Establishment dynamics of structured populations: Insights from individual-based integral projection models

Background/Question/Methods: Population establishment, including biological invasions, recovery from a disturbance, and outbreaks of an infectious disease, often begin with small numbers of individuals exhibiting a mixture of continuous (e.g. size) and discrete (e.g. developmental stage) traits. Vital rates, such as growth, survivorship, and reproduction, of each individual may depend on it’s traits. Consequently, not only the initial size of a founding population but also the distribution of traits among individuals within the population is likely to determine the likelihood of establishment. Recently, integral projection models (IPMs) have become a popular means of describing dynamics of continuously and discretely structured populations. These IPMs provide important insights into long-term population growth, asymptotic trait distributions, and trait-dependent reproductive values. However, they do not account for finite population sizes and, consequently, they cannot provide detailed insights into the dynamics of establishment. To address this limitation, we introduce a general framework for stochastic, individual based IPMs to address three questions (i) how does the probability of establishment depend on the initial size and trait distribution of the founding population? (ii) how quickly does the risk of establishment failure diminish over time?, and (iii)  how sensitive are these probabilities to the functional relationship between individual traits and vital rates?

Results/Conclusions: To answer these three questions, we developed (i) an analytic characterization of the probability of establishment success and the temporal distribution of establishment failures that depends on the initial size of the population and its trait distribution, (ii) analytic formulas for the sensitivities of these probabilities to the trait-dependent growth, survival, and reproduction kernels, and (iii) numerically efficient methods in R to compute the establishment probabilities, temporal distributions of establishment failures, and their sensitivities. Furthermore, we have extended these methods for individual based IPMs in temporally heterogeneous environments. To illustrate the implementation of these methods and how they provide insights into the establishment dynamics of structured populations, we applied our methods to an individual based IPM of the endangered, perennial monocarp Humboldt Bay Wallflower (Erysimum menziesii) using 9 years of quarterly data collected by the US Fish and Wildlife Service and the Bureau for Land Management. Among several things, our preliminary analysis reveals that despite strong signs of recovery of this plant species, as measured by lambda of the mean-field IPM, large variation in lifetime seed production makes establishment in new habitats difficult.