The golden rule of complementary feedback

This work demonstrates the occurrence of the Fibonacci sequence in the qualitative analysis of Lotka---Volterra dynamical systems. Herein we show the golden ratio to govern reciprocal effects between neighboring variables in simple food web models. Impacts to the entire community resulting from perturbation of a population variable can be predicted from the adjoint of the community (Jacobian) matrix, which we render in qualitative terms of complementary feedback cycles. Sequences of complementary feedback cycles follow the Fibonacci sequence, and are also configured as multiples and overlapping harmonics thereof. We derive an absolute-feedback matrix that clarifies the sequence. Patterns of complementary feedback cycles are determined by community structure, which can be portrayed and understood in terms of signed digraph structure.