A graph theory approach to demographic loop analysis

A demographic analysis of the life-cycle graph can be used to quantify the separate contributions of different life-history types to the population growth rate. Loop analysis has been proposed (van Groenendael et al. 1994) as the appropriate method for partitioning the elasticity matrix to determine these contributions. However, in the analysis of complex demographic models it is difficult to derive the loops by simple inspection of the life-cycle graph. I show how graph theory can be used to describe a general and systematic procedure for deriving the loops from the structure of the life-cycle graph. I demonstrate that the concept of nullity (from graph theory) can be applied in this context to correctly determine the number of loops for any graph. Using examples from Campanula americana, Dipsacus sylvestris, and Caretta caretta, I illustrate the relationship of the loops to biologically relevant life-history contrasts. This relationship is crucial for the application of loop analysis to life-history evolution for the purpose of partitioning the separate effects on the population growth rate among different life-history components.