Control theory and the management of ecosystems: A threshold policy with hysteresis is robust

摘要

Control theory is much used in engineering to stabilize a given dynamical system at a desired equilibrium point or to confine its trajectories to a region. Extinction of species is one of the most serious problems facing fishery and, to avoid it, different policies are applied. The threshold policy (TP) is a harvesting strategy commonly used in fisheries all over the world, and also is a special and simple case of the variable structure control (VSC). In this paper, the concept of virtual equilibrium point is used to design three different kinds of threshold policies; the standard one (TP), one with delay (TPD) and finally with hysteresis and delay (TPHD), for logistic models subject to Euler as well as two different nonstandard discretization schemes. Uncertainties in the intrinsic growth rate, in the carrying capacity, in the population density, and in the effort policy as well as an overexploitation situation are considered. A time lag in the control policies is also introduced. The important novel characteristic of the TPD and TPHD is that both ensure that, even though the system is subjected to uncertainties and a period of overexploitation, the system eventually stabilizes in bounded oscillations in a desired safe region of the state space. In addition, the sustainable yield under the proposed policies is compared with that of the proportional policy proposed in [1], in order to discuss economic aspects of the proposed threshold policies.