The Control Of Some Singular Population Ecosystems

It is well known that harvesting has a strong impact on the dynamic behavior of a population. With the increasing of people's demand, people began to scale on the harvest effort on the ecosystems. Some renewable resources are exploited by commercial purpose. Thus the ecological systems are damaged and the original balance is lost.Based on this background, according to the principles of economic and ecology, using the control theory, two singular system models are proposed in this paper, the economic interest of harvesting is considered using an algebraic equation. The feedback controllers which make the systems stable at positive equilibriums are designed. The paper not only investigates the interaction mechanism of the species, but also offers a simple way to study the dynamical behavior from economic interest of harvesting.First, based on a prey-predator system with harvest on predator, a singular system model is proposed. By using the differential-algebraic system theory and bifurcation theory, the local stability of the model around the positive equilibrium is investigated. We know that there is a phenomenon of singularity induced bifurcation around the positive equilibrium in the case of zero economic interest of harvesting, and the model system becomes unstable when economic interest of harvesting is positive. The singularity induced bifurcation can result in impulse phenomenon, which may lead to the collapse of model system. It could lead to the extinction of species. Such a result is that we do not want to see. Therefore, this paper then gives the state feedback control, which makes the system in the positive equilibrium stable, and also makes the economic interests reached an ideal level, and then makes a reasonable biological explanation.Second, in a complex ecological-economic system, selective harvesting is not feasible, thus, based on this background, a two-dimension differential-algebraic model which with harvest effort on two groups is proposed, then gives the state feedback control, which makes the system in the positive equilibrium stable.This paper not only considered to the impact of control on the dynamic evolution of a population, taking into account people's economic interests, which is realistic to produce more practical value.