is < 0.9% of the steady-state value (when αan = 0.001 year−1, = 0.0648 g cm−2 year−1, and all other parameter values were set to the defaults – see Table 1), although most model results are within 0.05% of the steady-state values. This finding suggests that 5000 simulated years is indeed long enough to gain a representative indication of model behaviour.

Model 2 (Link I)

Model 2 is very similar to that of Hilbert, Roulet & Moore (2000), so a detailed analysis is not repeated here. What is important is the difference in behaviour between Models 1 and 2, caused by the inclusion of Link I. The dependence of p upon Z causes Model 2 to exhibit bistability, predicting one of two possible peatland forms depending on net rainfall rate: either (i) a wet peatland with water tables at the surface (Fig. 4a), and in which increases in net rainfall have no effect upon peat thickness (Fig. 4b), or (ii) a drier peatland, with water tables c. 40–50 cm below the surface (Fig. 4a), in which peat thickness responds strongly to rainfall rate (Fig. 4b). The behaviour of Model 2 is more complex than that of Model 1, and the state variables B and Z respond in a non-monotonic manner to changes in U. Figure 4b shows that, as U increases up to c. 9 cm year−1, B increases approximately linearly, to a maximum height of between 328 and 382 cm (substantially lower than for the same rainfall rate in Model 1 – see Fig. 3a, above). However, for rainfall rates above 9 cm year−1, B either drops abruptly to between 337 and 348 cm (for αox = 0.005 and 0.015 year−1) or levels off at c. 328 cm (for αox = 0.05 year−1). Bog height appears to be invariant in response to further increases in U above this threshold. This behaviour is in sharp contrast to Model 1’s predictions that bog height continues to increase with increasing net rainfall for all assumed values of U (Fig. 3a). Indeed, Model 1 predicts bog heights of between 1133 and 1268 cm when = 100 cm year−1, equivalent to between three and four times higher than for the same parameter sets in Model 2.

Figure 4.

 Bistable response of Model 2 to assumed net rainfall, U.

Our use of Belyea & Clymo’s (2001) empirical equation for water tables below the peatland surface (eqn 8) means that we parameterized our model to represent raised bogs only. Therefore, unlike Hilbert, Roulet & Moore (2000), we are unable to interpret the wet and dry model states as fens and raised bogs, respectively. Furthermore, we did not represent any of the biogeochemical changes associated with fen-bog transition (Hughes 2000). Instead, we take our values of B and Z for the wet model state to represent bogs dominated by hollows or wet lawns, and those for the dry state to represent bogs with high proportions of hummocks. While we did not undertake a full exploration of model parameter space (see Parameter Values and Model Simulations, above), the work presented here, strongly supported by Hilbert, Roulet & Moore (2000), suggests that no more than two model states are possible for any combination of αan and αox.

Model 3 (Links I and II)

The results from Model 3 show that the responses of bog height and oxic-zone thickness to increasing net rainfall rate are similar to those of Model 1; i.e. B increases, at a decreasing rate, with increases in U (Fig. 5a), whereas Z is largely unresponsive to rainfall rate (Fig. 5b). However, the responses to increasing anoxic decay rate αan differ from those of both Models 1 and 2. After 5000 simulated years, bog height, B, peaks at c. 700 cm, at an intermediate anoxic decay rate of αan = 0.00025 year−1 (Fig. 5c). For αan above or below this value, B decreases to < 100 cm. Oxic-zone thickness, Z, is maintained at c. 40 cm for most anoxic decay rates used, but drops to < 5 cm as anoxic decay rate increases from 0.0005 to 0.0006 year−1 (Fig. 5d). Bog height also exhibits a similar – although less pronounced – response to the oxic decay rate, αox (Fig. 5a). For any given net rainfall rate, U, the intermediate oxic decay rate of αox = 0.015 year−1 leads to a thicker peat deposit than either the higher rate of αox = 0.05 year−1 or the lower rate of 0.005 year−1.

Figure 5.

 Response of Model 3 to net rainfall rate, U, anoxic specific decay rate, αan, and oxic specific decay rate, αox.

These nonlinear responses arise from the dual biogeochemical and biogeophysical role of decay. At low anoxic decay rates, mineralization of organic matter occurs slowly, but the high K of poorly decomposed peat (high θ; eqn 10) allows rapid drainage (eqn 11). As a result, bog height is limited by slow growth of the groundwater mound, and a thick oxic zone is maintained. Conversely, at high anoxic decay rates, bog height is limited by rapid mineralization of organic matter, but the low K of highly decomposed peat (low θ; eqn 10) inhibits drainage. In this case, only a thin oxic zone can be maintained. Hence, inclusion of Link II leads to qualitative and quantitative differences in behaviour compared to Model 2.

