Estimating effects of constraints on plant performance with regression quantiles

Rates of change in final summer densities of two desert annuals, Eriogunum abertianum and Haplopappus gracilis, as constrained by their initial winter germination densities were estimated with regression quantiles and compared with mechanistic fits based on a self-thinning rule proposed by Guo et al. (1998); Oikos 83: 237-245). The allometric relation used was equivalent to S = N-f (N-i)(-1) = c(f) (N-l)(-1), where S is the ratio of final to initial densities (survivorship), c(f) is a constant that is a final density specific to the species and environment. N-i is the initial plant density, and N-f is final plant density. We used regression quantiles to estimate c(f) assuming the exponent of -1 was fixed (model 1, N-f (N-i)(-1) = c(f) (N-i)(-1)) and also obtained estimates by treating the exponent as a parameter to estimate (model 2, N-f (N-i)(-1) = c(f) (N-i)(lambda)). Regression quantiles allow rates of change to be estimated through any part of a data distribution conditional on some linear function of covariates. We focused on estimates for upper (90-99th) quantiles near the boundary of the summer density distributions where we expected effects of self-thinning to operate as the primary constraint on plant performance. Allometric functions estimated with regression quantiles were similar to Functions fit by Guo et al. (1998) when the exponent was constrained to - 1. However, the data were more consistent with estimates for model (2), where exponents were closer to - 0.4 than - 1, although model fit was not as good at higher initial plant densities as when the exponent was fixed at - 1. An exponential form (model 3, N-f (N-i)(-1) = c(f) (N-i)(lambda)e(gamma Ni)) that is a generalization of the discrete logistic growth function, where estimates of lambda were - 0.23 to - 0.28 and estimates of gamma were - 0.003 to - 0.006, provided better lit from low to high initial germination densities. Model 3 predictions were consistent with an interpretation that final summer densities were constrained by initial germination densities when these were low ( < 40 per 0.25 m(2) for Eriogonum and < 100 per 0.25 m(2) for Haplopappus) and were constrained by the self-thinning process at higher germination densities. Our exponential model(3) estimated with regression quantiles had similar form to the mechanistic relation of Guo et al. (1998) when plotted as a survivorship function, but avoided the unrealistic assumption that all populations attained a similar final density, and was based on a statistical model that has formal rules for estimation and inference.