Predicting modes of spatial change from state-and-transition models

Abstract

State-and-transition models (STMs) can represent many different types of landscape change, from simple gradient-driven transitions to complex, (pseudo-) random patterns. While previous applications of STMs have focused on individual states and transitions, this study addresses broader-scale modes of spatial change based on the entire network of states and transitions. STMs are treated as mathematical graphs, and several metrics from algebraic graph theory are applied—spectral radius, algebraic connectivity, and the S-metric. These indicate, respectively, the amplification of environmental change by state transitions, the relative rate of propagation of state changes through the landscape, and the degree of system structural constraints on the spatial propagation of state transitions. The analysis is illustrated by application to the Gualalupe/San Antonio River delta, Texas, with soil types as representations of system states. Concepts of change in deltaic environments are typically based on successional patterns in response to forcings such as sea level change or river inflows. However, results indicate more complex modes of change associated with amplification of changes in system states, relatively rapid spatial propagation of state transitions, and some structural constraints within the system. The implications are that complex, spatially variable state transitions are likely, constrained by local (within-delta) environmental gradients and initial conditions. As in most applications, the STM used in this study is a representation of observed state transitions. While the usual predictive application of STMs is identification of local state changes associated with, e.g., management strategies, the methods presented here show how STMs can be used at a broader scale to identify landscape scale modes of spatial change.

Introduction

Environmental changes in landscapes are often spatially complex, due to multiple forcings, spatial heterogeneity of initial conditions, and interactions among components within the landscapes. The purpose of this paper is to introduce methods for identifying the modes of changes in ecosystems and landscapes, as an independent tool for interpreting observed changes, as a guide for the selection of appropriate predictive models, and as a means for predicting modes of change from knowledge of networks of environmental state-changes.

“Mode” has various definitions, but this project is concerned with mode defined as how something happens, or as a particular functioning condition or arrangement. Modes of environmental change are qualitatively different forms, styles, or genres of change. Thus, for example, infiltration-excess or saturation-excess overland flow are different modes of surface runoff generation; and C₃, C₄, and CAM plants represent different modes of carbon fixation.

In the spatial context, modes are characterized by different spatial patterns of change. The response of coastal wetlands to sea-level rise, for example, may be framed in terms of several different modes, from several different perspectives. Geomorphologically, the areal extent of wetlands may increase, decrease, or remain constant depending on the balance between net vertical accretion and coastal submergence. In terms of vegetation communities, transformations could occur along environmental gradients of elevation, salinity, or hydroperiod, with community transitions occurring when critical thresholds are transgressed. Or, transitions could occur in (pseudo-) random patterns as complex interactions among biota, hydrology, geomorphology, and soils create a spatial mosaic, rather than an advancing front, of changes. Conceivably, then, representations of change could be based on linear succession or gradient-type models (e.g., Brinson et al., 1995), random models (c.f. Erfanzadeh et al., 2010), or nonlinear dynamical systems models (e.g., Phillips, 1992).

This study is framed in terms of state-and-transition models (STMs), most commonly used in range ecology (see overviews by Briske et al., 2005, Bestelmeyer et al., 2009), but increasingly applied in ecosystem science more generally (e.g., van der Wal, 2006, Hernstrom et al., 2007, Czembor and Vesk, 2009, Zweig and Kitchens, 2009). Essentially, a STM identifies potential system states, commonly vegetation communities, and the possible transformations among them. The conditions under which these transformations occur are typically further refined based on theoretical or empirically determined probabilities, or on ecologically based rules or principles. While STMs were conceived as an alternative to classical deterministic succession models, the latter are special cases of STMs, where the state transitions occur in a single linear sequence. Random models are another special case of STM, where transition between any two states is equally likely. Zweig and Kitchens (2009) discuss the application of STMs to address complex succession patterns in wetlands, particularly where multiple stable states are possible.

STMs are typically used to link ecological theory and/or observations to ecosystem management and restoration, or as tools to model, predict, or describe ecological changes based on prior knowledge of processes or phenomena (Bestelmeyer et al., 2009, Zweig and Kitchens, 2009). However, the predictive applications have focused on individual states and transitions. For example, given a particular semi-arid vegetation community, what will be the changes in community composition in response to grazing strategies, fire regimes, or brush control? In this study, the concern is with assessing broader-scale modes of spatial change based on the entire network of states and transitions represented by a STM.

