Age-Structured PDEs in Economics, Ecology, and Demography: Optimal Control and Sustainability

Abstract

Optimal control of partial differential equations arises in population ecology, economics, and demography. The consistency of mathematical treatment is demonstrated for the Lotka-McKendrick model and its nonlinear modifications of increasing complexity. The obtained qualitative optimal dynamics show that the models have either the bang-bang structure of optimal controls or follow balanced growth dynamics.age-structured populations, harvesting, medical capital investment, partial differential equations, size-structured populations, vintage capital models,

Bang-Bang, Impulse, and Sustainable Harvesting in Age-Structured Populations

“…A growing number of publications [1][2][3][4][5][6][7][8][9][10][11] is devoted to analysis of their mathematical properties (existence, uniqueness, persistence, etc.). However, a little study is provided in using these models to portray features of the related biological populations.…”

Optimal harvesting in forestry: steady-state analysis and climate change impact

“…Speaking mathematically, a bang-bang structure of harvesting efforts u reflects selective logging when all trees with a certain diameter are to be harvested. Bang-bang principles for age-structured models are proven in [6,13,18] and for size-structured models in [14,17]. Following [17], the strong bang-bang principle is valid for (1)- (6): If Equation (17) holds and…”

“…where γ (t) is defined by Equation (14). The first two integrals produce Equations (9) and (10) and the last two integral terms lead to the dual system (11).…”

Modelling and optimization for palm oil plantation management

“…1 u and 2 u are control variables. In Equation (1), 1 u is the rate of planting new young palm oil, and in Equation (2), 2 u is the rate of felling inefficient old palm oil trees. As mentioned above, it is assumed that the amount of produced oil is proportional with the amount of biomass therefore,…”

“…In general, the population of trees varies during their lifetime and depends on age, size, or stage of species development [1]. A comprehensive survey on the mathematical modelling and analysis of the population of individuals is given in [2].…”