Fig. 4: Relationship between host density and Bd-driven population depression (δ) in R. darwinii in a 12-month period, predicted by a spatial IBM.

a, The influence of \({\lambda }_{\text{rd}}\) and \({\lambda }_{\text{syntopic}}\), which denote the density (frogs m⁻²) of R. darwinii and a tolerant host, respectively, on \(\delta\) is shown for two combinations of infection parameters (\({{\rm{\alpha }}}_{{p}_{\inf ,\text{rd}}}\) and \({\beta }_{{p}_{\inf ,\text{rd}}}\)), representing the mean estimates of these parameters from RFC and HUI. b,c, The relationship between \(\delta\) and \({\lambda }_{\text{syntopic}}\), which is well described by the Gompertz function \(\delta ({\lambda }_{\mathrm{syntopic}})=L\exp (-b\exp\)\((-c{\lambda }_{\mathrm{syntopic}}))\) (b) and its derivative \((\frac{{\rm{d}}\delta }{{\rm{d}}{\lambda }_{\text{syntopic}}})\) (which indicates the sensitivity of \(\delta\) to changes in \({\lambda }_{\text{syntopic}}\)—that is, how much the population depression changes when \({\lambda }_{\text{syntopic}}\) is increased by a small amount), for three values of \({\lambda }_{\text{rd}}\) (c). The nonlinear relationship observed in the contour lines in a arises because at very low \({\lambda }_{\text{rd}}\), stochastic infection events cause large relative impacts on \(\delta\), leading to high population depression when compared with a no-Bd baseline. As \({\lambda }_{\text{rd}}\) increases, the effect of stochasticity is reduced, resulting in a temporary plateau in depression. However, once the host density is high for the pathogen to invade, sustained Bd transmission drives further population depression, and as \({\lambda }_{\text{rd}}\) increases and Bd transmission approaches saturation a lower value of \({\lambda }_{\text{syntopic}}\) is required to produce an equivalent population depression. The results in a and b represent the median value from 1,000 simulations for each parameter combination. Contour lines in a indicate \(\delta\) values of 0.25, 0.5, 0.75 and 1. Value \(\delta =0\) means no depression, while \(\delta =1\) indicates total extinction due to Bd-induced mortality. Intermediate values reflect increasing levels of population depression.