Projection matrices in variable environments: λ1 in theory and practice

► The true λ1 is the dominant eigenvalue of the reproductive submatrix. ► Routine calculation of λ1 may result in a false growth rate instead of the true λ1. ► The false growth rate can only be <1, yet it still overestimates the true λ1. ► “Geometric mean” of annual projection matrices depends on the order of cofactors. ► Simulation method to estimate the stochastic growth rate avoids the both problems.