Fig. 1
Equilibria, thresholds and optima of the BSIT model. Density of males at stable (solid lines) and unstable (dashed lines) equilibria given release rate (R) under SIT (a) and BSIT (b). A bifurcation, where stable and unstable equilibria converge, provides an elimination threshold for SIT \(\left( {R_{{\mathrm{Thresh}}}^{{\mathrm{SIT}}} = 1414.7} \right)\). Pyriproxifen reduces the threshold \(\left( {R_{{\mathrm{Thresh}}}^{{\mathrm{BSIT}}} = 281.6} \right)\) and the distance between stable and unstable equilibria (b). With SIT, elimination time grows asymptotically at \(R_{{\mathrm{Thresh}}}^{{\mathrm{SIT}}}\) (blue dashed), whereas boosting can shift this asymptote below \(R_{{\mathrm{Thresh}}}^{{\mathrm{BSIT}}}\) (red dashed) (c). With R fixed (R = 1414), elimination time responds asymptotically to competitiveness (d), and boosting shifts the threshold (hThresh) towards zero. Thresholds for the eventual elimination of any initial population (solid), and for eliminating from carrying capacity in one (dashed) or two (dotted) years respond non-linearly to release rate (R) and competitiveness (h) (e). Two years (dotted) and 1 year (dashed) elimination thresholds for BSIT (red) are indistinguishable. The total release required for elimination (RTotal) is minimised at \(1.8 \times R_{{\mathrm{Thresh}}}^{{\mathrm{SIT}}}\) and \(0.7 \times R_{{\mathrm{Thresh}}}^{{\mathrm{BSIT}}}\) for SIT and BSIT, respectively (f, dashed lines). All simulations were initialised at carrying capacity, with M0 the initial density of males