The profile of θ with depth (age) for the run with = 30 cm year−1 shows three distinct phases (Fig. 6a, solid line). During approximately the first 500 years of development of the model peatland (i.e. peat age 5000 to 4500 years), a brief period of accumulation of poorly decomposed peat was followed by a rapid shift to accumulation of more highly decomposed peat. This shift is due to development of the oxic zone after peat initiation (Fig. 7); in real peatlands, a narrow zone of poorly decomposed peat may not exist near the base of the bog because of the enhancement of decay by minerotrophic water (e.g. Yu et al. 2003; Frolking et al. 2010). In peat between the ages of c. 4900 and 500 years there is a nonlinear decrease in degree of decomposition with decreasing age, owing to more gradual changes in oxic zone thickness. The rapid shift in degree of decomposition that occurs at the top of the profile (i.e. youngest peat of < 100 or so years) shows the effects of fast decay in the oxic zone. These changes in degree of decomposition arise from cross-scale interactions between simple rule sets, under conditions of constant net rainfall rate.

Figure 6.

 (a) Down-profile changes in proportional mass remaining, θ; and (b) temporal changes in bog height B; for Model 3 simulation with default parameter values (solid black line), and Model 4 simulations in which lateral extent, L, was assumed to increase as linear, logarithmic and step-wise functions of time.

Figure 7.

 Development of bog height B (solid line) and water-table height H (broken line) through model time for a single Model 3 run with all default values assumed (see Table 1). The development of an oxic zone is evident after about 500 years.

Model 4 (Links I, II and III)

In the peatland expansion simulations (Model 4), all model runs with increasing L predict a greater bog height than does the model with constant L (Fig. 6b). The model with linearly increasing L gives the greatest net rate of peat accumulation over the 5000 simulated years, with a final bog height of 659 cm, compared to 538 cm for the model with constant L. The functions used to describe L are clearly expressed in the temporal changes in bog height: in particular, the model run in which L increases in a step-wise manner exhibits step-wise increases in bog height (Fig. 6b) and sharp spikes in the final θ profile (Fig. 6a). Step-wise expansion of a bog’s lateral extent might occur in situations where the mineral substrate onto which the bog expands is topographically undulating or otherwise variable, or where adjacent bog domes coalesce, thereby causing abrupt changes in the effective drainage boundary positions (cf.Comas, Slater & Reeve 2005). Some of the changes in degree of decomposition that are clearly attributable to changes in L in these simulations could, if observed in peat cores, be mistaken for palaeoclimatic signals.

Discussion

Feedback Interaction leads to unexpected, and sometimes complex, model behaviour

By sequentially adding ecohydrological links to a very simple model, we have shown that the effect of a particular link depends on how it connects to form a distinct feedback loop and on how that loop interacts with the wider feedback network. For example, the addition of Link I (Model 2) completed a loop that on its own could show either positive or negative feedback. When connected to the rest of the network, the inclusion of this one link changed system behaviour to allow bistability. The addition of Link II (Model 3), in contrast, completed a loop that on its own could show only negative feedback. The inclusion of this link eliminated the bistability, although Model 3 does not exhibit the simple, monotonic response of bog height to anoxic decay rate seen in Model 1. These results show that the inclusion of simple links can sometimes have substantial and unexpected consequences for system behaviour.

For the models used in this article, we focussed on three ecohydrological links which are theoretically sound and for which there is at least some empirical evidence. Other links may also have important effects on system behaviour. For example, rates of evapotranspiration are commonly thought to decrease nonlinearly with depth to water table (equivalent to Z) (Ingram 1983; although see Lafleur et al. 2005). As this link would complete a negative feedback also involving storage change, we expect that its inclusion would tend to stabilize system behaviour.

Our models are 1-D and consider only vertical changes in peat properties. Horizontal heterogeneity in peat properties and process rates can promote localized transfers of water, nutrients and energy (e.g. Eppinga et al. 2008; Kettridge & Baird 2010), which in turn may create new feedback loops operating at larger spatial (and perhaps longer temporal) scales. For example, an increase in biomass of vascular plants leads to an increase in transpiration and advective transport of dissolved nutrients, which may in turn enhance nutrient uptake by, and growth of, the vascular plants (e.g. Eppinga et al. 2009a). This localized positive feedback cannot be represented in 1-D models, such as those presented here, but may interact in unexpected ways with the rest of the feedback network.

