State-and-Transition Models in Geomorphology
Final version published in Catena 153: 168-181,
Jonathan D. Phillipsa*
Chris Van Dykeb
aEarth Surface Systems Program
Department of Geography
University of Kentucky
Lexington, KY 40506-0027
bKentucky Transportation Center
University of Kentucky
Lexington, KY, 40506-0281
State-and-transition models (STM) are used to describe, model, interpret, and predict
when landscapes will undergo a qualitative state change. Although rangeland ecologists
pioneered STMs, geomorphological STM-type models were developed prior to and
independently of ecological STMs. This study categorized 47 geomorphological STMs
according to whether they were: based on single or multiple study areas; primarily for
description and interpretation or predictive and prescriptive use; explicitly concerned with
complex system dynamics; and the role of biogeomorphic interactions in the model.
Each STM was represented as a graph and the structure identified. Spectral radii were
calculated to measure the complexity of each STM. Although STMs are associated with
conceptual frameworks that recognize the possibility of nonequilibrium, alternative
states, and path dependency, results show that an explicit concern with complexity does
not necessarily lead to the identification of more states and transitions, or a more
complex transition pattern. The purpose for which a STM was created, as well as the
number of study sites it can be applied to, also had little bearing on the models’
complexity. This review suggests that geomorphic STMs, rather than being used to fit
explanations about landscape evolution into predefined theoretical categories, are
veridical representations of empirical observations. Although STMs are particularly
useful for grasping the biogeomorphological dynamics of landscapes, this review
indicates their utility is not limited to biogeomorphology or to systems with a strong
ecological imprint. Time scales involved in geomorphic change can make it difficult to
observe a large number of states and transitions, which may constrain what types of
STM structure can be identified, as the number of observed states and transitions
required to develop particular graph structures varies widely.
Key words: state-and-transition models; geomorphic systems; graph theory; geomorphic
There is a well-established consensus among geomorphologists, ecologists, and other
biophysical scientists which holds that understanding the dynamics of landscape change
demands knowledge of the recursive, multi-scale interactions among abiotic and biotic
states and processes. What complicates efforts to interpret landscape adjustments is the
fact that landscape states and biophysical processes operate and interact at multiple,
sometimes disparate, spatial and temporal scales, either naturally or through human
interventions (e.g., Ashmore, 2015; Bestelmeyer et al., 2015; Lane and Richards, 1997;
Okin et al., 2015; Van Dyke, 2015; Wainwright et al., 2011). Recognizing this,
researchers have devised a number of conceptual strategies to delineate the
relationships among processes and states to predict geomorphic transitions. Some of
these have been straightforward, relying on linear sequence of stages to explain
landscape adjustments. For example, models depicting channel evolution,
biogeomorphic succession, or the cyclical stages implicated in landscape evolution have
represented change as a single-path, predictable process (e.g., Corenblit et al, 2009;
Schumm et al., 1984; Simon and Rinaldi, 2006). Other researchers have incorporated
greater complexity into their models to present a fuller picture of geomorphic dynamics.
These models represent landscapes as complex networks that support multiple states
and transitional pathways — although some transitions occur linearly, others can arise
due to nonlinear, spatially disaggregated relationships among the landscape’s structural
and functional components (e.g., Rountree et al., 2000; Gurnell and Petts, 2002; Hesp,
2002; Bestelmeyer et al., 2003; Wyrick and Pasternak, 2014).
State in general refers to the condition or configuration (i.e., morphology) of a system.
Geomorphic state transitions occur when changes in form-process dynamics produce a
qualitatively different landform, landscape unit, or geomorphic environment. For
instance, a simple increase or decrease in coastal erosion rates would not qualify as a
state transition. However, if the system shifted from a stable or net accretional to an
eroding condition — or vice versa — it would count as a state transition. In this scenario,
other zones or landforms (e.g., nearshore, beach, dunes, marshes) may undergo state
transitions due to changing erosion-deposition process regimes.
Over the past 30 years, rangeland ecologists and other biophysical scientists have
increasingly turned to state-and transition models (STMs) to represent and analyze state
changes in ecological systems (Bestelmeyer et al., 2003; Twidwell et al. 2013; Westoby
et al., 1989). They consist of box-and-arrow diagrams coupled with expository narratives
that offer detailed accounts of possible landscape states and the underlying biophysical
dynamics that drive state transitions. These models are underpinned by in-depth
fieldwork and expert knowledge of landscape dynamics, and lend themselves to
qualitative or quantitative interpretation using graph and network analysis; interaction,
transition, or adjacency matrices; or causal models (Phillips, 2011a; Phillips et al., 2015;
Thompson et al., 2016). STMs have quickly become an essential tool for resource
management agencies in the United States and around the world (Twidwell et al., 2013).
As such, it is imperative to understand the role state-transition thinking has played in
geomorphology. Accordingly, this study reviews the application of STMs in
geomorphology by describing the range of geomorphic models that have either been
identified by their authors as STMs or have clear structural and epistemological affinities
with STMs. In conducting a meta-analysis of published models, our intention is to
demonstrate that state-transition frameworks are tools that have commonly been used
by geomorphologists; scrutinize the capacity of STMs to represent a range of simple and
complex forms of adjustment in geomorphic systems; characterize the graph structures
associated with our case studies; determine the number of observations of states and
transitions needed to produce specific graph structures; and argue for the expanded use
of STMs by geomorphologists to facilitate interdisciplinary collaborations.
1.1. Ecological state transition models
Rangeland ecologists developed STMs after recognizing the limitations of classical
ecological or range succession models to explain vegetation change. The latter
proposed that a rangeland — in the absence of grazing — adjusts toward a single climax
state (Westoby et al., 1989). Theoretical and empirical work has demonstrated that
rangeland sites may exhibit multiple vegetation states owing to complex, multivariate
interactions among existing landscape features and abiotic and biotic processes (see
Westoby et al., 1989; Bestelmeyer et al., 2003; Peters et al., 2015; for overviews of
STMs, see also Hobbs, 1994; Briske et al., 2008; Van Dyke, 2015). The adoption of
STMs has emerged from a growing body of empirical work that has demonstrated the
profound impacts nonlinear ecological dynamics have had on ecosystem management
practices, both within the United States and globally. Agencies within the U.S.
