State-and-Transition Models: Conceptual Versus Simulation Perspectives, Usefulness and Breadth of Use, and Land Management Applications

Abstract

State-and-Transition Simulation Modeling (STSM) is a quantitative analysis method that can consolidate a wide array of resource management issues under a “what-if” scenario exercise. STSM can be seen as an ensemble of models, such as climate models, ecological models, and economic models that incorporate human dimensions and management options. This chapter presents STSM as a tool to help synthesize information on social–ecological systems and to investigate some of the management issues associated with exotic annual Bromus species, which have been described elsewhere in this book. Definitions, terminology, and perspectives on conceptual and computer-simulated stochastic state-and-transition models are given first, followed by a brief review of past STSM studies relevant to the management of Bromus species. A detailed case study illustrates the usefulness of STSM for land management. As a whole, this chapter is intended to demonstrate how STSM can help both managers and scientists: (a) determine efficient resource allocation for monitoring nonnative grasses; (b) evaluate sources of uncertainty in model simulation results involving expert opinion, and their consequences for management decisions; and (c) provide insight into the consequences of predicted local climate change effects on ecological systems invaded by exotic annual Bromus species.

Appendix

1.1 Calculating the Palmer Drought Severity Index

The Palmer Drought Severity Index (PDSI) time series was used to calculate the temporal multipliers for replacement fire, drought, annual grass invasion, and tree invasion. Drought is a major influence for these disturbances. PDSI measures long-term soil drought and is updated monthly (Palmer 1965; Heddinghaus and Sabol 1991). Positive values indicate above average soil moisture (>3 is very wet), whereas negative values represent droughty soil (<−3 is very dry). A PDSI of zero is average soil moisture. The formula for PDSI at time t (month) is as follows:

$$ {\mathrm{PDSI}}_t=0.897\times {\mathrm{PDSI}}_{t-1}+\left({k}_t/3\right)\times \left({P}_t-{\underset{\_}{P}}_t\right) $$

(13.1)

where P _t is precipitation during month t, P _t is average (historic) precipitation for month t, and k is a monthly climatic coefficient that weighs the local importance of (P _t  − P _t ) (Palmer 1965). For example, k might imply that (P _t  − P _t ) in January does not contribute as much to PDSI as the same deviation in precipitation observed in August (Palmer 1965). Although we downloaded monthly precipitation values and obtained monthly P _t from historic precipitation data (respectively, month, precipitation [mm/day]: January, 0.8004; February, 0.8368; March, 1.0234; April, 0.9310; May, 0.9612; June, 0.6130; July, 0.6356; August, 0.7394; September, 0.6876; October, 0.7502; November, 0.7476; December, 0.6858), the value of k _t is unknown and requires complicated field estimation based, among others, on evapotranspiration (Palmer 1965). (To remove this complication and need for a heuristic equation, future projects will use the Standard Precipitation Index [Hayes et al. 1999]). Therefore, we made several arbitrary assumptions to imitate k using the month’s temperature differential. Specifically,

$$ {k}_t/3=1.5\times \left(1-{e}^{-0.15\times \left( MaxT-Tt\right)}\right) $$

(13.2)

where MaxT = 31 (°C) is the maximum temperature observed, and T _t is the average temperature during month t. In this heuristic equation, higher temperatures cause smaller values to multiply (P _t  − P _t ) when monthly precipitation is higher and thus PDSI becomes smaller (more evapotranspiration). The coefficients 1.5 and −0.15 are fitting constants we iteratively selected that allow the PDSI to vary within the observed range and be responsive to changes in precipitation, primarily, and secondarily to temperature. Using the latest observed monthly PDSI from March 2012 as the first PDSI _t−1 , we estimated future monthly PSDIs per replicate for 75 years using Eqs. (13.1) and (13.2) for both without and with climate change. Compared to the PDSI replicates without climate change, it is noticeable that three temporal replicates of PDSI estimated for climate change effects were drier during certain decades only (replicates #1, 4, and 5), whereas the third replicate was wetter and the second replicate neutral (Fig. 13.6).

Because PDSI can be negative and the STSM software requires positive values, heuristic functions (arbitrary coefficients) were developed for drought, replacement fire, invasive annual grass expansion, and tree expansion that transformed negative values into positive values while maintaining the role of PDSI on the intensity of the disturbance. Not many flexible functions allow the conversion of negative values into positive ones while also accepting positive values; therefore, these curve fitting requirements led us to adopt functions with exponential components that could be easily calibrated. These functions do not calculate the rate of the disturbance, which is found in the STSM, but the temporal variability of the disturbance. All equations generated non-dimensional values and the final temporal multipliers were also non-dimensional.

