Qualitative Stability and Ambiguity in Model Ecosystems

Abstract

Qualitative analysis of stability in model ecosystems has previously been limited to determining whether a community matrix is sign stable or not with little analytical means to assess the impact of complexity on system stability. Systems are seen as either unconditionally or conditionally stable with little distinction and therefore much ambiguity in the likelihood of stability. First, we reexamine Hurwitz's principal theorem for stability and propose two "Hurwitz criteria" that address different aspects of instability: positive feedback and insufficient lower-level feedback. Second, we derive two qualitative metrics based on these criteria: weighted feedback (wF(n)) and weighted determinants (wDelta(n)). Third, we test the utility of these qualitative metrics through quantitative simulations in a random and evenly distributed parameter space in models of various sizes and complexities. Taken together they provide a practical means to assess the relative degree to which ambiguity has entered into calculations of stability as a result of system structure and complexity. From these metrics we identify two classes of models that may have significant relevance to system research and management. This work helps to resolve some of the impasse between theoretical and empirical discussions on the complexity and stability of natural communities.

The persistence of subsistence: qualitative social-ecological modeling of indigenous aquatic hunting and gathering in tropical Australia

“…Another approach emphasizes a graphical analysis, whereby the topology of an interaction network is assessed to understand the causal influence of interaction pathways, branches, and feedback cycles to a system's dynamics (Roberts 1971, Levins 1974, Coyle 2000. Both of these approaches are based on the same mathematical foundations and can be seen to yield equivalent results (Puccia and Levins 1985, Dambacher et al 2002, 2003b. A third approach is based on qualitative simulations of a set of ordinary differential equations.…”

Modelling climate-change effects on Australian and Pacific aquatic ecosystems: a review of analytical tools and management implications

“…The feedback properties of this model system (Fig. 4) indicate a moderate potential for stability (Dambacher et al 2003).…”

Loop analysis of causal feedback in epidemiology: An illustration relating to urban neighborhoods and resident depressive experiences

“…Moreover, the usefulness of the method generally diminishes with system size, because the findings are much more likely to be ambiguous when more variables or more causal links between them are added to a system. (May, 1973;Dambacher, 2003) Because the mechanics of loop analysis pinpoint precisely where ambiguous results arise, the method can be used to identify where further information can be injected into the analysis to resolve such ambiguities. (Puccia and Levins, 1985;Justus, 2006) For example, an ambiguity arising from opposite signs for two different paths, if we can say we are certain that the magnitude of one of the paths is greater than the other.…”

“…However, because the levels of feedback are predominantly of one sign, and because the model does not distinguish between relative strengths of the linkages, we should expect that the effect of vacancy upon itself is negative using weighted feedback: a method used to resolve ambiguity in loop model predictions. (See Dambacher et al, 2003) By contrast, the ambiguity of the effect of neighborhood exit rate upon vacancy results from negative feedback on the path from exit to depressive experience to death to vacancy, but positive feedback on the direct path from exit rate to vacancy. The predicted effect of neighborhood death rate on vacancy is uncertain for similar reasons, but weighted feedback indicates a positive expectation.…”