Mathematics, Walailak University, Nakhon Si Thammarat, Thailand.
Mathematics, Ohio University, Athens, OH, USA.
J Biol Dyn. 2021 Dec;15(1):455-522. doi: 10.1080/17513758.2021.1971781.
We study a population model where cells in one part of the cell cycle may affect the progress of cells in another part. If the influence, or feedback, from one part to another is negative, simulations of the model almost always result in multiple temporal clusters formed by groups of cells. We study regions in parameter space where periodic '-cyclic' solutions are stable. The regions of stability coincide with sub-triangles on which certain events occur in a fixed order. For boundary sub-triangles with order '', we prove that the -cyclic periodic solution is asymptotically stable if the index of the sub-triangle is relatively prime with respect to the number of clusters and neutrally stable otherwise. For negative linear feedback, we prove that the interior of the parameter set is covered by stable sub-triangles, i.e. a stable -cyclic solution always exists for some . We observe numerically that the result also holds for many forms of nonlinear feedback, but may break down in extreme cases.
我们研究了一个群体模型，其中细胞周期的一部分可能会影响另一部分细胞的进程。如果来自一个部分到另一个部分的影响或反馈是负面的，那么对模型的模拟几乎总是会导致由细胞群形成的多个时间集群。我们研究参数空间中的区域，其中周期性“-循环”解决方案是稳定的。稳定区域与特定事件以固定顺序发生的子三角形重合。对于边界子三角形，我们证明，如果子三角形的阶数相对于集群数是互质的，则 -循环周期性解是渐近稳定的，否则是中性稳定的。对于负线性反馈，我们证明参数集的内部由稳定的子三角形覆盖，即对于某些，总是存在稳定的 -循环解。我们通过数值观察到，对于许多形式的非线性反馈，该结果也成立，但在极端情况下可能会失效。