1.2 Drought Disturbance

Because PDSI can be negative, therefore incompatible with PATH’s format for temporal multipliers, we chose a negative exponential function for drought to create positive values that increased exponentially with more negative (drier) PDSI values:

$$ \mathrm{Yearly}\;\mathrm{drought}\;\mathrm{variability}\;\mathrm{factor}=0.6\times {e^{-0.6}}^{\times \mathrm{PDSI}} $$

(13.3)

The parameters of this function (0.6 and −0.6) were chosen such that PDSI values close to −3 (very dry) were slightly greater than 3 (actually, 3.63) and that very severe droughts with PDSI of −5 (extreme drought) translated into slightly more than doubling of the function (12). Another consideration for curve fitting was that a mild drought characterized by a PDSI of −1 would be about equal to a neutral value of 1. Equation 13.3 is not the final temporal multiplier, however, because it is not divided by its average. In the absence of climate change effects, yearly values of Eq. (13.1) were divided by their temporal average over 75 years, whereas each yearly value of Eq. (13.3) with climate change was divided by the no-climate change average to reflect the hypothesis of altered levels.

1.3 Annual Grass Invasion and Tree Invasion Disturbances

The temporal multipliers for invasive annual grass expansion and tree expansion were calculated from two heuristic Gompertz equations (not including the CO₂ fertilization). The Gompertz equation is highly flexible for curve fitting and a special case of it is the negative exponential:

$$ \begin{array}{l}\mathrm{Yearly}\;\mathrm{annual}\;\mathrm{grass}\;\mathrm{expansion}\;\mathrm{variability}\;\mathrm{factor}=\hfill \\ {} 4.5\times {e}^{-2\times \exp \left(-0.75\times \left(\mathrm{PDSI}+1\right)\right)}\times {\mathrm{TMCO}}_2\hfill \end{array} $$

(13.4)

$$ \begin{array}{l}\mathrm{Yearly}\;\mathrm{tree}\;\mathrm{expansion}\;\mathrm{variability}\;\mathrm{factor}=\hfill \\ {} 2.5\times {e}^{-2\times \exp \left(-0.75\times \left(\mathrm{PDSI}+1\right)\right)}\times {\left({\mathrm{TMCO}}_2\right)}^{0.5}\hfill \end{array} $$

(13.5)

where TMCO₂ is the temporal multiplier for CO₂ levels, which is <2 for any yearly value with climate change and equal to one without climate change. In accordance with our hypothesized relationship between species expansion and soil moisture and CO₂ levels, the effect of CO₂ levels as expressed by its temporal multiplier (between 0 and 1) on variability is proportional, whereas the effect of PDSI is exponential (i.e., greater). We arbitrarily dampened the effect of CO₂ fertilization on trees by taking the square root of the CO₂ temporal multiplier. The Gompertz equations allow for some expansion during even dry years (PDSI < 0), average expansion (temporal multiplier close to 1) during average moisture years, and a rapid rise of expansion (multiplier increasing to 4.5 and 2.5), respectively, for invasive annual grass expansion and tree expansion during very wet years. The parameters 4.5 and 2.5 were chosen to match values from the initial Park’s study by Provencher et al. (2013). Equations 13.4 and 13.5 are not temporal multipliers, however, because they are not divided by their averages. In the absence of climate change effects, yearly values of Eqs. (13.4) and (13.5), respectively, were each divided by their temporal average over 75 years, whereas each yearly value of Eqs. (13.4) and (13.5) with climate change, respectively, was divided by the no-climate change average to reflect the hypothesis of altered levels.

1.4 Fire

The shrubland–woodland fire temporal multipliers considered the roles of 3 years of PDSI, more specifically that fine fuels will more likely burn in the current dry year immediately following two previous and consecutive wetter-than-average years where fine fuels accumulated. The equation to calculate the temporal multipliers from shrubland fire contained two Gompertz functions to account for 3 years of PDSI:

$$ \begin{array}{l}\mathrm{Yearly}\;\mathrm{shrubland}-\mathrm{woodland}\;\mathrm{area}\;\mathrm{burned}\;\mathrm{variability}\;\mathrm{factor}\hfill \\ {} =\mathrm{MaxFire}\times {e^{-3\times \exp -(0.7\times \mathrm{PDSI}t-1}}^{+\left(1-0.7\right)\times \mathrm{PDSI}t-2)}\times \left(1-{e}^{-3\times \exp \left(-2\times \mathrm{PDSI}t\right)}\right)\hfill \end{array} $$

(13.6)

where MaxFire = 1547 hectares and is 10 % of the area sum of all shrubland–woodland ecological systems. Equation 13.6 combines two Gompertz functions to accommodate negative and positive values of PDSI. The first part of Eq. (13.6) after MaxFire, representing fine fuels production, is a classic Gompertz function where a weighted sum is applied to soil moisture during 2 previous years (70 % of PDSI in year t−1 and 30 % of PDSI in year t−2). Wetter years (PDSI > 0) increase the value of this function (fine fuels accumulation) to a maximum of one. The first part is multiplied by the second function representing the current year, which is one minus another Gompertz function bound between zero and one. Increasingly drier soil moisture (PDSI < 0) causes the second part of Eq. (13.6) to increase to a maximum of one (maximum ignition probability). The PDSI values from the scenarios without and with climate change were used to calculate future area burned. Equation 13.6 is not the final temporal multiplier, however, because it is not divided by its average. In the absence of climate change effects, yearly values of Eq. (13.6) were divided by their temporal average over 75 years, whereas each yearly value of Eq. (13.6) with climate change was divided by the no-climate change average to reflect the hypothesis of altered levels.