A mathematical foundation of modelling thermal injury and repair dynamics in ectotherms

Andreas Havbro Faber , Peter Borgen, Bodil Kirstine Ehlers, Johannes Overgaard, Michael Ørsted

As global temperatures rise and extreme heat events impair ectotherm performance and survival, it is becoming increasingly important to predict how organisms accumulate and repair thermal injury under realistic benign and stressful temperatures. The thermal death time (TDT) model quantifies how heat events translate into thermal injury, but under natural temperature fluctuations the TDT model is inadequate as net injury reflects the balance between injury accumulation during stressful temperatures and repair during permissive temperatures. The relative rates of these antagonistic processes both depend on temperature, and repair also depends on the present level of injury. Empirical evidence of the thermal dependency of both rates remains scarce, and so far, they have not been integrated into mathematical foundational frameworks. Here, we develop two separate modelling approaches to infer these latent processes from thermal failure data for two model organisms: the aquatic plant duckweed (Lemna gibba) and the spotted wing fruit fly (Drosophila suzukii). First, we introduce a non-parametric TDT model that estimates how temperature exposure contributes to net injury or repair without assuming a predefined temperature response. The model assumes additivity of injury and repair and performs well at identifying permissive and stressful temperatures for the relatively simple L. gibba dataset. However, for the more complex D. suzukii data, it fails to estimate repair magnitudes because it cannot represent its antagonistic injury dependence. To address this limitation, we introduce a parametric thermal injury-repair model that jointly estimates injury accumulation and repair as antagonistic, temperature dependent latent processes in a probabilistic framework. Here, injury increases exponentially with temperature, while repair follows a unimodal thermal response and depends on accumulated injury through a first order decay process. When applied to the D. suzukii data, the model shows that repair can significantly alter rates of injury accumulation under fluctuating temperatures. Together, these approaches provide a framework for translating any temperature history into trajectories of injury and repair. When expanded across different life-stages, more traits and taxa, this will enable predictions of how fluctuating thermal environments shape ectotherm responses to heat extremes and ultimately global warming vulnerability.