Department of Agriculture use Ecological Site Descriptions (ESDs), which include STMs,
to assist with the identification, monitoring, evaluation, and management of rangelands
(Tidwell, 2013). While STMs have been viewed as increasingly authoritative tools within
a variety of institutional settings, they have also grown in popularity among ecologists
and other academic researchers because they are useful for cataloguing and
synthesizing large quantities of information about landscape dynamics, which in turn can
inform management and restoration decision making (e.g., van der Wal, 2006;
Hernstrom et al., 2007; Czembor and Vesk, 2009; Zweig and Kitchens, 2009;
Creutzberg et al., 2015).
Before highlighting the ways in which STMs have been applied to the analysis of
geomorphic systems, we take a closer look at an example STM from the U.S. Natural
Resources Conservation Service to clarify the epistemological underpinnings of state-
transition thinking and provide a better understanding of the data used to inform their
development. Ecological sites are classified based on physiographic factors such as soil
properties, slope, climate, and geomorphology, and on the vegetation assemblages they
support (Bestelmeyer et al., 2003; Caudle et al., 2013). Essentially, ecological sites
describe the relationship between environmental factors and plant community
composition (Caudle et al., 2013, p. 12). Ecological site descriptions include STMs that
describe what ecological states (mainly defined in terms of vegetation communities) are
possible on a given site, as well as the drivers of state transitions (e.g., overgrazing,
drought, mismanagement, other human interventions that impact form-process
dynamics). States, transitions, and their drivers are defined using inventories of soil and
vegetation, long-term monitoring data, historical data and paleoenvironmental
reconstructions, site dynamics revealed by recent monitoring, and expert and local
knowledge (Caudle et al., 2013; Knapp et al., 2009, 2011).
Figure (1) is a provisional STM for the Shallow Droughty ecological site, located in
northwestern Montana (LRU 43A-A, NRCS, 2009). This ecological site consists of three
states — Taller Bunchgrass State, Altered Bunchgrass State, and Invaded State, with
the former two states encompassing two distinct community types. States differ from one
another in terms of characteristic vegetation structure and composition, and the rates of
biogeomorphic processes (Bestelmeyer et al., 2003; Briske et al., 2008). Two types of
transitions are possible — within-state and between-state. Within-state transitions occur
when a shift from one vegetation community to another occurs, but with no change in the
dominant species. An undesirable within-state transition can be reversed through
modest adjustments to resource management. Between-state transitions are threshold-
crossing events that negatively impact ecological resilience and cannot be reversed in a
short time without significant management interventions (cf. Lawley et al., 2013). In this
example, overgrazing, soil erosion, and the introduction of weedy propagules catalyze
transitions away from the reference community (Taller Bunchgrass State), whereas
proper weed and grazing management or range seeding can produce a transition from
the Altered Bunchgrass State or Invaded State back to the Taller Bunchgrass State.
Although the STM diagram is a high-level representation of ecological sites, full ESDs
includes explanatory narratives which explain the dynamics of communities and states
and assist resource managers in analyzing where and when qualitative changes in
landscape states are likely to occur. Ecologists have typically used STMs to integrate
ecological theory and observations into ecosystem management and restoration, or as
tools to model or predict ecological changes (Bestelmeyer et al., 2009; Zweig and
Figure 1. STM for the Shallow Droughty ecological site in Major Land Resource Area
043A (Northern Rocky Mountains) — this site is found throughout western Montana.
Three states potentially exist on this site: 1) Taller Bunchgrass State; 2) Altered
Bunchgrass State; and 3) Invaded State. Nested boxes indicate community phases (i.e.,
within-state transitions). Legend: 1.1A Improper grazing management, soil erosion; 1.2A
Proper grazing management; 2.1A Improper grazing management, soil erosion; 2.2A
Proper grazing management; T1A Overgrazing, soil erosion; T1B Introduction of weedy
propagules, overgrazing; T2A Introduction of weedy propagules; R2A Range seeding,
proper grazing management; R3A Weed management, proper grazing management,
range seeding; R3B Weed management
Predictive applications have mainly focused on individual states and transitions with a
view toward predicting conditions under which different states will emerge, or on
identifying management practices that can promote desirable or inhibit undesirable state
changes. More recently, however, STMs have been applied to examine the dynamics
and complexity of potential state transitions as well as the spatial patterning of different
states (e.g., Bestelmeyer et al., 2009; Phillips, 2011a; b)
1.2. Geomorphological STMs
While there are relatively few examples of geomorphology studies that explicitly use
state-transition frameworks that characterized as such (exceptions: Phillips, 2011a;
2014; Van Dyke, 2016), a brief look at historical scholarship reveals that a number of
geomorphologists have conceptualized landscape dynamics and evolution in terms of
states and transitions. For example, Raymond Dugrand, a French geographer, deduced
state changes in soils and vegetation communities in scrublands driven in part by
erosion and pedogenesis in 1964 (Dugrand, 1964). Wainwright (1994) introduced
Dugrand's ideas to a larger audience of geomorphologists. Smart's (1988) model of
fluviokarst landscape changes exhibits state transitions. Geomorphic channel evolution
models (CEMs) appeared at least as early as 1984 (Schumm et al., 1984; Simon, 1989;
Van Dyke, 2013). These models decompose the evolution of river morphodynamics into
stages, each characterized by distinctive form-process relationships,. Early CEMs
emphasized a linear succession-like sequence of stages rather than acknowledging that
multiple evolutionary pathways may exist depending on the interrelations among
disturbance, sediment dynamics, hydrologic behavior, vegetation, and human actions
(e.g., Hawley et al., 2012).
A number of ecological STMs have been developed to models systems in which
geomorphic dynamics exert a strong influence over the development of ecosystems
(e.g., Brinson et al., 1995; Zweig and Kitchens, 2009). Other studies have either
represented or analyzed patterns of geomorphic change in terms of networks of
transitions among system states without using STM terminology (e.g., Kocurek and
Lancaster, 1999; Rountree et al., 2000; Cluer and Thorne, 2014). In our review we
include geomorphology studies (and some ecological and hydrological studies where
geomorphology plays a key role in defining states and driving transitions) where
changes are represented as a signed digraph, box-and-arrow diagram, or similar
construct in which distinct system states are connected according to observed or likely
transitions among them.
Three examples of studies include explicit STMs are Zweig and Kitchens (2009)
investigation of ecological succession and geomorphic change in the Florida Everglades
wetlands; Phillips (2011a) work on soil-geomorphology changes in a river delta; and Van
Dyke’s (2016) study of fluvial state transitions.
