AI 中文总结
该研究在无限性状空间的种群模型中，结合突变、选择与密度依赖调节，推导确定性近似系统，明确有界与约束死亡率的差异，证明约束死亡率下非零平衡态的存在及唯一性，数值实验验证了相关理论结果。
AI 中文摘要
我们研究了一个在可数无限性状空间上结合突变、选择和密度依赖调节的性状结构化种群模型。该基础随机过程是一个连续时间马尔可夫链，其中个体以与性状无关的速率繁殖，后代性状由定义在$\boldsymbol{Z}$上的突变核决定，死亡率依赖于性状，且当种群接近固定的种群上限时，出生率会逐渐受到抑制。利用具有可数多种类型的密度依赖马尔可夫种群过程的相关结果，我们推导出一个无穷非线性微分方程组形式的确定性近似。我们证明了解的存在性与正不变性，并研究了该确定性系统的平衡态结构。研究中发现了有界死亡率和约束死亡率剖面之间的根本区别：当死亡率保持有界时，突变可能会持续将质量传递到整个性状空间，且不一定存在稳态性状分布；相反，当死亡率随性状指数的绝对值增大而无界增长时，算子$L=D^{-1}P$（其中$P$和$D$分别控制突变和死亡率）是紧算子。通过将紧性与巴拿赫格上紧正算子的Kreǐn-Rutman理论相结合，我们证明$L$具有代数简单的主特征值和严格正的特征向量，并推导出非零平衡态存在的阈值条件。在该情况下平衡态是唯一的，其性状分布由$L$的主特征向量决定。数值实验支持了理论结果，并说明了有界死亡率和约束死亡率剖面对应的不同行为。
英文摘要
We study a trait-structured population model incorporating mutation, selection and density-dependent regulation on a countably infinite trait space. The underlying stochastic process is a continuous-time Markov chain in which individuals reproduce at a trait-independent rate, offspring traits are determined by a mutation kernel on $\mathbb Z$, mortality depends on trait, and births are progressively suppressed as the population approaches a fixed population ceiling. Using results for density-dependent Markov population processes with countably many types, we derive a deterministic approximation in the form of an infinite system of nonlinear differential equations. We establish existence and positive invariance of solutions, and investigate the equilibrium structure of the deterministic system. A fundamental distinction emerges between bounded and confining mortality profiles. When mortality remains bounded, mutation may continually transport mass through the trait space and a stationary trait distribution need not exist. In contrast, when mortality increases without bound as the absolute value of the trait index becomes large, the operator $L=D^{-1}P$, where $P$ and $D$ govern mutation and mortality, is compact. By combining compactness with Kre\uın-Rutman theory for compact positive operators on Banach lattices, we show that $L$ has an algebraically simple principal eigenvalue with a strictly positive eigenvector, and we derive a threshold condition for the existence of a non-zero equilibrium. In this regime the equilibrium is unique, and its trait distribution is determined by the principal eigenvector of $L$. Numerical experiments support the theoretical results and illustrate the contrasting behaviours associated with bounded and confining mortality profiles.
Comments42 pages, 10 figures
