An evolutionary approach by second derivatives of the population growth rate of Castilleja tenuiflora, a hemiparasitic plant with and without hosts

Abstract

The first partial derivatives of λ with respect to demographic parameters quantify linear responses of population growth to vital rate perturbations, but sensitivity itself is situational and can show non-linear responses. We use the second derivative of λ with respect to the demographic parameters to explore this type of responses in a population of a root hemiparasite (Castilleja tenuiflora). For plants growing with and without hosts, population transition matrices were built for the 2016–2018 period, and sensitivities and elasticities together with pure derivatives of the Hessian matrix were calculated, together with the second derivatives to survival and fecundity. We transformed the Hessian matrix to a logarithmic scale to avoid values with no biological meaning. Germination showed high sensitivity values, while permanence of individuals with dry aerial tissue that re-sprout had the highest elasticity. Negative values on the second derivative and on the logarithmic representation were obtained for germination with and without hosts in both years. Survival and fecundity have a similar behaviour in both conditions and periods, with negative values for survival for the first category and no negative values for fecundity. High negative values of the second derivative of the transition from seed to seedling indicate an evolutionary stable strategy and imply seeds are important to the permanence of C. tenuiflora population. A stochastic analysis of the second derivative of the transition from seed to seedling was performed with a Gamma distribution analysis, and showed negative values in all cases, corroborating this result.