Modelling of hardwood forest in Quebec under dynamic disturbance regimes: a time-inhomogeneous Markov chain approach

Summary

1. We investigate the consequences of global warming scenarios in Quebec forests using an inhomogeneous Markov chain model. This allows us to unify predictions from climate change models and mechanistic models of forest disturbance and growth and allows predicting the potential impacts of climate change on Quebec forests. The model predicts changes in fire rate in Quebec hardwood forests as well as possible growth enhancements due to increasing and temperature.
2. Our original method consists of three steps. (i) We estimate biomass transition matrices from forest inventories using the Bayesian method. (ii) We incorporate dynamic disturbance and forest growth scenarios, and (iii) we simulate transient dynamics and stationary states. This modelling approach allows for sensitivity analysis and quantitative assessment effects of variability of climate change scenarios.
3. We have considered published climate change scenarios for Quebec and conducted simulations for the most extreme predictions (the smallest and largest predicted changes). None of the considered scenarios is able to counterbalance the currently observed trend of increasing biomass in the next 40 years. By the beginning of 2090, the extreme scenarios diverge within about ±5% mean biomass.
4. Synthesis. In this work, we have developed an original modelling approach incorporating time- inhomogeneous effects within the Markov chain framework. We applied this approach to examine effects of climate change in Quebec's forests. The results demonstrate that the current trend of increase in forest biomass is robust with respect to a broad range of climate change scenarios. This study was not possible with previously employed homogeneous Markov chain models. The model can also be extended to include different harvesting methods and land-use practices, enabling better long-term management of Quebec's forest.

Introduction

Understanding and predicting ecosystem dynamics in the non-stationary world is one of the major challenges of modern ecology (Levin 1999). Complex adaptive systems such as forested ecosystems are especially difficult to model because of their hierarchical organization, nonlinear dynamics and their longevity (Chen & Popadiouk 2002; Levin 2003; Loudermilk et al. 2013). Forest complexity includes multidimensional dynamics and self-organization occurring across several spatial and temporal scales in response to various natural and anthropogenic disturbances (Dietze & Clark 2008; Strigul 2012). Interplays between forest succession and disturbance regimes create complicated patch-mosaic and landscape patterns (Watt 1947; Chen & Popadiouk 2002; Pastor, Sharp & Wolter 2005; Liénard, Florescu & Strigul 2015a). The hierarchical patch-mosaic concept provides a conceptual framework to understand these complex phenomena (Wu & Loucks 1996). Non-stationarity of the disturbance regimes, caused by climatic changes and land-use practices, creates additional challenges in forest modelling as the changes occur at several hierarchical levels of biological organization and generate nonlinear responses (Levin 1999). Additionally, future climatic factors and disturbance regimes cannot be predicted precisely, leading to additional challenges for forest modelling (Wotton, Nock & Flannigan 2010; Chmura et al. 2011).

Existing national forest inventory programmes collect a large number of individual tree records on permanent plots and sample forested ecosystems uniformly across the landscape. Particular examples of such forest inventories in North America include the USDA Forest Inventory and Analysis Program (http://www.fia.fs.fed.us/) and the Quebec provincial forest inventory data base (http://www.mffp.gouv.qc.ca/forets/inventaire/), both of which have collected data since the 1970s. These databases provide unique opportunities to quantify and examine forest disturbances using a data-intensive approach that involves data mining and the development of stochastic models (Liénard, Gravel & Strigul 2015b).

In particular, one approach relying on Markov chain models has recently been developed to capture stand-level dynamics from forest inventories (Strigul et al. 2012). These models operate with probabilities of forest state transitions. Overall, Markov chain models for the forest stand dynamics represent a discrete time-space generalization of the classic patch- mosaic ecological models represented as continuous type models and traditionally formulated as advection equations (Levin & Paine 1974). The major advantage of using the Markov chain models is that they encompass small-, intermediate- and large-scale forests disturbances, while the continuous time models typically consider only major disturbances (Strigul et al. 2012). Markov chain models can be naturally linked with forest inventory data by considering every permanent forest plot as an independent realization of the underlying Markov chain process. However, estimation of transition probabilities using forest inventory data requires data mining and presents a substantial challenge due to the irregularities of forest surveys, as different time intervals often lapse between successive measurements of the same plot. Strigul et al. (2012) suggested two approaches to estimate forest transition probabilities, the first being to invert the stationary distribution and the second being to use repeated measurements; however, both approaches have practical limitations in application to available forest surveys. Liénard, Gravel & Strigul (2015b) have developed a Bayesian methodology to overcome problems related to irregularities in the time interval between successive measurements. This method allows estimating transition matrices of different macroscopic forest characteristics including biomass, basal area, stand age and shade tolerance index, and it employs multivariate statistics for studying multidimensional forest dynamics (Liénard, Gravel & Strigul 2015b) and forest succession (Liénard, Florescu & Strigul 2015a).

