Abstract
This article is concerned with an optimal harvesting problem over an infinite horizon for age-dependent predator-prey system and the analysis of long-term behaviors of the optimal-controlled system. Instead of usual concepts, overtaking (or catching-up) optimality is adopted. First-order necessary conditions for optimal controller are carefully derived by means of Dubovitskii–Milyutin functional analytical extremum theory, which differs from the Lagrangian multipliers and the variation of calculus. Another focus of the article is the investigation of turnpike properties of population densities. Our results demonstrate the convergence of the optimal trajectories to an equilibrium in the sense of integral average, weak and point-wise, respectively. This research extend the results for linear equations of single species to situations of interacting species, and provide a unified framework for the treatment of multi-dimensional problems.
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