Abstract Basic leucine zipper (bZIP) is a conserved transcription factor (TF) widely present in eukaryotes, and it plays an important role in regulating plant growth and stress responses. To better understand the white pear bZIP gene family, comprehensive bioinformatics analysis of the pear genome was performed. A total of 84 PbbZIP genes were identified, which were divided into 13 subfamilies by phylogenetic analysis. The 84 PbbZIP genes were all located in the nucleus, and 77 of those genes were unevenly distributed across the 17 chromosomes of white pear. The other 7 PbbZIP genes were located on the scaffold. Subsequent expression profile analysis showed that PbbZIP genes in exocarp were significantly upregulated or downregulated in ‘Huangguan’ pear with brown spot (BS) compared with healthy pear and in response to hormonal treatment with gibberellin A3 (GA3). These results provide helpful insights into the characteristics of PbbZIP genes and their responses to BS in ‘Huangguan’ pear.

The van der Pol equation with a distributed time delay and a strong kernel is analyzed. Its linear stability is investigated by employing the generalized Nyquist stability and Routh-Hurwitz criteria. Moreover, local asymptotic stability conditions are also derived in the case of the strong kernel. By using the mean time delay as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcations. The direction and the stability criteria of the bifurcating periodic solutions are obtained by the graphical Hopf bifurcation theory. Some numerical simulation examples for justifying the theoretical analysis are also given.