Markov chain models are traditionally used to predict future forest changes under the time-homogeneity (stationarity) assumption, meaning that transitional probabilities remain the same over the focal time horizon (Usher 1969, 1979; Waggoner & Stephens 1970; Stephens & Waggoner 1980; Logofet & Lesnaya 2000; Pastor, Sharp & Wolter 2005). This assumption could probably be justified on small and intermediate time-scales that might span from years to decades provided that climatic variables and disturbance regimes are relatively unchanged (Stephens & Waggoner 1980). However, time-homogeneous Markov chains as well as other models based on this assumption will likely provide unrealistic predictions in case disturbance or growth regimes change substantially (Shugart, Crow & Hett 1973; Pastor, Sharp & Wolter 2005). In particular, homogeneous Markov chain models may not be suitable for forest modelling in Quebec as general circulation models (GCMs) predict substantial and temperature increases in Canada over the next several decades (Flato et al. 2000). Models of fire rate variations typically couple historical data of fire occurrence with climatic variables, rely on GCMs to predict climatic variables in the future and use the past relationships to infer future fire rate (Flannigan et al. 2005; Bergeron et al. 2006; Drever et al. 2009; Hély et al. 2010; Wotton, Nock & Flannigan 2010; Loudermilk et al. 2013). Thus, burn rate predictions are highly dependent (i) on the historical fire data and atmospheric variables and (ii) on the GCM used to derive future environmental conditions. The uncertainties in these variables result in substantially different predictions among published models (Flannigan et al. 2009). Independent from burn rate changes, the greater atmospheric concentration of and increased temperatures have also been reported to boost forest growth (Curtis & Wang 1998; Norby et al. 1999; Ainsworth & Long 2005; Norby & Zak 2011; Dieleman et al. 2012; De Kauwe et al. 2014; Walker et al. 2014).

The goal of this paper was to assess the possible impact of climate change on forest biomass dynamics in Quebec. In order to achieve this goal, we extend data-intensive Markov chain models to include non-stationary disturbance and growth regimes. In this paper, we apply an original methodology to predict forest changes that will occur under different scenarios of future disturbance regimes, in particular forest fire and forest growth (Figs S1 and S2 in Supporting Information). We employ forest biomass as this stand characteristic is directly affected by disturbances of different magnitudes (Strigul et al. 2012). We first parameterize biomass transition matrices for different forest types in Quebec using the Gibbs sampling method developed by Liénard, Gravel & Strigul (2015b), and we assess the current disturbance rate through the analysis of these unconstrained matrices. We finally restrict our application to Quebec hardwood forests and consider two extreme models of future fire occurrence changes proposed in the literature (Bergeron et al. 2006; Drever et al. 2009). These two models were investigated with and without the potential growth enhancement resulting from higher temperature and concentration.

Materials and methods

Quebec Inventory

The Quebec inventory data set was started in 1970 and is still ongoing. Part of it consists of measurements of saplings and full-grown trees in permanent and temporary plots. We relied on mature tree measurements to assess the plot biomass based on individual tree diameter at breast height and Jenkins’ regression formula (Jenkins et al. 2003, please refer to Appendix S2 for details on the calculation).

Quebec encompasses several forest types which can be subdivided into hardwood forests as well as mixed forests in the northern temperate zone and into continuous boreal forests in the boreal zone (Saucier et al. 2003; Paquette & Messier 2011). Permanent plots cover these subzones in the Quebec forest inventory data base with 9621, 7663 and 11 969 plots belonging to hardwood, mixed and continuous boreal forests, respectively (Fig. 1). We computed biomass transition matrices for each of these subzones as well as the overall transition matrix.

