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Optimal harvesting and stability for fishing models with stage structure in inshore-offshore areas. (English) Zbl 1043.34058

This paper considers the following fishing model with stage structure in inshore-offshore areas, \[ \begin{matrix} \frac{dx_1}{dt}&=&\alpha y_1-\gamma x_1-cx_1,\\ \frac{dy_1}{dt}&=&cx_1-\beta y_1^2+\sigma(y_2-y_1)-q_1E_1y_1,\\ \frac{dy_2}{dt}&=&-sy_2+\sigma(y_1-y_2)-q_2E_2y_2, \end{matrix} \] where \(\alpha,\gamma,c,\beta,\sigma,s\) are all positive constants, \(q_1\) and \( q_2\) are catchability coefficients for inshore and offshore population, respectively; \(E_1\) and \(E_2\) are the harvest effort for inshore and offshore population, respectively. Steady states of the dynamical system representing fishery are derived and their local, as well as global stability is discussed. It is shown that if the fishing effort in inshore area, where is the breeding palace, does not exceed some catching threshold and the dispersal between inshore and offshore area satisfies some parametric relation, then the selective harvest for two subpopulations both in inshore and offshore areas will be sustainable; otherwise, the biological steady state will be broken and fishing activity can not be sustainable. The optimal harvest policy is formulated and solved as a control problem. It is obtained that the optimal equilibrium levels of each subpopulation both in inshore and offshore areas are less than the equilibrium levels. Finally, a numerical example is given to illustrate the solution procedure.

MSC:

34D23 Global stability of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
49N90 Applications of optimal control and differential games
92D40 Ecology
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