Catastrophe Theory: A Unified Paradigm for Rangeland Ecosystem Dynamics

Abstract

Rangeland ecologists have elucidated 2 apparently distinct processes underlying rangeland dynamics. In some cases, disturbed or recovering rangelands move through a gradual, continuous series of changes which has been termed succession. In other illsfances, rangeland dynamics are typlfied by sudden, discontinuous changes in the vegetation, and this has been called state-and-transition. Catastrophe theory is a mathematical framework designed for the study of discontinuous phenomena, but it also generates models that permit continuous dynamics. Based on available literature, it appears that rangeland ecosystems conform to the mathematics of catastrophe theory. Rangelands exhibit the 5 essential symptoms of catastrophe systems: modality (distinct conditions or states of existence), inaccessibility (conditions which are very unstable), sudden changes (relatively rapid movement between states), hysteresis (processes associated with degradation or recovery are not readily reversible by simply inverting the sequence of events), and divergence (relatively small changes in initial conditions can result in dramatically different outcomes with time). Catastrophe theory has been successfully used to model rangeland grasshopper population dynamics, and it appears that many of the same control variables affecting insects (e.g., temperature and precipitation) may also underlie vegetative community dynamics. Application of catastrophe theory to empirical data sets will require relatively long-term but low-intensity research efforts.

Biotic and Abiotic Changes in Ecosystem Structure over a Shrub-Encroachment Gradient in the Southwestern USA

“…One form of land 58 degradation in the southwestern USA is the invasion of native grasslands by woody shrubs 59 (Buffington and Herbel, 1965;Schlesinger et al, 1990). It is well established that vegetation 60 in semi-arid and arid areas has the potential to transition rapidly from one vegetation state 61 to another; a transition that is not easily reversible on human timescales (Westoby et al 62 1989; Laycock, 1991; Lockwood and Lockwood, 1993;Scheffer and Carpenter, 2003a). …”

Retrospective assessment of dryland soil stability in relation to grazing and climate change

“…The theoretical framework for assessing the response of the SSI to both grazing and droughts is Catastrophe theory (Lockwood and Lockwood 1993). Westoby et al (1989) observed that drylands exhibited linear dynamics and nonlinear dynamics including positive feedbacks, multiple stable states, and thresholds or abrupt changes in vegetation physiognomic composition in response to different disturbances including commercial livestock grazing and droughts.…”

Sahelian Rangeland Development; A Catastrophe?

“…Jumps at distinct values of the control variables, when the latter follow either an increasing or a decreasing path, is referred to as hysteresis. There are 2 ways to formulate this property in relation to the rangeland ecosystem (Lockwood and Lockwood 1993). The first is the formulation that transitions between states are virtually irreversible without substantial human intervention (Friedel 1991).…”

“…It should be emphasized that the 5 catastrophe flags occur if the values of the control variables are within the bifurcation set. The definitions of the catastrophe flags used in this section are based on Giiore (1981) and Lockwood and Lockwood (1993). 1) Bimodality.…”

“…However, the analysis of that example remained rather cursory and therefore Jameson's work merely indicates the potential applicability of the cusp catastrophe to modelling rangeland ecosystem dynamics (Lockwood and Lockwood 1993). Lockwood and Lockwood (1993) pointed out that catastrophe theory has the potential of creating a unified paradigm for rangeland ecosystem dynamics. They based their analysis on the existence of succession-based concepts and of state-and-transition concepts (Westoby et al 1989) in rangeland ecology and management.…”