Figure 1

Map of permanent forest plots in the Quebec inventory. The dot colors show the vegetation subzone into which each plot is classified.

Inhomogeneous Markov Chains for Forest Patch Dynamics

The patch dynamics model for forest biomass can be represented as a discrete stage-structured Markov chain model operating with biomass vector , where the biomass stages {1,2,…,m} characterize the forest stand (patch) development up to a certain maturity stage m as the biomass increases within the stand (Strigul et al. 2012). In this paper, we generalize this model by considering inhomogeneous Markov chains (Fig. 2), where transition probabilities change with time (eqn. 1)

(eqn 1)

where the elements are the probabilities of transition to the next biomass stage; are the probabilities of staying at the same stage; and are the probabilities of decreasing stand biomass, for one time step Δt. These transitions reflect the balance between biomass increase for growing trees and biomass loss due to disturbances. The zero coefficients above the superdiagonal reflect the assumption that the forest growth is slow relative to the frequency of surveys, so that biomass can only increase to its next stage (Strigul et al. 2012). The elements have simple mechanistic interpretation as they correspond to total disturbance probabilities, where a complete biomass loss occurs from clear-cuts, severe forest fires and other catastrophic events. The elements correspond to a broader class of events that cause the removal of a portion of the trees and which we refer to as diffused disturbance; such events range from stand thinning, partial forest fire, tree diseases, droughts, floods, windstorms and other semi-catastrophic events. Overall, the model does not distinguish between causes and reflects all natural and anthropogenic disturbances depending on how they impact forest development (Strigul et al. 2012).

Figure 2

A graph of stage transitions for one time step.

This inhomogeneous model preserves some of the properties of homogeneous Markov chain models, for example the Markov property, the general form of the Chapman–Kolmogorov equation and matrix multiplication for computing further states, that is . However, there are substantial differences, most noticeably, stationary states can no longer be investigated using the Perron–Frobenius theorem for irreducible non-negative matrices by simply computing the first eigenvector as in the homogeneous case (Strigul et al. 2012). In general, the theory of inhomogeneous Markov chains is much less developed than the theory of homogeneous Markov chains (Levin, Peres & Wilmer 2009; Florescu 2014), and our overall strategy in this paper is to approximate the time-dependent transition matrix T(t) by a sequence of matrices simulating future changes in disturbance regimes. Each matrix in the sequence represents a homogeneous Markov chain, which is employed to predict changes in the forest patch mosaic within a given time step (see also the 4 section).

Transition Matrix Estimation

Direct estimation of the transition probabilities of forest stands is challenging due to the highly irregular sampling intervals and varying number of permanent plots (Strigul et al. 2012). Following the methodology established in a previous work on a multidimensional case (Liénard, Gravel & Strigul 2015b), we use a Markov chain Monte Carlo (MCMC) algorithm to estimate the transition matrices. Specifically, Gibbs sampling was used as in the so-called ignorable data collection mechanism case of Pasanisi, Fu & Bousquet (2012). We provide in Appendix S3 a concise description of the algorithm sufficient for implementation purposes and redirect the reader to Liénard, Gravel & Strigul (2015b) for more details and justifications.

Disturbance Scenarios

We consider here an estimated n-year transition matrix, M, obtained using Gibbs sampling. The matrix M is entirely deduced from empirical data, and its shape is not constrained during MCMC application. Therefore, there is no guarantee that the matrix adopts the theoretical shape of T ( eqn 1), and the matrix coefficients are not assumed to be null in the matrix upper triangle (see also the 4 section).

We model temporal changes in total disturbance probabilities with an increment τ(t) (eqn. 2)

(eqn 2)

In  eqn 2, the original coefficients of the transition matrix, , are modified to reflect the altered probability of total disturbance. Specifically, τ(t) is introduced as an annual difference averaged over n, the number of years for which the matrix is computed, thus, the additional probability in the matrix is nτ(t). τ(t) is positive when total disturbance probability increases and negative for decreases.

Similarly, we model a change in diffused disturbances with an increment δ(t) (eqn. 3)

(eqn 3)

In  eqn 3, the additional terms change the probability of going back to any earlier stage. The terms indicating probability of diffused disturbance over n years, nδ(t), are removed from the diagonal and are reported as added fractions in the left-side columns of the matrix.

