In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.
Quorum quenching (QQ)-based strategies are efficient for biofouling control. However, the feasibility of using QQ bacteria in antibiotic-stressed membrane bioreactors (MBRs) remains unknown. In this study, we isolated three novel QQ strains (Bacillus sp. QX01 and QX03, Delftia sp. QX14) from the activated sludge of an actual MBR. They can degrade 11 N-acyl-homoserine lactones (AHLs) with high efficiencies and rates through intracellular QQ pathways involving putative acylases and lactonases. Running two lab-scale MBRs, we found that introducing antibiotics (sulfamethoxazole, azithromycin, and ciprofloxacin, each at 100 μg/L) shortened the fouling cycle by 71.4 %. However, the immobilized inoculation of QX01 into one MBR extended the fouling cycle by 1.5-2.0 times. Quantitative detection revealed that QX01 significantly reduced the concentrations of two AHLs (C4-HSL and C8-HSL), which were positively correlated with the contents of extracellular polymeric substances (EPS) (Pearson's r = 0.62-0.83, P < 0.01). This suggests that QX01 could perform its QQ activity robustly under antibiotic stress, thereby inhibiting EPS production (proteins especially) and biofilm formation. Moreover, QX01 notably altered the succession patterns of both sludge and fouling communities, with more pronounced effects on abundant taxa. Genera associated with AHL synthesis and EPS production, such as Terrimonas and Rhodobacter, were significantly depleted, contributing to the mitigated biofouling. Additionally, QX01 increased the bacterial community diversity (evenness especially), which was inhibited by antibiotics. Overall, we demonstrate that the novel QQ bacteria could be effective for biofouling control in antibiotic-stressed MBRs, though future work is needed to develop practical approaches for prolonging QQ activity.