Integral projection models for populations in temporally varying environments

Most plant and animal populations have substantial interannual variability in survival, growth rate, and fecundity. They also exhibit substantial variability among individuals in traits such as size, age, condition, and disease status that have large impacts on individual fates and consequently on the future of the population. We present here methods for constructing and analyzing a stochastic integral projection model (IPM) incorporating both of these forms of variability, illustrated through a case study of the monocarpic thistle Carlina vulgaris. We show how model construction can exploit the close correspondence between stochastic IPMs and statistical analysis of trait–fate relationships in a “mixed” or “hierarchical” models framework. This correspondence means that IPMs can be parameterized straightforwardly from data using established statistical techniques and software (vs. the largely ad hoc methods for stochastic matrix models), properly accounting for sampling error and between‐year sample size variation and with vastly fewer parameters than a conventional stochastic matrix model. We show that the many tools available for analyzing stochastic matrix models (such as stochastic growth rate, λS, small variance approximations, elasticity/sensitivity analysis, and life table response experiment [LTRE] analysis) can be used for IPMs, and we give computational formulas for elasticity/sensitivity analyses. We develop evolutionary analyses based on the connection between growth rate sensitivity and selection gradients and present a new method using techniques from functional data analysis to study the evolution of function‐valued traits such as size‐dependent flowering probability. For Carlina we found consistent selection against variability in both state‐specific transition rates and the fitted functions describing state dependence in demographic rates. For most of the regression parameters defining the IPM there was also selection against temporal variance; however, in some cases the effects of nonlinear averaging were big enough to favor increased temporal variation. The LTRE analysis identified year‐to‐year variation in survival as the dominant factor in population growth variability. Evolutionary analysis of flowering strategy showed that the entire functional relationship between plant size and flowering probability is at or near an evolutionarily stable strategy (ESS) shaped by the size‐specific trade‐off between the benefit (fecundity) and cost (mortality) of flowering in a temporally varying environment.