A fuzzy-constrained cellular automata model of forest insect infestations

Abstract

Geographical and ecological processes are complex systems where individual elements interact to create complex behaviour. These systems can be examined with spatially explicit models such as cellular automata (CA) that explain how interactions at the local level lead to global patterns. Tree mortality patterns caused by forest insects provide a good case for CA as local interactions lead to changes at the landscape level. However, problems exist with defining aspects of insect–host relationships that explain the susceptibility of a tree to insect attack. The main objective of this study is to develop a GIS-based CA model of forest insect infestations that incorporates fuzzy set theory in order to obtain information from high resolution remote sensing (RS) images. The model is based on tree mortality patterns caused by outbreaks of mountain pine beetle (MPB), Dendroctonus ponderosae Hopkins, in the central interior of British Columbia, Canada. Fuzzy sets are used in order to represent the susceptibility of trees to MPB attack and to acknowledge the uncertainty inherent in dealing with geospatial data. Fuzzy values provide the input for the CA Sub-Model where MPB attack behaviour is constrained by the susceptibility level of trees. The results from the model reflect the process of MPB infestations as described in the literature and exhibit non-linear dynamics expected in ecological processes. This study reveals that fuzzy-constrained CA modelling can provide useful information for forest management in the presence of insect outbreaks.

Introduction

Geographical and ecological processes are complex dynamic systems with an inherently spatial nature. The complexity is manifest in the numerous elements that interact locally to produce global patterns that are difficult to predict, while the spatial nature is apparent in the significance of scale, distance and spatial arrangement of the interacting elements. Complex systems theory is suitable for incorporating both the complexity and spatial significance in ecological processes, and can provide results that enhance ecological knowledge for decision support systems. One class of complex system models that has recently gained recognition in ecology is cellular automata (CA) (Cannas et al., 1999, Grist, 1999).

CA are dynamic models that are discrete in time, space, and state (Baltzer et al., 1998). CA models typically consist of five main components: (1) a grid of cells, (2) cell states, (3) the neighbourhood, (4) transition rules that determine how cells change from one state to another at each time step, and (5) the number of time steps for which the model is run (White and Engelen, 2000). The grid is composed of a number of cells that are typically identical in size and shape. Cells at initial time T_i can take on an infinite number of states that are traditionally represented as discrete. The neighbourhood refers to the cells in a defined area surrounding each individual cell that will have an influence on the state of that cell at the next moment in time (i.e. T_i+1). The transition rules express how the state of each cell in the neighbourhood influences the future state of a cell from one time step to another. A CA model can be formulated as

$$s_{xy}^{T_{i+1}}=f(s_{xy}^{T_i},N_{xy}^{T_i}),$$

where $s_{xy}^{T_i}$ and $s_{xy}^{T_{i+1}}$ are the states of cell at a location described with x and y coordinates at time T_i and T_i+1, respectively; $N_{xy}^{T_i}$ represents the neighbourhood surrounding cell xy; f represents the transition rules that explain how the initial state will change in the next time step. The number of time steps refers to the temporal extent of the model.

The discrete nature of cell states makes CA attractive for spatial-temporal modelling in a geographic information system (GIS) raster-based environment, which describes the world as a static representation based on a discrete array of cells. GIS and CA are complementary with regards to spatio-temporal modelling as the former provides the spatial framework for geographic data while the latter contributes the temporal dimension for describing change. Furthermore, the ability to develop realistic spatial models within a GIS environment has progressed due to the increasing availability of remote sensing (RS) data. In geography, GIS-based CA have proven especially successful in simulations of urban dynamics (White and Engelen, 1993, Batty and Xie, 1994, Couclelis, 1997, Clarke and Gaydos, 1998), rural residential settlement patterns (Deadman et al., 1993), and socio-environmental systems (Engelen et al., 1995) among others. CA models have also gained popularity in the field of ecology as discrete cell states can represent the presence of organisms at a given location which can change over time due to competition and resource allocation (Cannas et al., 1999, Grist, 1999). Baltzer et al. (1998) explain that discrete cell states are advantageous for modelling ecological processes because discrete state transition can be governed by a probability distribution based on the initial state of each cell.

While CA are applicable for modelling numerous ecological scenarios, problems exist when examining complex processes where cell states cannot be readily defined as discrete. A good example is representing a tree in a forest by its susceptibility to attack by an insect, whereby susceptibility is defined by numerous variables of the insect–host relationship. In such cases, two main problems exist with providing a binary definition (i.e. susceptible or not susceptible to attack by an insect).