The additive inclusion of a disturbance term is in general well suited to model small variations in the probability of a specific class of catastrophic or semi-catastrophic events, without altering other disturbances. However, changes in forest growth are more easily understood with weighting existing transition probabilities, as they are usually expressed relative to baseline growth. We adopt the following formalism to model a relative change γ(t) in the forest growth rate (eqn. 4)

(eqn 4)

In this paper, we first study the stability of Markov chains long-term equilibriums under several alterations. In particular, we investigate the following cases:

1. small alterations of total disturbances (τ = ±0.01% in  eqn 4) and diffused disturbances (δ = ±0.01% in  eqn 3),
2. complete removal of the total disturbance () and doubling of their probabilities (),
3. small relative decrease (γ = 0.95) and increase (γ = 1.05) of growth rate.

We then select upper and lower bound scenarios from recently published studies that provide future estimates of forest fire and growth rate alterations caused by climate change (see an illustrative scheme at Fig. S2). Highly variable burn rate scenarios have been proposed in different forest types depending on underlying assumptions and models employed (see Flannigan et al. 2009 for a review). We selected two scenarios (Fig. 3a) which provide lower and upper bounds among the published predictions in Quebec hardwood forests (see also 4):

Figure 3

Climate-change scenarios and predicted average biomass changes. (a) Forest fires scenarios (black, left y-axis) and possible growth enhancement (green, right y-axis) in hardwood forests. The two burn rate scenarios were derived from Bergeron et al. (2006); Drever et al. (2009) (see Methods for details). (b) Average biomass predicted with no change in current disturbance regime (solid line) and in the two future burn rate scenarios computed with or without the addition of enhanced growth. Historical biomass records from the forest inventory are displayed in 10-year bins with standard error of the mean.

1. ‘Low burn rate’ scenario: the first scenario has been extracted from the study of Bergeron and colleagues (Site 6 from table 2 of Bergeron et al. 2006). It forecasts a decrease of 0.006% in the annual fire rate when the atmospheric concentration of is doubled (fire rate decreasing from 0.036 to 0.030%), and a further decrease of 0.010% when the concentration of is tripled (from 0.030 to 0.020%). The predicted changes of are thought to happen around 2040-2060 for the scenario, and around 2080–2100 for the scenario (Flato et al. 2000). In the absence of precise predictions, except for these two landmark points, we choose to model the annual fire rate decrease as a piecewise linear function:
   (eqn 5)
2. ‘High burn rate’ scenario: the second scenario has been chosen as the most dramatic increase from the pool of models of Drever and colleagues (model ‘HADCM3-A2’ from figure 3c in Drever et al. 2009). This scenario predicts an increase of the annual area burned of roughly 3 times by 2050, and up to 10 times by 2090. Coupling these predictions with the current burn rate of 0.036% (the same as in the ‘low burn rate’ scenario), we model the annual fire rate increase as:
   (eqn 6)

We also investigated polynomial and steps progression of forest fires, which led to overall similar results compared to the piecewise linear increase described above (Appendix S7).

Finally, we include a possible enhanced growth effect resulting from higher temperature and greater availability of atmospheric (Chmura et al. 2011). Unlike the fire rate, we could not find quantitative predictions specific to hardwood forests in Quebec, so we extrapolated results from various sources to generate a quantitative estimate. We relied in particular on meta-analyses of data gathered from Free-Air Enrichment (FACE) experiments that report average growth increases around 20–30% (Ainsworth & Long 2005; Norby & Zak 2011). This is in general accordance with prior experiments on seedlings or open-top setups (Curtis & Wang 1998; Norby et al. 1999) and with simulations predicting increases of 10–30% depending on species (Matala et al. 2005; Garcia-Gonzalo et al. 2007). The meta-analysis of Dieleman et al. (2012) investigating the dual effects of temperature and greater concentrations further reports changes in total biomass around 10–80%, with an average increase of about 30%. In accordance with all these results, and to avoid overestimation of boosted growth, we settled on a rather conservative growth enhancement of 20% occurring through a linear increase until the concentration is reached around 2090. Formally, this higher growth rate condition corresponds to:

(eqn 7)

Software Availability

All procedures were written in R (R Core Team 2015), and the source code is available in supplementary materials.