Zweig and Kitchens (2009) developed a generalized, non-spatial STM for a large portion
of the Florida Everglades to evaluate the post-restoration response of vegetation
communities to increased hydroperiods and water depths (Figure 2). They found that
state transitions were driven by hydrologic alterations, and in some cases geomorphic
feedbacks. For example, in sawgrass communities, transitions resulted from different
combinations of short- and long-term hydrologic forcings (e.g., year-to-year variability in
maximum water depths), although increasing peat depth was implicated as well — with
species composition being related to total peat depth. In slough communities, high winds
associated with hurricanes led to the displacement of Utricularia spp. from sloughs into
sawgrass strands, prompting a state transition. Overall, Zweig and Kitchens found that
water depth was a key control over community state composition, but also that
transitions were relatively infrequent. Based on their findings, they argued that STMs
have particular utility for understanding bio-hydrogeomorphic dynamics in landscapes
characterized by subtle environmental gradients, and that they can aid scientists in
unpacking the complex, multivariate, and nonlinear feedbacks between pattern and
process that govern ecosystem behavior. In such environments, the thresholds and
boundaries (e.g., elevation or hydroperiod) that control transitional behaviors are often
not readily apparent, and the STM framework helps researchers detect them.
Figure 2. One of three STMs for sawgrass communities in the Florida Everglades.
Transitions are driven by interactions among plant assemblages, hydrogeomorphic
dynamics (especially peat deposition and the variability in water depth), and disturbance
(modified slightly from Zweig and Kitchens, 2009)
Phillips (2011a) devised an STM for soils in the Guadalupe/San Antonio River delta
(GSARD) to understand the spatial complexity of environmental change as well as the
relative importance of universal controls and local environmental gradients in shaping
the developmental transitions of various soil types. The model (Figure 3) represented
system states by soil types, which are differentiated in the GSARD based on substrates
(related to depositional environments), topography, soil chemistry, and the age and
stability of the geomorphic surfaces on which they occur. Transitions among soil states
were driven primarily by the combination of sea-level rise and land subsidence,
variations in freshwater inflow, local topographic change due to deposition and surface
scour, river avulsions and cutoffs, and water diversions and withdrawals. After
developing the STM, Phillips (2011a) used algebraic graph theory to analyze the drivers
of pedological transitions. Three measures — spectral radius, algebraic connectivity, and
the S-metric — were used to interpret the STM with an eye toward understanding the
landscape’s propensity to amplify transitional behaviors or support spatially complex
state transitions. Most conceptual models of deltaic evolution assume that successional
patterns of change occur in response to events such as sea-level fluctuations or river
inflow. However, graph theoretic analysis demonstrated a more complex story — there
were complex modes of change that stemmed from the amplification of changes in
system states, relatively rapid spatial propagation of state transitions, and the system’s
inbuilt structural constraints. Phillips (2011a) concluded that the GSARD accommodates
spatially variable, complex but nonrandom geomorphic state transitions, and that state
transitions are driven by local environmental gradients and initial conditions.
Figure 3. STM for the Guadalupe-San Antonio River Delta, from Phillips (2011a). Lines
indicate transitions in either direction. Each state is represented by a soil type, which in
turn represents a particular geomorphic situation, as follows: Aransas-- Low floodplain
surfaces in lower deltatic fluvial-estuarine transition zone, influenced by saltwater
inundation: Austwell-- Low floodplain surfaces in lower delta, occasionally influenced by
saltwater; Degola--Natural levees & convex ridges on slowly accreting floodplain
surfaces; Meguin--Stable or slowly accreting flat or slightly convex surfaces on inner
portions of active floodplain; Placedo-- Infilled or partly infilled paleochanels; Rydolph--
Actively accreting floodplains, upper delta; Sinton-- Channel belts abandoned by
avulsions; crevasse fillings; Trinity--Low, relatively stable floodplain surfaces on clayey
overbank & channel fill deposits; Zalco--Recent sandy fluvial channel and point bar
Van Dyke (2016) created a diagnostic STM framework to catalog observed transitional
behaviors and predict future state changes along the newly restored Clark Fork River in
western Montana. This model (Figure 4) focused mainly on the river’s secondary
channels and floodplains, demonstrating that landscapes which have had their
biogeomorphic templates reset due to restoration and construction are particularly
vulnerable to state transitions. In the five years since restoration, seven out of the
model’s 11 hypothesized state transitions had occurred. While the diagnostic STM was
not spatially explicit, Van Dyke (2016) emphasized that fluxes of matter and energy
throughout the study area produced a set of complex feedbacks that led to considerable
spatial variability in channel and floodplain response — in some cases, multiple state
transitions were observed along individual secondary channels. Flooding and sediment
pulses have, and will continue to be, the most critical determinants of transitional
behavior. For example, while moderate flooding may induce sediment deposition along
secondary channels and encourage the recruitment of new vegetation (e.g., Salix spp.),
more sustained, high-magnitude flows are likely to act as resetting events by stripping
out vegetation and significantly altering channel morphologies. Diagnostic approaches to
restoration and river management, Van Dyke (2016) argued, can help practitioners
anticipate a landscape’s range of responses to various disturbance regimes.
Figure 4. Diagnostic STM for the secondary channels and floodplain of the restored
Clark Fork River. Transitions shaded green have been observed since the introduction of
flow in late 2010. Where Qw = discharge, Qs = bed-material load, V = vegetation cover
and density, Ef = floodplain erosion, and Ec = channel erosion (bed and bank). ! =
significant increase, = increase, — = no appreciable change; ○ = decrease, and □ =
The three examples above rely explicitly on STM terminology. There are other examples
of work we consider to be STM-based but that do not use STM terminology. These
include Rountree et al.’s (2000) study of landscape change in the Sabie River, South
Africa; Barchyn and Hugenholtz’s (2013) work on dune fields; Pietresiak et al.’s (2014)
research on Mojave Desert biogeomorphology; and a considerable body of work on
fluvial channel evolution models (synthesized by Van Dyke, 2013).
Rountree et al. (2000) studied riparian landscape changes from a prolonged low-flow
period that was followed by a large flood in the Sabie River, South Africa. They identified
seven landscape states in this semi-arid, mixed bedrock-alluvial channel, based mainly
on dominant substrate and vegetation cover. For each of three major channel types
(braided, pool/rapid, and bedrock anastomosing) they developed transition diagrams for
the seven states. The resulting directed graphs included weighted transitions based on
their probability of occurrence. They also produced similar graphs for three different time
periods between the mid- and late-20th century for a segment of the river within Kruger
National Park. Similar to other semi-arid rivers, Rountree et al. (2000) identified episodic
behavior along portions of the Sabie, with infrequent flooding events periodically
resetting the river’s biogeomorphic template and producing attendant state changes.
