Balancing: A solution to our distorted indices of transient dynamics

Because populations are at the mercy of random disturbances large and small, they rarely, if ever, converge on predicted long‐term behaviors. Therefore, when employing matrix population models, ecologists study the dynamics of populations that depart from stable distributions. Necessary for such studies are indices of transient dynamics that measure the size of short‐term population fluctuations. These indices advance our understanding of population dynamics by revealing that population growth rate in a single timestep can far exceed the stable population growth rate. Despite their value, indices of transient behavior possess two major shortcomings: they are scale dependent and easily distorted by outsized population classes. Distortion occurs whenever immature classes, due to their sheer size, carry greater weight in the calculation of population size than mature classes. Beluga sturgeon (Huso huso), for example, have an immature age class (eggs) that is several orders of magnitude larger than its mature age classes. To remove the undue influence of outsized classes, I use balancing, which rescales classes by the stable population distribution and makes the indices of transient dynamics scale invariant. I apply balancing to 1800 population projection matrices for various species across the Animal Kingdom, using reactivity and the Henrici metric of non‐normality as indices of transient dynamics. I found that balancing profoundly changes the picture of which populations have the greatest or least potential transient dynamics. Using a population projection matrix for northern pike (Esox lucius), I demonstrate how balancing influences pseudospectra contour plots that are used to infer transient dynamics.