The cusp catastrophe has been the most frequently applied catastrophe manifold in ecology. The reasons probably lie somewhere between (1) the recognition that the states of ecological processes or systems are likely to be multidimensional and hence of greater dimension than the fold catastrophe manifold, and (2) the empirical and mathematical difficulties encountered in identifying, measuring, and interpreting multiple independent ecological or environmental parameters. Nevertheless, the cusp manifold may be used effectively to understand the relationship between ecosystem or community states, and perturbative mechanisms.
For example, consider again the case of a coral-dominated tropical reef.A typical characteristic of this state is the maintenance of coral dominance by active microalgal foraging by herbivores such as echinoids and grazing fish. An alternative state of algal dominance is often achieved by the combination of a decline in herbivory, and the storm-driven destruction of coral colonies. While herbivore populations are healthy, the coral-dominated community exhibits resilience against storm-driven state transitions. Historical overfishing and recent epidemic events, however, have served to reduce herbivore levels in many Caribbean reefs, thereby undermining community resilience, and facilitating storm-precipitated transitions to macroalgae-dominated reefs. These dynamic relationships are explained by the cusp manifold, where the control parameters are the intensity of herbivory and storm intensity, and the response variable is the degree of coral or macroalgal dominance (Figure 7). It is obvious from the manifold that coral-dominated reefs with healthy levels of herbivory are robust against storm perturbations, even at high storm intensity. Coral dominance is maintained, even as herbivory declines, if perturbation intensity is low. Coral-dominated reefs with low herbivory that are subjected to high or increasing levels of perturbation will, however, encounter the fold catastrophe and transition to macroalgal dominance. Moreover, coral-dominated reefs with only slightly different levels of herbivory can diverge in their states at increasingly greater levels of perturbation; that is, one will transition catastrophically at the cusp to a state of macroalgal dominance, while the other will remain dominated by coral. The hysteresis effect suggests that recovery of coral dominance can occur only when levels of herbivory have improved significantly, and have done so under conditions of relatively low storm intensity. The cusp manifold may be applied as a qualitative explanation to other similar types of transitions between apparently stable community states, the most familiar example being lake eutrophication in response to increasing nutrient loads. Here again there are two states, usually measured with water clarity, between which the system will shift catastrophically as the result of
external perturbation, and internal changes in factors such as macrophyte abundance and the population densities of bottom-feeding fish.
In all these examples it should be noted, however, that 'catastrophe theory' is used as an explanation of observed events; its application as a predictive tool is more problematic for several reasons. First, ecological parameters are often difficult to quantify, and are sometimes measured indirectly (using proxy parameters), or without proper constraint on sources of error. Second, the relationships among supposed control parameters can be obscure, or not known at all, and when those relationships encompass the nonlinearities present in the neighborhood of a folded manifold, simple linear modeling of empirical data could fail dangerously. These two reasons position the application of 'catastrophe theory' to ecological problems very differently compared to its applications in the physical sciences. For example, the bending beam experiment outlined earlier is described by the well-known 'Euler-Bernoulli beam theory', the equations of which present a well-defined control space for catastrophe manifolds. A third reason is the complexity of ecological systems. There is every reason to believe that catastrophic transitions in communities will often be the result of positive feedback within those communities themselves. Transitions may therefore occur when internal states or interactions of the system have changed to the point where external perturbations are sufficient to precipitate a takeover of system behavior by internal feedback dynamics. This fits very well with catastrophe manifold topologies, but in order to actually predict such points of transition, successful recognition, modeling, and measurement of the feedback mechanisms might be required. The predictive capabilities of the theory improve significantly when external parameters are dominant drivers of the system. For example, epidemic outbreaks of grasshopper communities in the western United States correspond to temperature and precipitation regimes, but sudden transitions from normal to pest population levels place prediction outside the abilities of standard linear models. Catastrophe theory has been applied to this and similar situations to model the system with considerable explanatory and predictive success.
In spite of these difficulties, it should be kept in mind that Thom's theory establishes the governance of most multiparameter systems by catastrophe manifolds. While ecological systems can indeed be complex, current elementary 'catastrophe theory' is capable of dealing with up to five input or control parameters and at the very least, identification of the appropriate manifold, and qualitative exploration of that manifold, will reveal potential pathways to catastrophe in the system. Those manifolds also serve to explain observed transitions between states, when the appropriate control parameters have been identified. Finally, in situations where external drivers are important to systems dynamics, or perhaps as understanding of the complex feedback processes in ecological systems increases, catastrophe theory could become a useful predictive tool.
See also-. Bifurcation; Chaos; Driver-Pressure-State-Impact-Response; Hysteresis; Mathematical Ecology; Parameters.
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