Dynamic Analysis Of A Nutrient-phytoplankton Model With Nutrient Cycling

Water is the indispensable life foundation for human survival,with the continuous improvement of living standards and economic level,a large number of industrial waste water and urban waste water,especially those containing nitrogen and phosphorus elements,are discharged without treatment,resulting in eutrophication of many freshwater and frequent algal blooms.When the algal blooms erupt,the dissolved oxygen in the water decreases dramatically,and a large number of fish suffocate to death,which brings great harm to humans and the environment.In order to solve the problem of cyanobacteria bloom,based on the chemostat model,the nutrient circulation phenomenon and two control measures in water are considered,and the properties of the corresponding kinetic model are constructed and studied.The overall structure of the paper is as follows:In Chapter 1,the introduction introduces the current situation of the water environment at this stage,the background knowledge related to cyanobacterial blooms,and the current work done by researchers on this problem and some basic theories to be used in this paper.In Chapter 2,considering the circulation of nutrients in water,a continuous model of nutrient-phytoplankton with nutrient circulation is constructed.The qualitative and stability properties of the model,including the existence and stability of equilibriums are studied.Finally,numerical simulation is used to verify the theoretical results.The results show that if the concentration of nutrients in the water is large,the effect of the nutrient circulation on the model is negligible.If the concentration of nutrients in the water body is very small,the effect of the nutrient circulation phenomenon on the model is the most prominent.In Chapter 3,considering the state feedback control measures,a semi-continuous dynamic model with state feedback control is established.Based on the properties of the corresponding continuous model,the existence,uniqueness and orbital asymptotic stability of the order-1 periodic solution of the model under different control thresholds are studied by using the semi-continuous dynamical system theory,the successor function,the compression mapping theorem,and the class Poincare criterion.In addition,the numerical results verify the theoretical analysis,and the results show that different control thresholds and control intensities produce different control effects.In Chapter 4,considering that the water quality of water purification is staged by water conservancy regulation,a nutrient-phytoplankton Filippov model with nutrient circulation under the action of state switching is constructed.The dynamic properties of the sub-model are given,and the sliding mode dynamic properties and global dynamic properties of the Filippov model are discussed based on the kinetic properties of the sub-model.Theoretical and numerical results show that by monitoring the concentration of phytoplankton in the water,the appropriate control threshold is selected for hydraulic regulation,so that the amount of phytoplankton in the water is maintained within a certain range,and the bloom phenomenon caused by the proliferation of phytoplankton is limited,which provides a certain theoretical reference for the treatment of cyanobacterial blooms.