However, flow reductions and sediment supply increases resulted in more stabilized
riparian landform-vegetation states. More stable vegetation states require larger floods
for state transitions.
Barchyn and Hugenholts (2013) developed a conceptual model to explain the
circumstances under which sediment supply-limited dune fields are reactivated. Dune
field stability is influenced by factors such as climate, vegetation cover, disturbance,
geomorphology, and sediment dynamics. The model assumes that disturbances unsettle
stabilized dune fields, which in turn produce blowouts capable of both advancing
downwind and reactivating downwind sediments. They identified four dune reactivation
states: stable, blowout-dominated, reactivating, and stable but disturbance-susceptible.
Reactivation potential hinges on blowout behavior, disturbance, and vegetation
dynamics. Blowouts can be either depth-limited or morphology-limited, while vegetation
stabilizes dune fields by establishing a protective surface cover that prevents the direct
erosion of sediment or arrests the downwind propagation of blowouts once they do
occur. States can be identified based on the relationship between vegetation dynamics
and the rate at which the vegetation’s protective function diminishes, and the apron
deposition rate and tolerance of vegetation to deposition.
Pietrasiak et al. (2014) developed a landform evolution model for piedmont landforms in
the Mojave Desert. Using a state-transition framework, they show that feedbacks
between abiotic and biotic landscape components are implicated in landform evolution
over a 500–50,000-year timeframe. Two divergent evolutionary trajectories are possible
in this setting — abiotic and biotic landform evolution. During abiotic landform evolution,
young alluvial deposits gradually transform into a desert pavement as topographic relief
declines, which is driven by geomorphic and hydrological processes that redistribute
sediment, weather surface clasts, and fill in rock crevices with weathered materials. This
evolutionary trajectory inhibits biotic activity, including vegetation recruitment and the
establishment of biological crusts. Conversely, biotic landform evolution stems from
positive biotic feedbacks which are facilitated by plants and small burrowing animals.
Pietrasiak et al. (2014) proposed that shrubs and burrowing mammals both operate as
ecosystem engineers. Ground squirrels, kangaroo rats, and pocket mice (ubiquitous in
the Mojave Desert) preferentially burrow near shrubs, hindering the development of
desert pavement while opening up interspaces in the rocks and soil, which supports the
emergence of biological crusts and facilitates the growth of vascular plants. This process
eventually results in the development of shrub islands. This study highlights the critical
relationship between abiotic and biotic forces in shaping landform evolution and how
they influence what biogeomorphic states are possible given localized contingencies.
In each of the case studies discussed above, biota significantly influence biogeomorphic
feedbacks. Below, we further explore whether state-transition frameworks are especially
well-suited to biogeomorphic-oriented studies, or if they are simply more common in
these subfields of geomorphology because STMs originated in ecology. It is important to
note, however, that some STMs developed for geomorphic systems do not explicitly
incorporate biological or ecological factors (e.g., Smart, 1988; Loureiro et al., 2013;
Phillips, 2013; Solgaard et al., 2013; Bollati et al., 2014; Wyrick and Pasternack, 2014).
As such, state-transition thinking need not be pigeonholed as an epistemological
inclination that only has relevance for understanding Earth surface systems in which
morphodynamics are controlled or significantly influenced by ecological factors.
2.1. Selection of case studies
We collected geomorphological case studies that represented or described geomorphic
change in terms of shifts among qualitatively distinct states. Using Web of Science, we
used search terms with various combinations of the terms system, state, transition, and
model, filtered by the terms geomorphology, geo*, or landform. Although few of the
authors describe their models as STMs, for convenience we refer to them as such. What
constitutes a state differed across studies — some studies classified states in terms of
developmental phases or stages (e.g., a channel evolution model), while others defined
states based on predefined taxonomic units such as soil types, wetland types, or stream
channel classification units. Other studies defined states ad hoc based on field
observations. We omitted studies concerned with purely classification or mapping that
did not directly address transitions among states. Nor did we include ecological STMs
unless they clearly foregrounded geomorphic processes as drivers of state transitions.
Our search uncovered 47 example STMs from 40 publications.
For this review, our focus was on studies explicitly concerned with state transitions or
complex spatial and temporal patterns of geomorphic change. We also included case
studies that used STM-like box-and-arrow models to represent observed, inferred, or
modeled historical sequences. The ultimate criterion for inclusion was that the study
include a STM-like representation, or results/descriptions that could be readily translated
into STM form without requiring any assumptions or opinions on our part. While our
analysis focuses on the 47 geomorphic STMs we identified, there are undoubtedly more
that meet our criteria. Nevertheless, our sample is large enough to draw meaningful
conclusions about geomorphic dynamics across a broad range of systems.
2.2. Graph analysis
We converted each STM to a box-and-arrow diagram if it was not already in that form. In
graph theory terminology, these graphs are equivalent to signed digraphs, where graph
nodes (vertices) represent geomorphic states and graph edges (links) denote transitions.
We restricted our analysis to connected graphs (i.e., no isolated nodes). All graphs were
simple and unweighted, meaning that quantitative values were not assigned to nodes or
edges to indicate their magnitude, intensity, frequency, probability, or importance.
Because the spectral radius value — a key measure of graph complexity (see below) —
was zero for some graph structures in signed digraph form, we also analyzed each
diagram as a simple, undirected graph. In an undirected graph, nodes are connected
(i.e., they share an edge) if a transition between those states in either or both directions
A STM graph with N states and m transitions has a corresponding N x N adjacency
matrix, A, in which each column and row denote a geomorphic state. If two states are
connected by a transition, the entry for ai,j = 1; if the states are not connected or for i = j,
ai,j = 0. The adjacency matrix has N eigenvalues
, such that
> . . . >
algebraic or spectral graph theory, the largest eigenvalue,
, is the spectral radius.
