Matrix scaling and tipping points

To assess which ecosystems are most vulnerable it is necessary to compare the resilience of complex interaction networks in a meaningful way. A fundamental problem for the comparative analysis of ecological stability is that the organisms in ecological networks operate on different time scales. A conventional solution to this problem has been to assume the intraspecific interaction strengths in the dynamical system (and diagonal elements in the community matrix) have the same value, ignoring the time scale differences, and therefore disregarding vital ecological information. In this paper, we consider two methods that have previously been developed to deal with community matrices arising from populations with widely different time scales and which contain differing self-regulation terms (diagonal entries). One approach considers the critical self-regulation in a system by proportionally adjusting the diagonal entries until the tipping point is found. The other is a scaling procedure that translates the intraspecific information on the diagonal on to the off-diagonal entries. We show the relation between the leading eigenvalue of the latter, and the numerical diagonal parameter of the former, which in many ecologically relevant networks is exact. In addition, we show for 3 \times 3 scaled competitive systems how the feedback determines whether the leading eigenvalue is real- or complex-valued, which is important for knowing when the scaling procedure remains ecologically sensible. While arising from an ecological setting, this work has wider implications in network theory and linear algebra.