Beyond Size: Matrix Projection Models for Populations Where Size Is an Incomplete Descriptor

Size-based matrix models have been an important analytical tool in the study of populations. Although size can be an important variable in determining demographic parameters such as survivorship, growth, and reproduction, other factors such as genetic variation and access to resources can play key roles. We determined that size was not the only state variable that explained the survivorship, reproduction and future growth of the annual kelp, Alaria nana. A. nana shows persistent differences among individuals in growth (growth autocorrelation) that could be characterized with a negative residual (relatively poor growth) or positive residual (relatively good growth) from the relationship between size and growth. In two separate years of demographic study, we show that, during good growth conditions, growth autocorrelation is relatively strong, while poorer growth conditions associated with the 1997 El Niño event resulted in less consistency among individuals in growth state. To quantitatively describe this population, we used a size based matrix model with an added state to reflect previous growth status. We show that a matrix projection model with growth state better captures the size structure of a natural population of A. nana and the fate of individuals than one that uses only size. Without growth state, a size-based model underestimates the number of individuals that get large and thus the potential growth rate of the population. When we used our model to approximate among-generations pattern, and included a loop analysis algorithm, we found that fast growth to the largest size class contributed greatly to population growth in the relatively good year but that there were not such obvious benefits to getting large in the El Niño year. The size-based model with growth state that we present here is general and applicable to any species where persistent individual differences in performance violate the first-order Markov assumption of stage- or size-based matrix models.