Applying symmetries of elasticities in matrix population models

Elasticity analysis is a key tool in the analysis of matrix population models, which describe the dynamics of stage-structured populations in ecology and evolution. Elasticities of the dominant eigenvalue of a matrix model to matrix entries obey certain symmetries. Yet not all consequences of these symmetries are fully appreciated, as they are sometimes hidden in mathematical detail. Here, we propose a method to reason about these symmetries directly by visual inspection of the life cycle graph that corresponds to the matrix model. We present two applications of this method, one in ecology and one in evolution. First, we prove several conjectures about elasticities that were obtained from purely numerical results and that can support population managers in decision-making under scarce demographic information. Second, we show how to identify candidates for invariant trade-offs in evolutionary optimal life cycles. The method extends to the elasticity analysis of non-dominant eigenvalues, of the stochastic growth rate and, in next-generation matrices, of the basic reproduction number.