标题： 动态问题中的分岔与季节更替 显示英文标题

标题： Bifurcation in dynamic problems with seasonal succession

摘要： 我们研究由季节长度参数$\tau$控制的一类非自治常微分方程中平衡点的分岔结构，该参数决定了增长和衰退动力学之间的交替。 这种结构模拟了表现出季节变化的生物系统，如昆虫种群动态或传染病传播。 利用Crandall-Rabinowitz分岔定理，我们建立了在临界阈值$\tau^*$处从灭绝平衡点发生的分岔。 我们还探讨了从一般情况下显式未知的非平凡平衡点产生的次级分岔，这些平衡点只能通过数值方法处理。 我们的结果通过一个两物种竞争的Lotka-Volterra增长季节模型和一个Malthusian衰退季节模型进行说明，其中可以解析地计算主分岔和次级分岔，从而验证数值近似。 我们的分析表明季节性如何驱动两种种群全部灭绝、仅一种灭绝以及两者共存之间的转换。 显示英文摘要

摘要： We investigate the bifurcation structure of equilibria in a class of non-autonomous ordinary differential equations governed by a season length parameter, $\tau$, which determines the alternation between growth and decline dynamics. This structure models biological systems exhibiting seasonal variation, such as insect population dynamics or infectious disease transmission. Using the Crandall-Rabinowitz bifurcation theorem, we establish the existence of a critical threshold $\tau^*$ at which a bifurcation from the extinction equilibrium occurs. We also explore the emergence of secondary bifurcations from, in general, explicitly unknown non-trivial equilibria which can only be treated numerically. Our results are illustrated with a two-species competitive Lotka-Volterra model for the growth season and a Malthusian model for the decline season for which primary and secondary bifurcations may be computed analytically, allowing the validation of numerical approximations. Our analysis shows how seasonality drives transitions between extinction of both populations, of only one, and coexistence of both populations.