Harvesting of a single-species system incorporating stage structure and toxicity
This paper treats a model of a single-species ecosystem which is divided into immature and mature populations \(x\) and \(y\), resp., where some toxicant attacks \(x\), and harvesting among the population \(y\) is admitted. The resulting two-dimensional differential system is composed of one linear equation with harvesting \(E\) as a parameter, and one cubic equation where the third-order term expresses the toxicant effect. First, the equilibria of this system are investigated, and it is shown that the unique equilibrium in the open positive quadrant is globally asymptotically stable. Secondly, it is assumed that harvesting \(E\) is not a constant and, thus, a three-dimensional vector field for \((x,y,E)\) arises. A sufficient condition for global asymptotic stability of the so-called bionomic equilibrium for \((x,y,E)\) is then derived by introducing a suitable Lyapunov function. Finally, an optimal control problem is considered, with the state variables \(x,y,\) the control variable \(E\), and the net gain as the performance criterion. However, the corresponding calculations seem to be incorrect since, when solving the adjoint equations of Pontryagin's maximum principle, the variables \(x,y,E\) are treated as constants. Some numerical results conclude the paper.
scientific article; zbMATH DE number 5701462