Results

Biomass Disturbance in Quebec Forests

Following the Gibbs sampling methodology presented in Methods and in Appendix S4, we obtained 3-year transition matrices describing the biomass dynamics in the specific subzones of hardwood forests, mixed forests and continuous boreal forests as well as the overall matrix of Quebec forests (Fig. S3). The transition matrices display similar patterns across Quebec. After 3 years, the probability of a stand remaining within the same biomass range is in the 40–70% range (high values in the diagonal in Fig. S3). The probability of a 2000–6000 kg increase of biomass is close to 20% (relatively high coefficients above the diagonal in Fig. S3). Disturbances leading to a biomass decrease are overall unlikely and sum up to less than 10% in most cases, except for the very high-biomass stands where they can reach higher values (overall small coefficients below the diagonal in Fig. S3). This pattern is in good agreement with the theoretical matrix (T, in eqn eqn 1) and in accordance with previous works (Strigul et al. 2012; Liénard, Gravel & Strigul 2015b). These matrices exhibit their biggest contrasts for high biomass transitions (bottom rows of the matrices); as high biomass stands are also the less numerous stands in the data base, the influence of these transition coefficients is less important in determining future states of the forests, mitigating the differences between the subzones. The matrices were overall remarkably similar, indicating that the disturbance regimes for biomass are rather similar throughout Quebec.

We assessed the disturbance rate in Quebec forests since 1970 by multiplying the biomass distribution with the total disturbance probability from the different transition matrices (Table 1), excluding plots in the first biomass state (with < 2000 ). Standard deviation of the current disturbance rates was assessed by bootstrapping (Efron 1979). The annual probabilities of total disturbances were in the interval 0.194–0.283% (Table 1). These probabilities are slightly higher than the current burn rates in most Quebec forests reported to be 0.036–0.258% (Bergeron et al. 2006), which is expected as the matrices include natural and anthropogenic disturbances.

Table 1. Annual rate of total disturbance (mean ± SD) per forest type and per decade

	1970–1979	1980–1989	1990–1999	2000–2007
Mixed	0.2832 ± 0.0009	0.2642 ± 0.0011	0.2497 ± 0.0007	0.2556 ± 0.0008
Boreal	0.2587 ± 0.0011	0.2376 ± 0.0010	0.2422 ± 0.0008	0.2420 ± 0.0011
Hardwood	0.2262 ± 0.0005	0.2006 ± 0.0006	0.1947 ± 0.0005	0:1944 ± 0.0006

Our study of the disturbance rate in different decades also showed that apart from a decrease from the 1970s to the 1980s, the total disturbance rate has remained stable over time (Table 1). This early decrease might correspond to short- and medium-term impacts of public policies protecting Quebec forests that started to take affect in the 1970s (Coulombe et al. 2004) or reflect advances in fire prevention and control. The disturbance rate was overall similar throughout subzones (Table 1) with a slightly higher disturbance rate in mixed and continuous boreal forests compared to hardwood forests. At the same time, the burn rates are highly variable for these forest types (Bergeron et al. 2006). This discrepancy could indicate that the total disturbance rate is capped and that fire disturbances are effectively in competition with other types of catastrophic disturbance, either natural or anthropogenic. For example, a stand destroyed by a forest fire cannot be destroyed by disease or insect outbreaks and vice versa.

Sensitivity Analysis

A stability analysis on long-term equilibriums was conducted with several hypothetical changes of disturbances regimes, and the details are presented in Appendix S5. In particular, we investigated the consequences on long-term distribution and convergence speed of diffused disturbances and total disturbances as well as growth rate alterations (Fig. S4). The long-term distributions and convergence rates were consistently stable with respect to alterations in the disturbance and growth regimes. These results support our further study of dynamics under climate-induced changes of similar magnitudes.

Global Warming Scenarios

The influence of climate change on forest burn rate scenarios was based on published studies (Bergeron et al. 2006; Drever et al. 2009), providing lower and upper bounds of future burn rates. In particular, we extracted two burn rate models: ‘low burn rate’ scenario (derived from Bergeron et al. 2006) and ‘high burn rate’ scenario (derived from Drever et al. 2009). We also investigated the possible coupling of these scenarios with an increase in growth rate, which we refer to as the ‘higher growth’ condition.