Ecosystem states and the transitions between them can be treated as a network, with states as the nodes of the network or vertices of the associated graph, and transitions as the links among the nodes. STMs are, indeed, typically represented as box-and-arrow diagrams directly translatable to mathematical graphs. Applications of graph theory in landscape ecology go back to at least the early 1990s (Cantwell and Forman, 1993), and earlier in other aspects of ecology, though the methods are unfamiliar enough to most ecologists that introductions to basic graph theory concepts are still presented in most recent papers (e.g., Tremi et al., 2008, Urban et al., 2009). Previous applications dealt primarily with issues of connectivity and centrality of landscape and habitat elements, and dispersal mechanisms or movement pathways (e.g., Cantwell and Forman, 1993, Bunn et al., 2000, Bode et al., 2008, Tremi et al., 2008, Urban et al., 2009, Padgaham and Webb, 2010). This paper is concerned with the synchronization properties of ecological systems represented as graphs, which has not been studied in landscape ecology or biogeography. Graph theory has also not previously been applied to STMs.

Several end-member or archetypal network structures of STMs can be identified. These provide a template for evaluating the structural connectivity of real-world landscapes and interpreting modes of landscape change. A linear sequential STM is a classic succession-type form. For example, in a four-state system with states A, B, C, D, A leads to B leads to C and finally to D (with reversals possible due to disturbance). A cyclical sequential variant may be identified, with state D leading back to A. Examples of this would include succession where disturbance returns the final state back to A. A second case is termed radiation. In this case a single state (A) can transition to or from any of several states (in this case B, C, D). This could correspond, for example, to a state A dominated by a highly successful invading species, whereby invasion of any other state (B, C, D) can result in a transition to A (and removal of the non-native may restore B, C, or D). The third extreme is termed maximum connectivity, in which all states are linked so that, e.g., A, B, C, or D could transition to any of the other states. This is equivalent to a random model, because if any transition is equally possible, then over a broad enough area all will be observed. These archetypes are illustrated in Fig. 1.

Depicting a STM as a network with an associated graph, three metrics are employed here from algebraic graph theory and spectral graph theory (Biggs, 1994) to characterize the modes of spatial change. Spectral radius is an indicator of the degree of amplification or filtering of changes or perturbations by the system. Fath and colleagues, for instance, have used this measure as an indicator of the intensity of cycling in food webs represented as graphs (Fath, 2007, Fath and Halnes, 2007, Fath et al., 2007). A second metric is algebraic connectivity, commonly used as an indicator of the synchronizability of networks (Biggs, 1994). In the context of this study, high levels of synchronization indicate rapid propagation of effects through the spatial network (and vice versa). The S-metric was derived by Li et al. (2005) as an indication of the extent to which a system is scale-free. The index is sensitive to the degree to which network nodes are hubs with multiple links. In the context of this study the S-metric is an indication of (system) structural constrains on state transitions. These metrics will be discussed further below.

Each STM can be considered as a network represented by a graph with N nodes (the states) and m edges or links (transitions between states). These graphs may be directed (transitions are only possible in one direction along any edge) or undirected (transitions are possible in both directions). Undirected graphs will be considered here, as most STMs allow two-way transitions among pairs of states. Graphs are connected if it is possible to follow a path of one or more edges between any two nodes. Any graph has a N × N adjacency matrix A, the entries of which are 1 if the row and column states or nodes are connected, and zero otherwise. For the case of an undirected, connected graph A is symmetric. The adjacency matrix has N eigenvalues λ, which may be complex numbers, the real parts of which are ordered such that λ₁ > λ₂… > λ_N−1 > λ_N. The largest eigenvalue is the spectral radius, and λ₁ is an important determinant of many system properties (Restrepo et al., 2007). The spectral radius is directly related to the number of paths or cycles in a network. λ₁ < 1 indicates damping or filtering behavior, so that changes are essentially absorbed by the system. If λ₁ > 1 amplification effects are indicated, with higher values indicating stronger amplification.

