生理结构种群的非局部运输-更新系统的严格分析

中文摘要 AI 辅助

我们针对一类具有两个内部状态变量、非局部生态反馈、动态资源、区间间转移和选择性收获的生理结构种群模型，建立了严格的分析框架。该完整模型是耦合的非线性偏微分方程-常微分方程运输-更新系统，在补充边界处具有内生流入，此情形中运输、非局部依赖和边界更新处于同一层面相互作用。针对该完整的非自治多区间系统，我们在基于正L¹的状态空间中证明了有限时间范围适定性，包括任意有界时间区间上的整体存在性、唯一性、非负性，以及对初始数据、环境强迫和收获努力的连续依赖性。随后，我们分离出灭绝时的自治单区间约化，并在补充者空间上构造了正的紧下一代算子。在进一步的非线性平稳约化中，我们证明基本再生数ℛ₀>1的超临界性，在编码密度依赖更新反馈的参数化紧算子假设下，存在非平凡平稳状态。最后，针对紧Lipschitz正则容许类上的有限时间范围收获目标，我们证明了最优控制的存在性。这些结果区分了可针对完整气候显式系统证明的结论与仅在自治约化后可证明的结论，从而阐明了结构种群阈值理论与控制理论的数学适用范围。

英文摘要

We develop a rigorous analytical framework for a class of physiologically structured population models with two internal state variables, nonlocal ecological feedbacks, dynamic resources, inter-zone transfer, and selective harvesting. The full model is a coupled nonlinear PDE--ODE transport--renewal system with endogenous inflow at the recruitment boundary, a setting in which transport, nonlocal dependence, and boundary renewal interact at the same level. For this full nonautonomous multi-zone system, we prove finite-horizon well-posedness in a positive $L^{1}$-based state space, including global existence on arbitrary bounded time intervals, uniqueness, nonnegativity, and continuous dependence on initial data, environmental forcing, and harvesting effort. We then isolate an autonomous single-zone reduction at extinction and construct a positive compact next-generation operator on the recruit space. In a further nonlinear stationary reduction, we prove that supercriticality of the basic reproduction number $\mathcal R_{0}>1$ yields existence of a nontrivial stationary state under a parametrized compact-operator hypothesis encoding density-dependent renewal feedback. Finally, for a finite-horizon harvest objective over a compact Lipschitz-regular admissible class, we establish existence of an optimal control. The results separate what can be proved for the full climate-explicit system from what can be justified only after autonomous reduction, thereby clarifying the mathematical scope of threshold and control theory for structured populations.

发表机构

• College of Engineering, Boston University（波士顿大学工程学院）
• Department of Mathematics, Northeastern University（东北大学数学系）
• College of Engineering, The University of Iowa（爱荷华大学工程学院）

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