N -1, with equality occurring only in a fully connected graph in which all nodes are
connected to all others. The spectral radius is influenced by the number of links or edges
(i.e., transition pathways), the number and length of cycles (paths through the graph that
start and end at the same node), and the degree distribution. The degree of a node is
equal to the number of edges it is connected to — in a directed graph this includes in-
and out-degree. Spectral radius is a standard measure of graph complexity, and
indicates the extent to which the effects of a state transition are potentially amplified and
propagated through the graph (Phillips, 2011a; 2011b; 2014). Graph entropy, a measure
of complexity more familiar to geomorphologists, is directly related to spectral radius. It
measures the number of paths that are possible within a graph (Geller et al., 2012):
Hg ~ ln
2.3. Classification of graph structures
We classified the signed digraphs taken or derived from the sample studies according to
several graph archetypes (Table 1). Because the mesh typology encompasses a great
deal of variability, we distinguished between sparse meshes, where m < 2N, and dense
meshes, where m > 2N. If graphs had characteristics of more than one structure, we
assigned it a hybrid classification. It is important to note is that simple divergent and
convergent types with the same number of states, N, have identical mathematical
properties, and that for N = 3 the cycle and fully connected forms are the same.
Table 1. Generic STM graph structures. Edges indicates the number of edges
(transitions) m for the graph type in simple, undirected form relative to the number of
nodes or states N.
Fixed sequence of transitions
m = N – 1
Fixed cycle of transitions
m = N
Multiple starting states transition to single end state
m = N – 1
Single starting state transitions to multiple possible
m = N – 1
All states connected to all others
m = (N2- N)/2
Single dominant high-degree node; other nodes
with degree < 3
m < 2N – 2
At least one node with degree > 3; no more than
one node with degree < 2
N + 1 < m <
Studies were also based on their extent and classified according to whether their
findings derived from observations at a single site, multiple sites, or the global level (i.e.,
non-site specific). STMs were further differentiated as interpretive or descriptive (i.e.,
their purpose was to summarize or contextualize findings) and also as those represented
as predictive or prescriptive models (or not). Finally, we distinguished among studies
with an explicit multi-pathway orientation versus a single-pathway orientation. The
former included studies conducted with an explicit STM or complexity-based framework
intended to address nonlinear geomorphic dynamics and evolutionary trajectories, but
was not limited to those with a stated nonlinear or evolutionary perspectives. The single-
pathway examples describe a particular historical sequence or response. Our initial
expectations were that, in general, multi-site studies, interpretative/descriptive models,
and multi-pathway orientations would yield more complex STMs, as measured by graph
type and metrics. Finally, we assessed the role of biogeomorphic or ecological
interactions with a simple three-part categorization: none (no direct role); present (biotic
factors included); and preeminent (biogeomorphic phenomena are the major concern of
the model and significantly influence system dynamics).
Our analysis is based on 47 geomorphic STMs, which were drawn from 40 separate
publications (Table 2). Although this is a representative sample, readers should not
construe this as an exhaustive census of all geomorphological studies that use state-
transition frameworks. This is an undercount, for at least two reasons. First,
geomorphologists rarely use state-and-transition terminology; thus our literature search
likely missed some work that would have fulfilled our criteria for inclusion. Second, our
criteria were relatively strict. For example, many ecological STMs have
geomorphological implications or feature geomorphic processes as drivers of state
transitions (e.g., soil erosion). We only included ecological studies in which geomorphic
dynamics played a central role in shaping ecosystem transitions (e.g., Brinson et al.,
1995; Zweig and Kitchens, 2009). Similarly, many authors have developed conceptual
models framed in terms of temporal stages or sequences that could be converted to a
graph or STM form, but which describe gradual changes in system morphodynamics.
We only included models that focused on qualitative transformations in system state, as
opposed to changes more generally (Phillips, 2014).
Table 2. Geomorphic state-and-transition models.
pedogenesis, land use
SCSF: Soil change,
massive limestone &
pedogenesis, land use
SCSF2: Soil change,
pedogenesis, land use
San Antonio River
delta, Texas, USA
sea-level rise, freshwater
SARD2: Soil change in
fluvial avulsion zone
San Antonio River
delta, Texas, USA
state transitions, delta
San Antonio River
delta, Texas, USA
LF: Soil transitions
water table change,
Thawing, freezing, lake
drainage, infilling, fire,
et al., 2000,
et al., 2001;
CEM: Evolution of
incised alluvial channels
Southern & central
Bed & bank erosion,
CEM2: Evolution of
incised alluvial channels
Southern & central
Bed & bank erosion,
CEM3: Evolution of
incised alluvial channels
Bed & bank erosion,
CEM4: Evolution of
gravel bed river channel
Bed & bank erosion &
Rio Puerco, NM,
Bed & bank erosion,
Elliott et al.,
SAEM: Arroyo evolution
Bed & bank erosion,
Elliott et al.,
Bed & bank erosion,
SEM: Stream Evolution
Channel narrowing &
GEM: Gully evolution
Sea cliffs, Isle of
Gully erosion, cliff retreat
Small streams, Blue
Upland & channel
Discharge, channel size,
BSC: Beach states
SW Portugal (6
controlled by ratio of
wave height to sediment
fall velocity & wave
period; & relative tidal
SDB: Sand dune
Bigstick Sand Hills,
GLIS: Greenland ice
Tectonic uplift, ice sheet
SRSA: Sabie River,
sediment stripping &
MDB: Mojave Desert
et al., 2014
sediment system state
LCW: coastal wetlands
Storms, sea level
DEWB: Dam effects,
wandering braided river
Reduced flow &
sediment supply due to
PRBD: Parabolic dunes
Wind velocity, sand
UF2E: Upland forest to
Climate, land use,
et al., 2015
Yuba R., California
CF: coastal foredunes
Waver erosion &
erosion & deposition;
NDR: Negev Desert
Grazing; erosion; soil
Stavi et al.,
CFR: Clark Fork River
LCM: Lockyer Creek
Channel erosion &
et al., 2016
VDR: Virginia dam
Bed & bank erosion,
USC: Urban semiarid
Bed & bank erosion,
Karst conduit growth;
widening, SW U.S.
