表型结构化群体在均匀竞争下适应度渐近无关

中文摘要 AI 辅助

我们证明，在均匀竞争下的表型结构化群体中，长期表型分布不依赖于增殖速率：它仅由切换动力学的平稳分布决定。具有表型x的细胞密度u(x,t)在表型空间中通过平流和扩散（切换）演化，并以依赖于表型的速率r(x,t)增殖，该速率受对所有表型共同的因子g(U)调节，其中U是总群体大小。这将对区室模型的一个近期结果推广到连续表型。在有界区间上，在无通量边界条件下，总群体单调收敛到承载容量U*，密度在L1中收敛到U*psi，其中psi是切换动力学的平稳密度，非线性模型渐近等价于线性模型；通过一个显式估计，适应度的印记以切换动力学的谱间隙速率衰减。在整个实直线上，我们在两个自然条件下证明同样的收敛性：约束和无逃逸至无穷，涵盖Ornstein-Uhlenbeck动力学、多阱景观和重尾密度。如果g'(U*) < 0且psi满足Poincare不等式，则在加权L2范数下收敛是指数的，而对于Ornstein-Uhlenbeck动力学，没有统一的L1速率成立。证明遵循区室模型的证明：在均匀竞争下，反应项具有一个符号，因此其大小是总群体的增长率，在时间上可积，切换动力学压缩其产生的零质量扰动。由于它仅使用由切换生成的马尔可夫半群，该证明可推广到一般机制；我们将其应用于跳跃的非局部切换和R^d有界域中具有非梯度漂移的扩散，其平稳密度不是Boltzmann形式且携带循环通量。

英文摘要

We prove that, in phenotype-structured populations under uniform competition, the long-term phenotypic distribution does not depend on the proliferation rates: it is the stationary distribution of the switching dynamics alone. The density u(x,t) of cells with phenotype x evolves by advection and diffusion in phenotype space (switching) and by proliferation at a phenotype-dependent rate r(x,t), modulated by a factor g(U) common to all phenotypes, where U is the total population. This extends to continuous phenotypes a recent result for compartmental models. On a bounded interval with no-flux boundary conditions, the total population converges monotonically to the carrying capacity U*, the density converges in L1 to U*psi, where psi is the stationary density of the switching dynamics, and the nonlinear model is asymptotically equivalent to the linear one; by an explicit estimate, the imprint of fitness decays at the spectral gap of the switching dynamics. On the whole real line we prove the same convergence under two natural conditions, confinement and no escape to infinity, covering Ornstein-Uhlenbeck dynamics, multi-well landscapes, and heavy-tailed densities. If g'(U*) < 0 and psi satisfies a Poincare inequality, convergence is exponential in a weighted L2 norm, while no uniform L1 rate holds for Ornstein-Uhlenbeck dynamics. The proof follows the compartmental one: under uniform competition the reaction term has one sign, so its size is the growth rate of the total population, integrable in time, and the switching dynamics contracts the zero-mass perturbations it produces. Since it uses only the Markov semigroup generated by switching, the proof extends to general mechanisms; we apply it to nonlocal switching by jumps and to diffusions with non-gradient drift in bounded domains of R^d, whose stationary density is not of Boltzmann form and carries a circulating flux.

发表机构

• Instituto de Matemática e Computação, Universidade Federal de Itajubá（伊塔朱巴联邦大学数学与计算研究所）
• Carl Gustav Carus School of Medicine, Technische Universität Dresden（德累斯顿工业大学卡尔·古斯塔夫·卡鲁斯医学院）

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