Variable Structure Control Of Red Tide Population Models And Stability Analysis

With the rapid development of the global industry and agriculture, a largenumber of domestic sewage and waste water are emitted into oceans or lakes, onthe one hand, the nitrogen, phosphorus and microelement in the pollutant aboveplay an important role in the growth of the phytoplankton; on the other hand,when the substances exceed a threshold, eutrophication in the ocean appears,which leads to the Red--Tide. Based on the above situation, several Red---Tidealgae are given, and analysised by differential equations; according to theprinciple of marine ecology, the corresponding control models are proposed;The controllers are designed to make our models stable at the ideal equilibriumusing Variable Structure Control principle. The studys provide the basis for thetreatment of Red—Tide, and keep marine ecosystem in blance.The main contents of this paper as follows:Firstly, based on a Red—Tide biological model, the stability of theequilibrium is analysised using the theory of differential equations. For theunstable situation, the corresponding control model is proposed wariableStructure Control; the controller is designed to make the model stable at theideal equilibrium quickly.Secondly, combing biodynamic principle with the theory of marineecosystem, the general logistic model of Red—Tide algae is proposed throughthe interaction of the populations. And this model is anslysised by the theory ofdifferential equation for the possibility of the Red—Tide’s breakout, thecorresponding control model is proposed wariable Structure Control; thecontroller is designed to make the model stable at the ideal equilibrium.Thirdly, Based on the principle of nutrition dynamics and Ecotoxicology, aHAB’s model with toxin is proposed. According to Lyapunov Stability Theory, the threshold of Red Tide outbreak is determined. Using the theory of VariableStructure Control, the controller is design to reduce the possibility of Red TideOutbreak, and make the model stability. Finally, he correctness of theconclusion is tested via numeral simulation.Based on these above, using Matlab, numerical simulation can be carriedout, the efficiency and feasibility of variable structure controllers is proved.