A spatially explicit hierarchical approach to modeling complex ecological systems: theory and applications
Introduction
Ecological systems are characterized by diversity, heterogeneity and complexity. Complexity often results from the nonlinear interactions among a large number of system components which frequently lead to emergent properties, unexpected dynamics, and characteristics of self-organization (Jørgensen, 1995; Prigogine, 1997; Levin, 1999). The study of complexity has a history of at least several decades, ranging from physical, biological, and to social sciences, and a recent resurgence of interest in complexity issues is evident as new theories and methods have mushroomed in the past few decades (see Wu and Marceau, this issue and references cited therein). One of the most intriguing and widely-cited theories in the science of complexity is self-organized criticality (SOC; Bak et al., 1988; Bak, 1996). According to the theory of SOC, large interactive systems naturally evolve toward a self-organized critical state in which a minor event can lead to a cascading catastrophe. When a system is at the self-organized critical state, the frequency and magnitude of events follow a power law distribution, and this may be viewed as a statistically stable, internally controlled state with no characteristic scale within the system. At this point, events are correlated across all scales exhibiting a statistical fractal pattern in spatial structure. Bak and Chen (1991) claimed that SOC may explain the dynamics of a wide range of natural and human-related phenomena, including earthquakes, ecosystems, and social and economic processes. Furthermore, the title, as well as the content, of Bak's (1996) book even suggested that SOC was the mechanism of ‘How Nature Works’.
However, while it is extremely intriguing, SOC does not seem adequate for explaining the great diversity of ecological phenomena (Levin, 1999). Of course, it is not surprising that simple statistical analyses may reveal that some ecological variables in certain ecosystems exhibit power–law relationships (e.g. Jørgensen et al., 1998; Solé et al., 1999). However, the existence of a power–law relationship alone is not adequate to prove that a system is at the self-organized critical state because diverse mechanisms may result in such a relationship in both physical and ecological systems (Raup, 1997; Jensen, 1998; Kirchner and Weil, 1998; Levin, 1999). SOC de-emphasizes or completely ignores the existence of multiple-scale constraints and their significance in influencing system dynamics. Bak (1996) asserted that all complex self-organizing systems move themselves to the self-organized critical state just like a sandpile. To the frantically enthusiastic SOC advocates, top-down constraints in controlling system dynamics do not seem to be important. In this regard, SOC appears to represent an extreme reductionist view. Interestingly, Bak and Chen (1991) claimed that SOC was ‘the only model or mathematical description that has led to a holistic theory for dynamic systems’. It is evident from the above discussion, however, that the implications of the theory of self-organized criticality for ecological systems are in sharp contrast with hierarchy theory or any holistic systems theory.
In general, ecological systems are not, and do not behave like, sandpiles. Levin (1999) argued that heterogeneity, nonlinearity, hierarchical organization, and flows are four key elements of complex adaptive systems, like ecosystems, that allow for self-organization to occur. That is, CAS typically become organized hierarchically into structural arrangements through non-linear between-component interactions, and these structural arrangements determine, and are reinforced by, the flows of energy, materials and information among the heterogeneous components. Levin (1999) further suggested that SOC and modular structure represent the two ends of a continuum along which most ecosystems are found in the middle. While we agree with Levin's postulation in general, we dare to speculate that the majority of ecosystems, especially once well-developed, are hierarchically structured, so that component diversity, spatial heterogeneity, process efficiency, and system stability are simultaneously accommodated. Simon (1962) convincingly argued that ‘complexity frequently takes the form of hierarchy, and that hierarchic systems have some common properties that are independent of their specific content.’ In other words, hierarchy is a central structural scheme of the architecture of complexity, and often manifests itself in the form of modularity in nature.
Why are complex systems usually hierarchically organized? For biological and ecological systems, a hierarchical architecture tends to evolve faster, allow for more stability, and thus is favored by natural selection (Simon, 1962; Whyte et al., 1969; Pattee, 1973; Salthe, 1985; O'Neill et al., 1986). Although not all hierarchical systems are stable, the construction of a complex system using a hierarchical approach is likely to be more successful than otherwise as suggested by the watchmaker parable (Simon, 1962; Müller, 1992; Wu, 1999). In evolutionary biology it is well documented that complexity is built upon existing complexity. This is also frequently the case in the business world, the political arena, and the engineering disciplines. For example, to build complex, yet stable and efficient software, computer software engineers have developed the object-oriented paradigm, which is based on the decomposition principle of hierarchy theory (Booch, 1994). In general, successful human problem-solving procedures are hierarchical, too. It has been argued that a non-hierarchical complex system cannot be fully described, and even if it could, it would be incomprehensible (Simon, 1962; Newell and Simon, 1972). In ecology, the hierarchies we construct inevitably result from the interactions between the inherent characteristics of the system under study and the observer who studies the system. While there is no absolute objectivity, how closely a constructed hierarchy corresponds to the structure of the real system significantly affects the usefulness and power of using a hierarchical approach.
