Individual‐based integral projection models: the role of size‐structure on extinction risk and establishment success

Summary Matrix models or integral projection models (IPMs) are commonly used to study the dynamics of structured populations, where discrete or continuous traits influence survival, growth or reproduction. When a population's size is small, as is often the case for threatened species or potentially invasive species arriving in novel habitats, extinction risk may be substantial due to demographic stochasticity. Branching processes, which are individual‐based counterparts to matrix models and IPMs, allow one to quantify these risks of extinction. For discretely structured populations, the theory of multitype branching processes provides analytic methods to compute how extinction risk changes over time and how it depends on the size and composition of the population. Building on prior work on continuous‐state branching processes, we extend these analytic methods to individual‐based models accounting for any mixture of discrete and continuous population structure. The individual‐based IPMs are defined by probabilistic update rules at the level of the individual which determine how each individual with a given trait value dies, changes trait values (e.g. grows in size) or produces individuals with the same or other trait values. We show that probabilities of extinction can be analytically determined by probability‐generating functionals associated with the individual‐based IPMs. In particular, we present analytical expressions for how extinction probabilities change over time and depend on the initial abundance and trait distribution of the population. We illustrate how to numerically implement these methods using data from the short‐lived desert shrub species Cryptantha flava and provide a more general discussion of how to implement these methods to other data sets including those involving fluctuating environmental conditions. As most IPM studies have the necessary data to parameterize individual‐based IPMs, these methods provide a computationally efficient means to explore how continuously structured populations differing in their evolutionary history and environmental context may differ in their vulnerability to extinction or ability to colonize new habitats.