The average biomass in Quebec hardwood forests displays distinct trajectories and dynamics under the different scenarios (Fig. 3b). The unaltered projections show a continuous increase in biomass until 2050 which is consistent with the trend observed in the data base (historical records in Fig. 3b). In the ‘low burn rate’ scenario, the average biomass is barely discernible from the unaltered predictions; the slight decrease in burn rate considered in this scenario does not significantly affect the average biomass dynamics. In the ‘high burn rate’ scenario, a departing from the unaltered predictions is apparent after reaching the threshold. The addition of the ‘higher growth’ condition substantially changed the dynamics with a marked increase in both scenarios. In the ‘low burn rate’ scenario, the boosted growth is able to sustain an increasing trend until 2090, while in the ‘high burn rate’ scenario the boosted growth negates the effects of increased burn rate for the average biomass.

Biomass distributions have been computed for years 2050 and 2090, as these dates correspond respectively to the doubling and tripling of atmospheric (Fig. 4 in main text and S5). In 2050, both scenarios predict biomass distributions nearly identical to the predicted biomass obtained without alteration of the transition matrix (Figs 4a and S5). This is in line with the average biomass displayed in Fig. 3b. However, in the ‘high burn rate’ scenarios, the biomass distribution displays a substantial increase of very low biomass stands which become 3–4 times more abundant. Although the average biomass in ‘high burn rate’ with ‘higher growth’ is similar to the unaltered predictions (Fig. 3b), the biomass distributions are different with a higher prevalence of very low biomass stands in the former case (Fig. 4b).

Figure 4

Biomass distributions at horizon 2050 (a) and 2090 (b) with no change and in the low and high burn rate scenarios. These predictions have been obtained using the ‘higher growth’ condition resulting from a greater availability (predictions without the higher growth are provided in Figure S5). Arrows indicate the mean biomass.

Discussion

Dynamic Disturbance Modelling with Markov Chains Process

Markov chains are widely used in forest ecology as they capture transitions between forest states and allow a simple mechanistic interpretation. Markov chains have been used to describe transitions between different forest stages at different scales including the landscape level (Logofet & Lesnaya 2000; Korotkov, Logofet & Loreau 2001), stand-level succession (Usher 1969, 1979), and species replacement and abundance within the forest stands (Waggoner & Stephens 1970; Stephens & Waggoner 1980). More recent developments include composite models where stochastic fire occurrences can reinitialize the chain to its first state (which is the equivalent of adding a constant, time-independent probability τ in our framework, see e.g. James et al. 2007). Most of these studies explicitly or implicitly assume that the Markov chain is homogeneous. Only a few studies expand this framework and include inhomogeneous effects such as age-dependent transitions of forest stands (Pastor, Sharp & Wolter 2005). The time-homogeneous assumption limits the scope of model application to small–intermediate time intervals where changes in environmental conditions are not significant. This is a well-known limitation of the Markov chain framework that was broadly discussed in the ecological literature (Shugart, Crow & Hett 1973; Usher 1979; Stephens & Waggoner 1980; Li 1995; Pastor, Sharp & Wolter 2005; James et al. 2007). In this study, we have developed an original approach to take into account time-inhomogeneous effects within the Markov chain modelling framework. Inhomogeneous Markov chain models have been employed in other areas such as biomedical research (Perez-Ocon, Ruiz-Castro & Gamiz-Perez 2001; Hubbard, Inoue & Fann 2008), landscape (Li 1995), pollution (Lagona et al. 2011), and climatic (Rajagopalan, Lall & Tarboton 1996; Hughes, Guttorp & Charles 1999) modelling. Overall, the broadly employed stationary Markov chain models work particularly well when the homogeneity assumption is justified, that is when the environmental conditions do not change substantially. However, the more quickly and significantly environmental conditions change the less accurate predictions for homogeneous Markov chains will be. The major advantage of the developed method consists of allowing predictions for how the forest will change in response to climate change scenarios that stationary models simply cannot predict. At the same time, our approach allows for understanding and quantitative evaluation of the errors that emerge when the homogeneous Markov chain models are employed to non-stationary ecological systems.