In a sense, a hierarchical approach is a way of breaking down complexity and a process of discovering or rendering order. To do so, a number of hierarchical modeling methods have been developed in different disciplines (see Wu 1999 for a review). However, the problem of spatial heterogeneity and the need for spatial explicitness present grand challenges to the application of hierarchy theory in modeling ecological systems. Based on the hierarchical patch dynamics (HPD) paradigm (Wu and Loucks, 1995; Wu, 1999), we present a spatially explicit hierarchical modeling approach to studying complex ecological systems and a modeling software platform that was designed to facilitate the development of HPD models. Not to be confused with specific modeling methods such as cellular automata, genetic algorithms, and Markov chains, the spatial HPD modeling approach is a multiple-scale methodology for studying complex systems that can bring different modeling techniques together in a coherent manner.
Section snippets
Theoretical basis for the spatially explicit hierarchical modeling approach
The theoretical basis for the spatially explicit hierarchical modeling approach is the hierarchical patch dynamics paradigm (HPDP), which emerges out of the integration between hierarchy theory and patch dynamics (Wu and Loucks, 1995; Wu, 1999). The following is a brief discussion of the major elements of HPDP and their ecological implications.
A hierarchical patch dynamics model of the Phoenix urban landscape (HPDM-PHX)
In this section, we demonstrate how to implement the hierarchical patch dynamics paradigm and the scaling ladder approach in modeling complex ecological systems through an example, the hierarchical patch dynamics model for the Phoenix urban landscape (HPDM-PHX). This example is a part of our on-going modeling efforts associated with the Central Arizona-Phoenix Long-Term Ecological Research (CAP-LTER) and related research projects. Although the system under study is an urban landscape, the
Developing a hierarchical patch dynamics modeling platform (HPD-MP)
Constructing and evaluating hierarchical patch dynamics models like HPDM-PHX can be technically complex in terms of programming, data handling, and model linkage and interface. To facilitate the development of such models, therefore, we have been building an HPD-based modeling platform (HPD-MP). In this section we describe the general structure of HPD-MP, and illustrate how it is being used in our effort to develop HPDM-PHX.
Discussion and conclusions
A distinctive feature of the prevailing theme in the science of complexity is that local interactions among components are essential for the organization and global dynamics of complex systems. As Mitchell et al. (1994) pointed out, ‘a central goal of the sciences of complex systems is to understand the laws and mechanisms by which complicated, coherent global behavior can emerge from the collective activities of relatively simple, locally interacting components.’ This view has been reinforced
Acknowledgements
We thank Darrel Jenerette, Habin Li, Matt Luck and Thomas Meyer for their comments on this paper. Darrin Thome and Hoski Schaasfma helped with the development of the spatial multi-species population dynamics model. JW would like to acknowledge the support for his research in hierarchical patch dynamics from US Environmental Protection Agency grant R827676-01-0 and US National Science Foundation grant DEB 97-14833 (CAP-LTER). Although the research described in this paper has been funded in part
References (81)
Scaling water vapor and carbon dioxide exchange from leaves to a canopy: Rules and tools
- et al.
Aggregation in model ecosystems: I. Perfect aggregation
Ecol. Model.
(1987) - et al.
Ecosystem as self-organizing critical systems
Ecol. Model.
(1998) Hierarchical approaches to ecosystem theory
Ecol. Model.
(1992)- et al.
Hierarchy theory as a guide to mycorrhizal research on large-scale problems
Environ. Pollut.
(1991) The ecosystem modeling approach in the coniferous forest biome
- et al.
Scaling ecophysiology from the plant to the ecosystem: a conceptual framework
- et al.
Criticality and scaling in evolutionary ecology
Trends Ecol. Evol.
(1999) - et al.
A patch-based spatial modeling approach: conceptual framework and simulation scheme
Ecol. Model.
(1997) - et al.
Effects of leaf area profiles and canopy stratification on simulated energy fluxes: the problem of vertical spatial scale
Ecol. Model.
(2000)
Self-organized criticality
Sci. Am.
Self-organized criticality
Phys. Rev. A
Software for portable scientific data management
Comput. Phys.
A Bayesian test of hierarchy theory: scaling up variability in plant cover from field to remotely sensed data
Landscape Ecol.
Cross-scale morphology, geometry, and dynamics of ecosystems
Ecol. Monogr.
Aggregation in model ecosystems: II. Approximate aggregation
IMA J. Math. Appl. Med. Biol.
Scaling processes and problems
Plant Cell Environ.
No fractals in fossil extinction statistics
Nature
Disturbance, patch formation and community structure
Proc. Natl. Acad. Sci. USA
Modeling effects of spatial pattern, drought, and grazing on rates of rangeland degradation: a combined Markov and cellular automata approach
A hierarchical model of lotic ecosystems
Ecol. Monogr.
Cited by (409)
Viewpoint: Composing complex earth system models
2023, Ecological ModellingThe network perspective: Vertical connections linking organizational levels
2022, Ecological ModellingAn urban hierarchy-based approach integrating ecosystem services into multiscale sustainable land use planning: The case of China
2022, Resources, Conservation and RecyclingCitation Excerpt :Urban agglomeration often includes a cluster consisting of one or more megacities with a compact spatial structure, close socioeconomic links, and well-developed infrastructure networks (Fang, 2015). According to previous studies, each urban agglomeration belongs to a nested hierarchical system and contains three administrative levels: the city proper, the metropolitan region, and the urban agglomeration (Li et al., 2013; Wu and David, 2002). Our study selected the three largest national-level urban agglomerations in China: the BTH, YRD, and PRD urban agglomerations.