Channel erosion &
withdrawal affects on
& flow, groundwater
GPRR: Great Plains
Missouri & Platte
Flow regulation, channel
IDR: Island dominated
Channel erosion &
IAC: Italian alluvial
Channel incision &
wetlands – wet prairie
Water table change; peat
Water table change; peat
Water table change; peat
Among our case studies, the sparse mesh graph structure occurred most often, as 45
percent of the STMs had this form (21 cases; see Table 3). Next most common was the
linear structure (11 cases, or 23 percent). Five of the STMs had a cyclical structure, four
had a dense mesh configuration, and three had a divergent form. A single STM fell into
the connected, convergent, and bipartite form classifications, respectively. Twenty-four
of the 45 studies examined fluvial systems. Of the 11 linear STMs eight came from the
fluvial geomorphology literature. Arguably, the tendency to view streams as evolving in a
linear manner (similarly to classical ecological succession models) reflects the impact of
classical CEMs in that subfield. It has only been within the last 10–15 years that fluvial
geomorphologists have begun to develop CEMs that account for complex forms of
channel evolution in which hydrogeomorphic fluxes and stream–vegetation interactions
produce multiple adjustment pathways (e.g., Gurnell and Petts, 2002; Hawley et al.,
2012; Van Dyke, 2016).
In 33 of the 47 examples (70 percent), we observed a predisposition toward complexity
and multiple states — as opposed to linear trends or cycles. An orientation toward
multiple states and complex systems did not necessarily preordain identification of a
mesh or divergent model; two of the linear structures and two of the cycles emerged
from such studies. We identified one fully connected structure, with N = 4. This, and the
small number of dense mesh structures, suggests that most geomorphic STMs are
strongly constrained with respect to possible state transitions.
Table 3. Characterization of geomorphological state-and-transition models. N is the
number of states or nodes, SR is spectral radius for the undirected graph version of the
STM, and SR/max is the calculated spectral radius relative to the maximum for a given
1S = applies to a single site, situation, or region; M = applies to multiple sites; G = global
or general model.
2Primary purpose. D = descriptive or interpretive; P = predictive or prescriptive.
3CS = did study have apparent a priori complex systems or multiple stable state
perspective? Y, N = yes, no.
4Graph structure. Bp = bipartite; C = cycle; Cv = convergent; DM = dense mesh; Dv =
divergent; FC = fully-connected; L = linear; SM = sparse mesh
3.2. STM Complexity
The median number of states (nodes) was 5 (mean = 6.0; SD = 2.3; range of 3–14).
Spectral radius values ranged from 1.414 to 4.702 (mean = 2.498; SD = 0.904) — this
excludes the bipartite graph, which has a largest eigenvalue of zero. The STM size
(number of states) explained approximately 26 percent of the variability in spectral radius
values (Figure 5). This is reflected in the variability of computed spectral radius divided
by the maximum for a given N (Table 3; Figure 5). There was a negative relationship
between N and spectral radius relative to the maximum possible spectral radius. This
indicates that the presence (or identification) of additional state transitions does not
proportionally increase STM complexity. This also suggests that complexity is not merely
an artifact of identifying more states in a system.
Figure 5. Relationships between spectral radius, and spectral radius relative to the
maximum possible for a given N, and the number of states. The regression equation for
the first graph is y = 0.2127x + 1.2179 (R2 = 0.26). For the second graph, the regression
equation is y = -0.0531x + 0.8781 (R2 = 0.39).
Although we had anticipated more complex STMs to emerge from global-scale and
multi-site studies than from single-area projects, this was not the case. While global-
scale studies had a slightly higher mean N both the absolute and relative spectral radius
was actually slightly lower than for single-site studies. None of the differences were
statistically significant, based on t-tests. There were also no statistically significant
differences between the 35 descriptive or interpretive studies, and the 12 predictive or
prescriptive STMs. The 35 examples we classified as arising from studies with an a priori
focus on multiple states and complex dynamics had more states (mean 6.364 vs. 5.214),
and larger spectral radii (2.765 vs. 1.870) than models from the 12 other studies. Due to
the large variation and high standard deviations in the former group, however, these
differences were not statistically significance at a 0.05 confidence level. There were also
no significant differences when comparing the 24 fluvial examples with the 23 from other
3.3. Role of Biogeomorphic Interactions
Nine of the STMs lacked a direct, explicit role for biota or ecological phenomena. In
some cases, such as the alluvial channel model of Phillips (2013; FCF) and the channel
evolution model of Bollati et al. (2014; CEM4), biotic factors certainly influenced some of
the phenomena included, but were not explicitly part of their STMs. Vegetation, for
instance, affects erosion and sedimentation processes included in the models, but plants
were not directly included. This was also the case for the SARD2, GEM, DEWB, and
GBMU models (see Table 1). Biotic influences are very important in the development of
karst landscapes, but were not directly implicated in Smart’s (1988 FKL) fluviokarst
landscape evolution model. Two of the nine abiotic STMs involved processes and
phenomena that are, or can be, completely abiotic — Solgaard’s (2013; GLIS) STM for
the Greenland ice sheet and Loureiro et al.’s (2013; BSC) classification of
morphodynamic beach states
Approximately half of the case studies (22) included state transitions that were at least
partially driven by biological or ecological processes. These models thus accorded a role
to biotic-abiotic feedbacks. Fifteen of the models conferred a significant role to plants in
fluvial systems (e.g., vegetation establishment working to stabilize floodplains or channel
bed, the removal of vegetation by erosional processes, and plant-root interactions
affecting boundary resistance). This generalization applies to the following models
(Table 1): CEM, CEM2, CEM3, RPCEM, SAEM, ACEM, SEM, BRCE, ARRO, LCM,
VDR, USC, SWAW, GPRR, and IAC. Three STMs described interactions among
vegetation, erosion, and deposition in aeolian dunes (ASSS, PRBD, and CF), while three
others included reciprocal interactions in wetland or riparian environments among
topography, surface and subsurface hydrological processes, and vegetation (SARD,
SARD3, GWRA), and one (LF) incorporated hillslope vegetation–erosion interactions.
We classified 16 of the STMs (34 percent) as biogeomorphically oriented, meaning they
were pointedly designed to explain, interpret or predict bio- or ecogeomorphic
phenomena, or where a majority of states and transitions were defined in terms of
biogeomorphic relationships or interactions. Six of these models integrated relationships
among hydrology, soil properties, vegetation, and erosion/deposition in dryland (desert,
semi-arid, and Mediterranean environments). These included three arising from
Dugrand’s pioneering work (1964; via Wainwright, 1994): VCSF, SCSF, and SCSF2.
Others in this category were the NDR, MBB, and SAVC models. Five of the
biogeomorphic STMs were focused on wetland dynamics, including three in the work of
Zweig and Kitchens (2009; EW1, EW2, EW3), as well as the LCW and UF2E STMs. The
remaining examples involve periglacial (PFTK) and aeolian dune (SDB) studies, and the
fluvial examples of Van Dyke (2016; CFR) and Rountree et al. (2000; SRSA) discussed
in Section 1.2.