In the presented model of Quebec forests, there are small discrepancies between the estimated transition matrices (Fig. S3) and the theoretical form (eqn eqn 1). In particular, eqn eqn 1 has zero coefficients in the upper triangle above the superdiagonal elements . These zero coefficients reflect the model assumption that the forest stand is observed frequently enough relative to the succession process that the forest does not grow through two consecutive measurements (Strigul et al. 2012). However, in the estimated matrices presented in this and other publications (e.g. tables 2 and 3 in Strigul et al. 2012 and figure 3 in Liénard, Gravel & Strigul 2015b), nonzero coefficients seldom occur in the upper part, above the diagonal. This discrepancy is mainly related to the methodology of forest surveys. Indeed, in Quebec forest service survey manuals, trees are recorded only if their diameter at breast height is larger than 9 cm (7 cm for the USDA forest inventory). The estimated biomass can experience a substantial jump in a case where many smaller trees existed but were not counted in a survey, and then were counted in the next survey as they passed the breast height diameter threshold. This scenario and the general noise inherent with experimental data lead to nonzero probabilities sometimes observed in the upper right matrices. One way to mitigate these discrepancies would be to smooth the transition matrix coefficients, another way would be to look for outliers in the incomplete temporal sequences and to remove them. Both methods involve choosing arbitrary parameters, that is a threshold for outlier removal and kernel parameters for matrix smoothing. However, our simulations indicate that the model predictions do not change significantly from small modifications of the matrix. In this paper, we thus opted to simply acknowledge this discrepancy and operate without modifying the unconstrained transition matrices.

Climate Change Scenarios Predictions

One of the substantial advantages of this model is that it allows us to conduct straightforward sensitivity analysis and quantitatively assess variability of climate change scenarios. This allowed us to predict biomass changes for the most extreme climate change scenarios and to assess the outcomes of their variability. Our study predicts relatively small effects of the climate change scenarios on biomass dynamics for the next 30 years, and all trends demonstrated biomass increases until 2050 (Figs 3 and 4). The dominance of increasing biomass until 2050 as a legacy of previous land-use practices has been shown in our previous analysis of the Quebec forest inventory (Liénard, Gravel & Strigul 2015b) and is consistent with other recent studies throughout North America (e.g. Loudermilk et al. 2013; Rhemtulla, Mladenoff & Clayton 2009; Thompson et al. 2011, respectively in southwestern, north-central and northeastern US). Predictions differ between considered scenarios at the horizon of 2090. Indeed, the model predicts overall negligible changes in the ‘low burn rate’ scenario and a relative decrease of biomass in the ‘high burn rate’ scenario. In both scenarios, the ‘higher growth’ condition results in substantially increased biomass. On average, this increase is counterbalanced in the ‘high burn rate’ scenario, but in this scenario very low biomass stands also become more abundant due to more frequent forest fires.

It is worth noting that other published forest fire prognoses provide forecasts within our low and high burn rate scenarios, under in Southern Quebec. The two considered scenarios thus represent the opposing ends of the spectrum for fire rate predictions in Quebec. For example, most scenarios from Drever et al. (2009) predict a milder increase of 1–3 times the current burn rate. Flannigan et al. (2005) predicts 1.62–1.72 times the current burn rate, in a broader geographical area encompassing both mixed and hardwood forests and using two different GCMs. Similarly, Wotton, Nock & Flannigan (2010) predicts in Quebec changes of 1.33–2.40 times the current burn rate at . Here, the ‘low burn rate’ scenario results in a decrease of roughly 1.5 times and the ‘high burn rate’ scenario results in an increase of 10 times; hence, we can expect a probable outcome in-between these two extreme scenarios. The main prediction of an overall weak impact of fire-induced changes by 2050 is similar in all scenarios studied here and can thus be generalized to intermediate burn rate scenarios. The predictions for year 2090 however differ noticeably among the scenarios, which assures the need for precise predictions.

The results are specific to Quebec hardwood forests and could vary substantially in areas where the impact of forest fire is more pronounced (Flannigan et al. 2009) leading to different outcomes as a result of climate change. However, our methodology is general and can be applied in any area where a data set with repeated consistent measurements is available. Climate changes are also predicted to move the climatic zones boundaries (McKenney et al. 2007). On the one hand, the presented biomass model is not spatially explicit, and the biomass dynamics revealed overall similar transitions (Fig. S3) and disturbance probability (Table 1) across Quebec domains, mitigating the possible bias due to moving boundaries. However, the included changes in burn rates are specific to forest types (Bergeron et al. 2006) as well as climate-related and anthropogenic actions (Wotton, Nock & Flannigan 2010), making our predictions specific to Quebec hardwood forests and not readily applicable to other vegetation subzones.

