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

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.

💡 Research Summary

This paper investigates optimal harvesting in age‑structured populations modeled by the McKendrick–von Foerster transport equation, focusing on two fundamentally different harvesting mechanisms: rate‑control and effort‑control. In the rate‑control formulation, harvesting appears as an additive removal term u(t,a) directly subtracting individuals of age a. The authors formulate an infinite‑horizon discounted profit maximization problem, introduce a Lagrange multiplier (the adjoint variable λ), and apply Pontryagin’s maximum principle. They derive the adjoint PDE ∂ₜλ + ∂ₐλ = (r + μ(t,a))λ − c(t,a)·𝟙_{c>λ}, together with boundary condition λ(t,0)=k(t) and a transversality condition λ→0 as t→∞. Switching rules for the distributed control u and the boundary inflow p are expressed in terms of the sign of c − λ, and complementary slackness handles the non‑negativity constraint on the state.

In contrast, the effort‑control model treats harvesting as a multiplicative mortality intensity w(t,a) that scales with the total stock E(t)=∫₀ᴬ x(t,a)da. Natural mortality μ depends on E, i.e., μ=μ(E,a), introducing a non‑local dependence into the state dynamics: ∂ₜx + ∂ₐx = −