发表机构
Imagination Corp.; Department of Applied Mathematics, Fictional University; School of Mathematics and Statistics, The University of Sydney（想象公司; 虚构大学应用数学系; 悉尼大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用几何奇异摄动理论和爆破方法，结合参数化方法，严格证明了肿瘤模型中的稳定松弛振荡和瞬态大偏移，并揭示了多种奇异分岔，为肿瘤复发和消除提供了数学基础。
AI 中文摘要
Kuznetsov等人提出的五室肿瘤-免疫模型捕捉了关键的临床现象，包括肿瘤休眠、免疫逃逸（“潜行通过”）和免疫刺激。Osojnik等人近期的研究强调了该模型的兴奋性和振荡动力学，为肿瘤复发提供了一种机制。在本文中，我们通过几何奇异摄动理论（GSPT）和爆破方法的视角重新审视这些发现，为这些现象建立了正式的数学基础。由于该模型具有多重时间尺度结构，动力学分离高达三个数量级，标准的Tikhonov-Fenichel理论不足以应对。为解决此问题，我们采用参数化方法系统地计算慢向量场的高阶近似。我们引入了一项方法论创新，证明参数化方法在几何爆破分析中不可或缺。具体而言，我们表明需要局部中心流形的高阶修正项来匹配相应慢流形上的向量场。利用这些组合工具，我们严格证明了稳定松弛振荡和瞬态大偏移的存在。我们发现了丰富的奇异分岔集合，包括不变圆上的鞍结分岔和两种不同的奇异Andronov-Hopf分岔，它们驱动完全和不完全的canard爆炸。在高度退化的原点处，我们的爆破分析揭示了奇异跨临界分岔和幂零伪奇点的空间碰撞。我们推测这种碰撞形成了一种新的奇异跨临界Bogdanov-Takens组织中心。最终，这些几何发现在效应细胞供应和死亡率的参数空间中产生了精确的分岔图。
英文摘要
The five-compartment tumour-immune model introduced by Kuznetsov et al. captures critical clinical phenomena, including tumour dormancy, immune evasion (`sneaking through'), and immunostimulation. Recent studies by Osojnik et al. highlighted the model's excitable and oscillatory dynamics, providing a mechanism for tumour relapse. In this paper, we recast these findings through the lens of geometric singular perturbation theory (GSPT) and the blow-up method, establishing a formal mathematical foundation for these phenomena. Because the model features a multiple-time-scale structure with dynamics separated by up to three orders of magnitude, standard Tikhonov-Fenichel theory is insufficient. To resolve this, we employ the parametrisation method to systematically compute higher-order approximations of the slow vector fields. We introduce a methodological novelty by demonstrating that the parametrisation method is indispensable within the geometric blow-up analysis. Specifically, we show that higher-order correction terms to the local centre manifolds are required to match the vector field on the corresponding slow manifold. Using these combined tools, we rigorously prove the existence of stable relaxation oscillations and transient large excursions. We uncover a rich collection of singular bifurcations, including a saddle-node on invariant circle and two distinct singular Andronov-Hopf bifurcations that drive complete and incomplete canard explosions. At the highly degenerate origin, our blow-up analysis reveals the spatial collision of a singular transcritical bifurcation and a nilpotent pseudo-singularity. We conjecture that this collision forms a novel singular transcritical Bogdanov-Takens organising centre. Ultimately, these geometric findings yield a precise bifurcation diagram in the parameter space of effector cell supply and death rates.
Comments29 pages, 19 figures
