AI 中文总结
该研究针对环境周期性下的两物种趋化-竞争系统，推导了第一物种临界趋化灵敏度的存在条件，揭示了周期态的暂态失稳与模式交替规律，并通过数值模拟验证了相关结论。
AI 中文摘要
本研究探讨了在两个物种的环境容纳量均随时间呈周期性变化的假设下，具有竞争动力学的两物种趋化系统中周期性模式的形成机制。我们将对应的非自治常微分方程（ODE）系统的全局吸引周期解作为空间齐次基态，推导得出了第一个物种存在临界趋化灵敏度（记为$χ_{\text{crit}}$）的充分条件。一旦超过该阈值，周期态会在每个环境周期内的非空时间集合上发生线性失稳，进而形成不稳定区间与稳定区间的周期性交替：空间扰动在不稳定区间内被放大，在稳定区间内被衰减。我们还在周期共存和竞争排除两种机制下开展了数值模拟，对解析推导结果进行了补充。最后，我们讨论了失稳集合的时间位置以及负失稳阈值存在的可能性。
英文摘要
This work investigates recurrent pattern formation in a two--species chemotaxis system with competitive dynamics under the assumption that the carrying capacities of both species are periodic in time. By considering a globally attracting periodic solution to the associated nonautonomous ODE system as a spatially homogeneous base state, we derive sufficient conditions for the existence of a critical chemotactic sensitivity of the first species, denoted by $χ_{\text{crit}}$. Whenever this threshold is exceeded, the periodic state becomes linearly unstable over a nonempty set of times within each environmental cycle. This gives rise to a recurrent alternation of unstable and stable intervals, during which spatial perturbations are respectively amplified and damped. The analytical derivation is complemented with numerical simulations, in both periodic coexistence and competitive exclusion regimes. Finally, we discuss the temporal location of the instability sets and the possibility of negative instability thresholds.