The first problem concerns the issue of uncertainty in defining susceptibility. It is difficult to use traditional approaches to this problem such as defining a tree with binary values like susceptible or not susceptible, or deriving the probability of a tree becoming attacked when lacking sufficient data on insect attacks. This is due to the fact that insect disturbances are driven by numerous components of the insect–host relationship that are difficult to understand. Appreciating this relationship is further complicated by the presence of numerous climatic variables such as temperature, wind, humidity and precipitation, which, coupled with the geographic variation of a species’ life cycle, produce varying results and incomplete knowledge on insect behaviour. Therefore, considering a raster-based representation of a forest landscape, significant uncertainty is present when attempting to assign a discrete binary or probability value to a cell describing a tree's susceptibility to attack. Furthermore, deriving probabilities requires sufficient data that illustrate the types of trees that are most likely to be attacked. However, the spatial and temporal resolutions of commonly used geospatial data (Fall et al., 2004, Nelson et al., 2004, Riel et al., 2004) hamper the ability to study and understand the forest infestation process at the individual tree level. This is because it is difficult to determine attack patterns at the tree level with large-scale images collected over a short or inappropriate time period.

The second problem is the inherent uncertainty in classifying RS data of forest landscapes in a GIS in order to obtain information on the susceptibility of trees in a forest (Lowell and Gold, 1995). As forests are continuously changing over space and time, the value given to a cell through classification procedures only represents that location for the moment in time when the data were acquired. Processes such as insect infestations operate at refined spatial and temporal scales. The process itself is difficult to capture by RS imagery due to the continuous change of an individual tree's appearance in the canopy at the local level. This leads to uncertain transition zones between forest stands of different sizes and different species where a discrete definition of a cell cannot be provided.

As a solution to these two problems, fuzzy set theory has been suggested in situations where the presence of uncertainty prevents a discrete definition of cell states (Brown, 1988, Robinson, 1988). Fuzzy sets, developed by Zadeh (1965), allow for membership to one and/or several sets, thus objects in space are represented by a fuzzy value between 0 and 1. The membership function of an element x belonging to a fuzzy set A (e.g. A = susceptibility of a cell to insect attack) is represented by μ(A): U → (0,1), where U is the universal set of x. This explains that the function associates a graded membership with each point x in U. Therefore, trees can be represented by ‘fuzzy’ values from 0 to 1 based on its membership to the set of susceptible trees.

Fuzzy set theory has been utilized in geography in order to explain the inherent spatial uncertainty manifest in both objects (Molenaar, 1996, Cross and Firat, 2000, Cheng, 2002) and regions (Fisher, 1996, Schneider, 2001, Tang and Kainz, 2002) – hence the terms fuzzy objects and fuzzy regions. The development and use of fuzzy membership functions are evaluated based on the representation of objects and regions and how uncertainty can be implemented into analysis of such spatial entities (Altman, 1994). Geographic applications with fuzzy sets include defining soil classes (Burrough, 1989, Oberthur et al., 2000), land suitability analysis (Hall et al., 1992, Davidson et al., 1994), RS classification (Wang, 1990, Foody, 1996), spatio-temporal interpolation (Dragicevic and Marceau, 1999), validating categorical maps (Hagen, 2003), among others. Fuzzy sets have also been utilized in forestry for distinguishing stand boundaries (Brown, 1998), digitising forest types (Lowell and Gold, 1995) and identifying individual trees from high resolution images (Brandtberg, 2002).

The objective of this paper was to develop a fuzzy set theory driven methodology for developing a GIS-based CA model of insect-induced tree mortality patterns. The methodology used derived values of tree susceptibility to attack by mountain pine beetle (MPB), Dendroctonus ponderosae Hopkins, in the central interior of British Columbia, Canada where MPB commonly attack susceptible lodgepole pine, Pinus contorta. Outbreaks of MPB in recent decades have generated interest in understanding the behaviour of the insect and the patterns of tree mortality that it inflicts on a landscape. In particular, studies have examined the use of partial differential equations (Bolstad et al., 1997, Logan et al., 1998, Powell et al., 2000), climate models (Jackson and Murphy, 2004), remote sensing techniques (Franklin et al., 2003, Roberts et al., 2003) landscape-scale spatial analysis (Fall et al., 2004, Nelson et al., 2004) and spatio-temporal models (Riel et al., 2004). This study employed the use of a fuzzy-constrained CA model because patterns of MPB-induced mortality are applicable to the modelling logic of CA, and also to incorporate existing uncertainty and incomplete knowledge within the definition of tree susceptibility to MPB attack.