Optimal Control in Age-Structured Populations: A Comparison of Rate-Control and Effort-Control

Abstract

This paper investigates the dynamics and optimal harvesting of age-structured populations governed by McKendrick--von Foerster equations, contrasting two distinct harvesting mechanisms: rate-control and effort-control. For the rate-control formulation, where harvesting acts as a direct additive removal term, we derive first-order necessary optimality conditions of Pontryagin type for the associated infinite-horizon optimal control problem, explicitly characterizing the adjoint system, transversality conditions, and control switching laws. In contrast, the effort-control formulation introduces harvesting as a multiplicative mortality intensity dependent on aggregate population size. We demonstrate that this aggregate dependence structurally alters the optimality system, formally generating a nonlocal coupling term in the adjoint equation that links all ages through the total stock. By combining rigorous variational control derivations, and explicit stationary profiles, this work clarifies the profound mathematical and bioeconomic distinctions between additive and multiplicative harvesting strategies.

Figures (7)

• Figure 1: Mechanism map of the paper. Starting from a unified age-structured transport framework, the analysis splits into rate-control and effort-control harvesting. The former yields a local Pontryagin system; the latter generates a nonlinear nonlocal adjoint through aggregate population dependence.
• Figure 2: Evolution of the age-structured population along characteristic curves in the $(t,a)$ plane. The state variable $x(t,a)$ propagates along the characteristics $da/dt = 1$. The boundary inflow control $p(t)$ is applied at $a=0$, and the initial profile $x_0(a)$ is given at $t=0$. The vertical shaded region illustrates the nonlocal aggregate population size $E(t^*)$, obtained by integrating the population density over all ages at a fixed time $t^*$.
• Figure 3: Information flow and system architecture of the infinite-horizon necessary optimality conditions. The system illustrates the fundamental coupling between the forward-in-time demographic state dynamics and the backward-in-time adjoint dynamics (originating from the transversality conditions at $T \to \infty$ and $a=A$). The shadow prices $\lambda$ generated by the adjoint system subsequently dictate the optimal distributed and boundary controls via the switching conditions.
• Figure 4: Stationary age profiles. (a) Rate-control: the direct removal $u(a)$ produces a visible change in slope at the harvesting window boundaries. (b) Effort-control: the multiplicative effort $w(a)$ preserves the exponential structure, yielding a smaller baseline population density. The shaded region indicates the age window $[3,7]$ where harvesting is active. Dashed curves show the unharvested baseline.
• Figure 5: Time-dependent dynamics of the rate-control model with periodic boundary inflow and ramped harvesting. (a) Space--time heatmap of the population density. (b) Age profiles at selected times. (c) Aggregate population $E(t)$ showing an initial transient followed by quasi-periodic oscillations.

Theorems & Definitions (15)

• Theorem 4.2: Necessary conditions
• proof
• Remark 4.3: Scope of the theorem
• Proposition 5.1: Formal adjoint system for the effort-control problem
• Remark 5.2
• Proposition 6.1: Steady-state ODE for the rate-control model
• proof
• Proposition 6.2: Steady-state ODE for the effort-control model
• proof
• Proposition 6.3: Explicit stationary profile for the rate-control model