The maximum spectral radius for a graph of a given number of nodes or states and edges or transitions is

$${\lambda }_{1,\max }={\left[2m\frac{(N-1)}{N}\right]}^{0.5}$$

Based on this, maximum λ₁ for the archetypal STM structures can be determined as follows:

• Linear sequential, radiation: ${\lambda }_{1,max}={[2{(N-1)}^2/N]}^{0.5}$

• Cyclical sequential: ${\lambda }_{1,max}={[(2N)(N-1)N^{-1}]}^{0.5}$

• Maximum connectivity: λ₁ = λ_1,max = N − 1

Using λ₁, λ_upper, λ_max, respectively, to signify the observed largest eigenvalue, the upper bound on λ₁ for given N and m, and the upper bound for a given N (Phillips, in press):

$$\frac{{\zeta }_{connection}}{{\zeta }_{total}}=\frac{({\lambda }_{max}-{\lambda }_{upper})}{({\lambda }_{max}-{\lambda }_1)}$$

$$\frac{{\zeta }_{wiring}}{{\zeta }_{total}}=1-\frac{{\zeta }_{connection}}{{\zeta }_{total}}$$

The contribution to reduction of λ₁ associated with having fewer transitions or edges than the maximum connectivity or random case is given by ζ_connection/ζ_total. The relative importance of the specific network of connections (i.e., how a given number of nodes are linked given a specific m) or “wiring” is indicated by ζ_wiring/ζ_total.

Algebraic connectivity is defined as the second-smallest eigenvalue (λ_N−1) of the Laplacian matrix L(A) of the adjacency matrix. The entries of L(A) are:

$$a_{ij}=\left\{\begin{array}{ll}\mathrm{deg}(v_i)\hfill & \text{if}i=j\hfill \\ -1\hfill & \text{if}i\ne j\text{and}{v}_i\text{adjacent to}{v}_j\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\right.$$

where deg(v_i) is the degree of vertex or node i. The degree is equal to the number of transitions or edges connected a node to other nodes.

Algebraic connectivity (λ_N−1) is a measure of the synchronizability of the system (Biggs, 1994, Duan et al., 2009), and is bounded by the vertex connectivity κ(A) and graph diameter D:

$$\frac{4}{ND}\le {\lambda }_{N-1}\le \kappa (\text{A})$$

Vertex connectivity is the minimum number of vertices or nodes that could be removed to disconnect the graph. D is the maximum shortest path (number of links or edges) between any two vertices. Vertex connectivity is bounded by edge connectivity of A such that κ(A) ≤ edge connectivity ≤ minimum degree. Edge connectivity is the minimum number of edges that could be removed to disconnect the graph. Minimum degree is the smallest number of edges associated with any node or vertex.

Vertex connectivity for the sequential and radiation archetypes is 1, while k(A) = N−1 for the maximum connectivity case. D = N−1 for the linear sequential and N −2 for the cyclical sequential cases. For the radiation type, D = 2, and D = 1 for maximum connectivity. Thus the following intervals can be defined for algebraic connectivity:

• linear sequential: 4/(N² − N) ≤ λ_N−1 ≤ 1

• cyclical sequential: 4/(N² − 2N) ≤ λ_N−1 ≤ 1

• radiation: 4/(2N) ≤ λ_N−1 ≤ 1

• maximum connectivity: 4/N ≤ λ_N−1 ≤ N − 1

The S-metric s(g) devised by Li et al. (2005) applies to undirected, simple, connected graphs with a fixed degree sequence:

$$s(g)=\sum _{i=1}^{(N-1)}(d_i{d}_{i+1})$$

where d is the degree of a given node or state (these appear as the diagonal elements of the Laplacian). By arranging the adjacency matrix in order of increasing or decreasing degree, the fixed degree sequence requirement can be met. The S-metric measures the extent to which g has a hub-like core. Maximum s(g) values occur when high-degree nodes are connected to other high-degree nodes. Further comments on the mathematical developments and implications are given by Li et al. (2005).

For the archetypal cases:

• linear sequential: $s(g)=\left\{{\sum }_{i=1}^{N-1}\left[(i-1)N-{(i-1)}^2\right]\right\}-N$

• cyclical sequential: $s(g)={\sum }_{i=1}^{N-1}\left[(i-1)N-{(i-1)}^2\right]$

• radiation: s(g) = (N − 1) + (N − 2)

• maximum connectivity: s(g) = (N − 1)³

The application of spectral radius, algebraic connectivity, and the S-metric is illustrated below.