Aggregation unveiled: A sequential modelling approach to bark beetle outbreaks.

Tree-killing bark beetle infestations are a cause of massive coniferous forest mortality impacting forest ecosystems and the ecosystem services they provide. Models predicting bark beetle outbreaks are crucial for forest management and conservation, necessitating studies of the effect of epidemiological traits on the probability and severity of outbreaks. Due to the aggregation behaviour of beetles and host tree defence, this epidemiological interaction is highly non-linear and outbreak behaviour remains poorly understood, motivating questions about when an outbreak can occur, what determines outbreak severity, and how aggregation behaviour modulates these quantities. Here, we apply the principle of distributed delays to create a novel and mathematically tractable model for beetle aggregation in an epidemiological framework. We derive the critical outbreak threshold for the beetle emergence rate, which is a quantity analogous to the basic reproductive ratio, R, for epidemics. Beetle aggregation qualitatively impacts outbreak potential from depending on the emergence rate alone in the absence of aggregation to depending on both emergence rate and initial beetle density when aggregation is required. Finally, we use a stochastic model to confirm that our deterministic model predictions are robust in finite populations.