Qualitative analysis of the community matrix

This work sets forth a reformulation of Levins' loop analysis for the qualitative modeling of complex dynamical systems. Relationships between members of ecological communities can be analyzed through a qualitatively specified community matrix, whereby +1, -1, or 0 represent effects of one species upon another. A contribution is made to the analysis of ambiguity in predictions of system response to disturbance and system stability. The equilibrium response of a perturbed model system is determined by the countervailing balance of complementary feedback cycles, which are composed of all direct and indirect effects. The degree to which the correct sign or direction of a response can be predicted is determined by the proportion of countervailing feedback, as detailed in a 'weighted predictions' matrix. Similarly, the potential for qualitative stability is determined by a countervailing balance of overall system feedback, through the measure of 'weighted stability'. These measures are determined by system structure, are independent of system size, and are derived through the use of the matrix permanent, and classical adjoint matrix. These qualitative techniques are tested against an array of ecological systems selected from the published literature, and are used to pose falsifiable hypotheses for previously unexplained results, and provide novel insights into system behavior. Further validation is accomplished through simulations that suggest the weighted measures of prediction and stability are a robust means to assess system ambiguity. A discovery was made of the occurrence of the Fibonacci number series embedded within the prediction matrices. The reciprocal relationship between community members can be described, in a dynamical sense, by a convergent value of Phi. This work supports Levins' original theme that a qualitative understanding of community structure can provide critical insights into biological complexity