The long-run distribution of births across environments under environmental stochasticity and its use in the calculation of unconditional life-history parameters

Matrix population models assume individuals develop in time along different stages that may include age, size or degree of maturity to name a few. Once in a given stage, an individual’s ability to survive, reproduce or move to another stage are fixed for that stage. Some demographic models consider that environmental conditions may change, and thus the chances of reproducing, dying or developing to another stage depend on the current stage and environmental conditions. That is, models have evolved from a single transition matrix to a set of several transition matrices, each accounting for the properties of a given environment. These models require information on the transition between environments, which is in general assumed to be Markovian. Although great progress has been made in the analysis of these models, they present new challenges and some new parameters need to be calculated, mainly the ones related to how births are distributed among environments. These parameters may help in population management and to calculate unconditional life history parameters. We derive for the first time an expression for the long-run distribution of births across environments, and show that it does not depend only on the long-range frequency of different environments, but also on the set of all transition and fertility matrices. We also derive the long-run distribution of deaths across environments. We provide an example using a real data set of the dynamics of Saiga antelope. Theoretical values closely match the observed values obtained in a large set of stochastic simulations.