Limitation of the Approach

There is an important variability associated with the prediction of climate change scenarios that go as far into the future as 2090 and directly affect biomass long-term predictions. Our strategy to deal with uncertainty in the climate change scenarios is twofold. First, we selected extreme scenarios among the estimates available in the literature and thus considered the lowest and highest burn rate projections from a pool of 10 scenarios (extracted from Flannigan et al. 2005; Bergeron et al. 2006; Drever et al. 2009; Wotton, Nock & Flannigan 2010), and excluded/included the possible forest growth boost due to increased concentration (in accordance with Curtis & Wang 1998; Norby et al. 1999; Ainsworth & Long 2005; Matala et al. 2005; Garcia-Gonzalo et al. 2007; Norby & Zak 2011; Dieleman et al. 2012). By considering only the combinations of these scenarios (same/higher growth and low/high burn rate, cf. Fig. S2), our predictions thus encompass the whole range of future conditions currently considered plausible. Secondly, we also conducted a theoretical sensitivity analysis on the transition matrix alterations. This analysis showed that the predictions of the model are robust to small and intermediate changes in the disturbance and growth terms (Fig. S4). The overall conclusion of this study should thus hold true even if the precise realization of the considered scenarios does not occur.

The presented approach is the first that allows researches to consider an evolving environment in a Markov chain framework. It is thus limited by our capacity to formalize climate change consequences in probabilistic terms. In this study, we modelled two well-studied anticipated changes with altered forest fire occurrences and growth rate through higher availability of . However, many other less-studied consequences of climate change are also anticipated, including varying soil moisture, evapotranspiration and other temperature-related effects, drought-related effects, epidemics of insect, disease outbreaks, invasive species proliferation and nitrogen cycle alterations (Huang et al. 2010; Chmura et al. 2011; Bernal et al. 2012; Hicke et al. 2012).

By operating on transitions, our modelling framework is able to unify any effect that can be understood in terms of probabilistic repercussions on biomass, and can thus be extended to predict biomass variations due to other climate change consequences. In particular, most of the climate-related consequences enumerated above could be included as diffused disturbances (eqn eqn 3) or altered growth (eqn eqn 4). When adding more climate-related effects in our framework, care must be exercised to avoid bias by properly formalizing the probabilities of correlated effects (Dieleman et al. 2012).

Forest Management Implications

The prognosis that even the most extreme climate change scenario will not alter the biomass of Quebec hardwood forests more than about 5% in the next century provides a scientific ground for long-term forest management decisions. In addition, this modelling approach has a potential to be developed as a practical forest management tool. Effects of different land-use practices and harvesting methods can be included in the biomass transition matrices and, therefore, simulate joint effects of climate change and different silvicultural operations. That would allow the comparison of the effects of different silvicultural operations on overall carbon footprint at different time horizons (short, intermediate and long), and for different forest types. Some silvicultural operations, such as clear cutting and shelter wood cutting, can be immediately simulated using non-homogeneous Markov chains. Other silvicultural operations such as selection and intermediate cuttings or prescribed fire would require additional research to estimate their effects on stand biomass before being incorporated in the model. The presented approach can also be used with other modelling tools, as for example stand-age dependent Markov chain models specifically developed to simulate forest harvesting regimes (Pastor, Sharp & Wolter 2005). In particular, the change of the matrix coefficients should allow to simulate intermediate cutting operations used in ecological forestry to improve forest quality.

Acknowledgements

This work was partially supported by a grant from the Simons Foundation (#283770 to N.S.) and a Washington State University New Faculty SEED grant. We are grateful to Leslie New, John Harrison, Kendra Lynn and John Bishop for useful comments to the manuscript.

Data accessibility

The Quebec data base is available online from the Ministére des Forêts, de la Faune et des Parcs: http://www.mffp.gouv.qc.ca/english/products-services/index.jsp (English version) and http://www.mffp.gouv.qc.ca/forets/inventaire/fiches/donnees-descriptives-pet-poe.jsp.