Rangeland ecologists, having observed the complex dynamics of arid and semi-arid
landscapes as well as the presence of multiple vegetation states, devised STMs as a
way to more realistically model ecological processes. Although monotonic, single-
pathway successional or developmental sequences accurately characterize adjustment
trajectories for some landscapes, their logic does not universally hold. Indeed, the state
terminology directly relates to the concept of alternative stable states: the idea that there
can be more than one attractor state in ecological systems — and as this research
shows, in geomorphic or biogeomorphic systems. The case studies we reviewed show
that entire landscapes or portions of landscapes transition to different states. Transitions
are the consequence of local interactions among topoedaphic factors, hydrology,
vegetation, geomorphic processes, underlying geology and lithology, and, potentially,
Even though we commonly trace STMs to the pioneering work of Westoby et al. (1989)
and other rangeland ecologists, it is worth noting that geomorphologists productively
used state-transition frameworks prior to and independently of ecologists. These early
attempts to conceptualize landscapes as consisting of interrelated states and transitions
(e.g., DuGrand, 1964; Smart, 1988; Wainwright, 1994) did not emerge from conscious
efforts to integrate complex systems thinking or concepts of alternative stable states into
geomorphic analysis. Rather, these studies, grounded in detailed empirical work,
illustrated that multiple evolutionary trajectories and transitions may exist in geomorphic
systems — this happened to involve multiple possible pathways and attractors. Similarly,
while classical CEMs adhered to linear or cyclical understandings of channel evolution
based on observations in a small number of settings, they were an important step for
discussing landscape evolution in terms of discrete states, with each state characterized
by a unique set of form-process relations (Schumm et al., 1984; Simon, 1989). While the
theoretical underpinnings of these studies assumed a stable-steady state attractor (e.g.,
a dynamic equilibrium condition — see Simon and Rinaldi, 2006), they did not presume,
a priori, that channel evolution pushed a system toward an inevitable climax through
deterministic successional progression.
Just as ecological STMs emerged alongside discussions about alternative stable state
concepts, the development of state-transition frameworks in geomorphology occurred as
geomorphologists began to explore the implications of analogous ideas for geomorphic
systems. Although their models did not always employ box-and-arrow diagrams or
transition matrix logic, geomorphologists such as Burkham (1972), Hey (1978, 1979),
Slingerland (1981), Thornes (1981; 1983; 1985), Scheidegger (1983), Trofimov and
Moskovkin (1984), Culling (1988), and Montgomery (1989) by 1989 had explicitly
addressed the presence of state changes in nonequilibrium geomorphic systems.
Further, the threshold concept (e.g., Schumm, 1969; 1979) had been well-established in
geomorphology literature by the 1970s — although earlier studies had studied
thresholds in connection with process mechanics. Crossing a threshold implies a
qualitative change in landscape state. Schumm’s work on river metamorphoses
(Schumm, 1969; 1979; Nadler and Schumm, 1981) demonstrated that, for example,
thresholds exist at which a meandering channel will transition to a braided one due to
increases in sediment loading and discharge. As such, the transition concept by
definition recognizes alternative stable states.
Beginning in the 1990s, there was a parallel development of STMs in ecology and
geomorphology, although STM terminology was generally reserved for the former. Over
the past 10 years there has been something of a merger, with ecological STMs moving
beyond rangeland ecology and more explicitly foregrounding geomorphic drivers and
controls (e.g., Zweig and Kitchens, 2009; Phillips, 2011b), and geomorphologists
embracing and adapting STM terminology (e.g., Phillips, 2011a; 2014; Van Dyke 2013;
2015; 2016). In the latter case, the consistency of STM approaches with complex
systems perspectives, and the utility of STMs in applied geomorphology and river
restoration were key motivators.
4.1. STMs and Complex Systems
The STM approach is well suited to conceptual frameworks involving nonequilibrium
dynamics, alternative stable states, and complex systems. STMs are also amenable to
the graph and network based approaches that have become increasingly prominent in
the geosciences (Heckmann et al., 2015; Phillips et al., 2015). As our synthesis reveals,
an explicit concern with issues such as complexity and path dependence does not
necessarily lead to the identification of more states and transitions, or of a more complex
transition pattern. Some studies grounded in frameworks attuned to complexity and
alternative states turned up relatively simple STM patterns, while at the same time there
were studies not explicitly framed in those terms that identified complex relationships
among states and transitions. There was no statistically significant difference in the size
or complexity of STMs arising from explicitly complexity-based studies versus other
The catalog of geomorphic STMs presented here is dominated by studies that examined
fluvial systems. There are three likely reasons for this. First, fluvial geomorphology is the
field’s largest subdiscipline. Second, the advent and development of CEMs strongly
influenced how geomorphologists and river scientists conceptualize stream adjustments
in the face of disturbances — this has firmly entrenched STM-type frameworks (Van
Dyke, 2013). Finally, fluvial (along with some coastal and aeolian) systems tend to
experience more rapid (and therefore easily observed) state transitions. In comparison,
transitions are often more gradual and subtle in glacial, periglacial, and karst and other
Biogeomorphology is concerned with reciprocal, mutually constitutive interactions
between biotic and abiotic components of geomorphic systems. This, and the obvious
connections with ecology, make the STM approach well suited to this subfield. About a
third of the identified studies were explicitly concerned with biogeomorphic phenomena,
while another 47 percent incorporated biotic effects on geomorphic processes and/or
vice versa. Only 19 percent did not include some biological or ecological influences. In
our view studies such as those of Wainwright (1994), Rountree et al. (2000), and
Barchyn et al. (2013) — along with the use of STM's in our own empirical biogeomorphic
studies — confirms the utility state-transition frameworks for biogeomorphology. The use
of STMs to analyze landscapes that lack the imprint of ecological processes indicates,
however, that the usefulness of the state-transition approach is by no means limited to
biogeomorphology or to problems with a strong ecological component.
4.3. Spatial and Temporal Extents
With a few exceptions, geomorphological state transitions within a single system have
recurrence intervals of years (commonly decades) or much longer. Long recurrence
intervals pose challenges for observing or otherwise identifying (e.g., via geomorphic
indicators or historical reconstructions) the full spectrum of state changes possible in a
landscape. Even systems which are vulnerable to landscape resetting events, such as
arid and semi-arid rivers (e.g. Graf, 1988; Huckleberry, 1994; Webb et al., 2014), may
appear quiescent over human time scales if a disturbance or series of disturbances of
sufficient magnitude do not appear and therefore cause a state transition. Our ability to
observe transitions firsthand or through the use of paleoenvironmental reconstructions in
turn can affect the type of STM graph structures that can be identified.
