Variable Structure Control Of Biological Population Models With Environmental Toxicants

With the development of industry and agriculture, the environmental pollution becomesincreasing serious. By continous or one-off discharge, a large of toxicants are emitted into theenvironment which have some side effects on the growth and reproduction of organisms. Inthis paper, based on some biological population models with environmental toxicants, thecorresponding control models are proposed. According to the principles of variable structurecontrol, controllers are designed to purify environment and ensure persistence of populations.The main contents of this paper are as follows:Firstly, for a biological population model with continuous discharge of environmentaltoxicants, an ideal state is given based on the persistence and extinction of the population.Then, a control model which can be changed to controlliability canonical form is proposed.According to the principles of variable structure control, switching function is chosen, and acontroller is designed to make the control model track the ideal state in finite time.Secondly, combining the priciples of biological dynamics with ecotoxicology, a singularbiological population model with one-off discharge of environmental toxicants is built up. Anideal state is given on the basis of the biological interpretations. Then, the correspondingcontrol model is proposed which can be changed to controlliability canonical form. Accordingto the principles of variable structure control, a controller is designed to make the controlmodel stable at the ideal state in finite time.Finally, for a biological population model with harmful mutant, the correspondingcontrol model which can be changed to controlliability canonical form is proposed based onthe stability of the equilibria. Using the principles of variable structure control, a controllerand a continous controller with approximation are designed, which make the control modelstable at the extinction state of the harmful mutant in finite time, respectively.Based on these above, using Matlab, numerical simulations are carried out to prove theefficiency of variable structure controllers which deal with the control problems of biologicalpopulation models with environmental toxicants.