It is clear, therefore, that even the most complicated of our models (Model 4) does not include all important internal feedbacks regulating peatland development, and it is reasonable to expect additional links and an expansion to 2- or 3-D to have a substantial effect on system behaviour. It is possible, however, that model behaviour is insensitive to some feedbacks. This leads us to ask: what further feedbacks must be included in models of peatland development in order to capture the essential behaviour of these systems? We suggest that the goal for future research in this area should be the formulation of a model that strikes a balance between parsimony (Goldenfeld & Kadanoff 1999) and predictive power (sensuGrimm et al. 2005).

Disproportionate model response to external influences

The behaviours of models that include some or all of Links I, II and III (i.e. Models 2, 3 and 4) help to explain empirical observations (e.g. Belyea & Malmer 2004) that peatlands may undergo abrupt changes in both structure (e.g. oxic-zone thickness, and by implication, dominant vegetation) and function (e.g. net rate of peat accumulation), even under weak external forcing. When positive feedback dominates, large changes in peat-accumulation and water-table regimes may occur in response to a small change in a single variable. In Model 2, for example, oxic-zone thickness and bog height exhibited sharp threshold responses to changing values of net rainfall rate. In Model 3, oxic-zone thickness showed a step-change response to anoxic decay rate whereas maximum bog height occurred at intermediate anoxic decay rates. Such behaviours, in which small changes in external influences can generate disproportionately large responses in the model’s state variables, highlight the role of positive feedback. These behaviours also clearly have implications for interpreting peat palaeo-records and for predicting peatland response to future climate change.

The lesser-known corollary – that large changes in the externally imposed variables assumed to drive peatland behaviour (e.g. temperature and rainfall regimes) may sometimes lead to only small responses in the model’s behaviour – was also evident in our results. In such situations negative feedback dominates, and the system’s state variables are largely resistant to external changes. In Model 3, for example, oxic-zone thickness was almost invariant in response to changes in net rainfall rate. Furthermore, increases in net rainfall caused almost no thinning of the oxic zone, and increases in anoxic decay rates did not always lead to lower net rates of peat accumulation. This argument reinforces our earlier point (see Feedback Interaction Leads to Unexpected, and Sometimes Complex, Model Behaviour, above) that additional links not included in our models, such as the dependence of peat decay rates upon temperature (as represented by Ise et al. 2008), may lead to further qualitative and quantitative changes in model behaviour.

The nonlinear responses shown by our models are intriguing, because they suggest that the strength, and in some cases even the direction, of peatland development may be only weakly related to external influences such as long-term trends in rainfall (cf. Belyea & Malmer 2004) or temperature. This possibility has theoretical and practical implications for palaeoclimatic studies that seek to reconstruct past climates using proxy data from peat cores, and for model projections of the possible response of peatlands to future climate change. Reliable indicators of past water-table depth (i.e. oxic-zone thickness), such as plant macrofossil and testate amoebae assemblages (Charman, Hendon & Packman 1999), might not always be good indicators of rainfall regime, while peat accumulation rates may be poor indicators of instantaneous decay rates if variations in oxic-zone thickness are not also accounted for. Link III, which was not explored fully, would appear to complicate the peatland palaeo-record further, although more research is necessary before this link can be satisfactorily represented in rule-based models such as ours. Moreover, model-based projections of peatland response to future climate change (e.g. Ise et al. 2008) should be interpreted with caution unless one can be certain that all relevant ecohydrological feedbacks have been accounted for. The sensitivity of such model projections to the independent and interacting effects of a range of ecohydrological feedbacks could be examined in future studies by using an extended version of our approach, in which links could be activated and deactivated in isolation and in factorial combination (e.g. Eppinga et al. 2009b).

Northern peatlands store several hundred Pg of carbon in the form of soil organic matter (Gorham 1991; Turunen et al. 2002), and it is currently unclear how stable or otherwise the peatland carbon store will be under a changing climate. Drier mid-latitude climates may lead to lowering of peatland water tables, which in turn may cause affected peatlands to become large net sources of atmospheric carbon dioxide, providing a positive feedback to global warming (Ise et al. 2008). On the other hand, new peatlands may develop rapidly to the north of their current range and act as strong sinks of atmospheric carbon. To improve understanding of these competing effects, efforts are being made to include peatlands in global climate models (GCMs) (cf. Baird, Belyea & Morris 2009; Frolking, Roulet & Lawrence 2009). Our work will be useful in identifying those ecohydrological features of peatlands that should be included in GCMs.

Acknowledgements

We thank the editorial board for helpful dialogue, and Nigel Roulet and Dicky Clymo for insightful and constructive suggestions for improvement on an earlier version of the manuscript. This work was funded by a Queen Mary University of London PhD scholarship awarded to Paul Morris.

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