The number of states and transitions that must be observed to infer particular graph
structures varies. A bare minimum of two states and one transition must be observed,
and with this minimum only a trivial linear structure is possible. If two states but >2
transitions are observed, a simple cycle (alternative stable state) structure can emerge.
For a convergent or divergent graph to be observed, at least three states and two
transitions must be identified. A non-binary cycle requires at least three states and three
transitions. It is only possible to identify mesh structures with observations of >4 states
and >5 transitions (for a sparse mesh), or at least 6 states and 12 transitions for a dense
mesh. A fully-connected graph could be discerned with as few as three states and
transitions, but the minimum number of transitions that would have to be observed to
produce a fully connected structure increases as a quadratic function of N since this
form requires m = (N2 - N)/2.
Table 4 summarizes the relationship between observed states and transitions and the
possibility of identifying STM types. The large number of observations required to
identify a dense mesh structure, or a fully connected graph with N > 4 is evident. This
may be at least partly responsible for the prevalence of linear, cyclical, and sparse mesh
structures in this review. This also suggests that geomorphic STMs should not be
viewed as temporally invariant descriptions or hypotheses about landscape evolution —
they are dynamic constructs partially contingent on our observations. As the record of
observations deepens and the number of states and transitions observed increases,
STMs can be refined to more faithfully represent the dynamics of geomorphic systems.
Table 4. Relationship between number of observed states (N) and transitions and
potential STM graph structures.
6 < q < 9
10 < q
7 < q < 12
12 < q < 15
15 < q
q = N -1
q = N
N < q < 2N
2N < q < (N2-N)/2
(N2-N)/2 < q
4.4. STMs and Applied Geomorphology
State-and-transition models have gained widespread currency among ecologists,
particularly those with research foci on arid and semi-arid environments in which state
transitions are common. Indeed, state-transition frameworks have been used to inform
ecological restoration projects, guide long-term monitoring experiments, assist with
routine management activities, and simulate ecosystem trajectories under a range of
climate change scenarios (e.g., Standish et al., 2009; Chambers et al., 2014;
Provencher et al., 2016; Williamson et al., 2016). Many ecologists now leverage the
vocabulary of STMs to frame their analyses and describe landscape dynamics.
However, because geomorphologists have tended not to use STMs, or have appealed
only implicitly to notions of states and transitions to characterize the behavior of
geomorphic systems, there is a discursive fissure between geomorphologists and
ecologists. Geomorphologists would potentially be well-served by adopting the
vocabulary and heuristics of STMs to describe what drives transitional behaviors in
systems that are governed either predominantly by geomorphic processes or in which
geomorphic processes strongly interact with vegetation to shape the landscape. Doing
so will facilitate more productive collaborations among geomorphologists, ecologists,
biologists, climatologists, and even social scientists to understand the complex
relationships among the multivariate drivers of state transitions (Sayre et al., 2013;
Bestelmeyer et al., 2015). While state-transition frameworks can certainly help
geomorphologists understand purely geomorphic systems, their promise lies even more
so in creating dialogues among researchers about how to develop more realistic models
of geomorphic system behavior — and potentially understand where human actions
intervene to reshape and potentially redirect landscape evolution. Additionally, STMs
offer a convenient way to facilitate communication among environmental managers,
restoration practitioners, and the general public on the potential implications of climate
change, disturbances, and alterations in resource use on landscape condition and
adjustment pathways (e.g., Knapp et al., 2011).
Landscapes and geomorphic systems transition to qualitatively different states, a notion
long implicit in geomorphology via the concepts of thresholds, river and landscape
metamorphosis, and channel evolution models. These transitions can be, but often are
not, linear or cyclical. The STM framework is an approach to describing, modeling,
interpreting, and predicting state transitions that allows for linear successions or cycles,
but more importantly, it also accommodates more complex, multidirectional patterns of
geomorphic change. While STM terminology emerged from rangeland ecology,
geomorphologists used state-transition frameworks prior to and independently of
ecologists. Like ecological STMs, which grew out of discussions about alternative stable
states, state-transition frameworks in geomorphology gained purchase as
geomorphologists began to explore the implications of analogous ideas for geomorphic
systems. State-and-transition models developed in parallel in ecology and
geomorphology, but only the former have typically garnered this label. Geomorphology
would benefit from adopting the vocabulary and heuristics of STMs to facilitate
collaborations among geomorphologists, ecologists, biologists, climatologists and social
scientists to address complex interrelationships among the multivariate natural and
anthropic drivers of state transitions.
STMs should find a natural home in biogeomorphology, however, the 47 geomorphic
STMs reviewed here indicate the approach’s utility is not limited to biogeomorphology or
problems with a strong ecological component. The STM approach also has strong
affinities with conceptual frameworks that implicitly or explicitly recognize the possibility
of nonequilibrium dynamics, alternative stable states, and path dependency in
geomorphic systems. But as our synthesis revealed, an explicit concern problems of
complexity, path dependence, and nonequilibrium dynamics does not necessarily lead to
the identification of more states and transitions or a more complex transition pattern.
Among the studies we analyzed, STM complexity was not strongly influenced by either
the general purpose of a study or the number of sites investigated as part of the study.
This is encouraging — it indicates that geomorphic STMs represent empirical findings in
a veridical, straightforward manner, and that researchers have not used them as an
instrument to justify or validate a priori normative concepts of how a particular system
ought to behave (e.g., linear successions, cycles, or complex systems).
The time scales typically involved in geomorphic change make it challenging to observe
or identify a large number of states and transitions. This can affect what types of STM
graph structures can be identified. At least six states and 12 transitions are required to
identify a dense mesh structure, for example, and for more than four states, a minimum
number of transitions at least double the number of states must be observed to detect a
fully-connected STM. The burdens of observation, and the relatively short time over
which systems are generally investigated, may partly explain the large proportion of
linear, cyclical, and sparse mesh structures that have been documented, and indicates
that geomorphic STMs are not static constructs. Rather, they should be treated as
dynamic constructs that may be elaborated — and improved upon — with additional
We appreciate the careful and constructive critiques of two anonymous reviewers.
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