The Modelling And Control Of A Singular Biological Economic System In A Polluted Environment

The influence of environmental pollution in biological systems has become more and more serious, in order to investigate the development and change rule of the population the pollution factor is necessary to be considered when the mathematical model is modeling. In addition, delay is also a kind of common phenomenon in reality and it has great influence on the dynamic behavior of system, then the delay differential equations is needed to describe the system when the influence of time delay is considered. On the basis of general system theory, the principles of economics, singularity induced bifurcation theory and the Hopf bifurcation theory, this paper investigated the general stability of the singular biological system in a polluted environment.First, the influence of the environmental pollution on the population was overviewed, the research significance and status of the singular economic model and the biological systems with time-delay were introduced, some basic knowledge related to the paper is given.A singular biological economy system affected by toxins was investigated, the stability near the equilibriums of the system was analyzed by using the Routh-hurwitz theorem, the singularity induced bifurcation occurred at the positive equilibrium. Then a state-feedback controller was designed to eliminate the bifurcation. Finally, according to the parameters of a practical example, a numerical simulations was given, the conclusion could be verified by the state response figures.A Singular biological economic model with time-delay was investigated, the system meets the conditions of Hopf bifurcation theorem at the positive equilibrium, then the Hopf bifurcation occurred. Investigate the properties of the Hopf bifurcation by the normal form method and center manifold theorem. Finally, the conclusion was verified by numerical simulations.