Rigorous Analysis of A Nonlocal Transport–Renewal System for Physiologically Structured Populations

ABSTRACT 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 ‐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 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.