Dynamic Analysis Of A Nutritional-phytoplankton Model With Variable Consumption

Protecting fresh water resources is one of the topics that human beings continue to pay attention to.In recent years,the water environment has changed,nitrogen,phosphorus and other elements are enriched in some lakes,resulting in excess nutrition in water,which leads to frequent cyanobacteria bloom,water quality decline,water odor,and even affect the safety of drinking water.Faced with this situation,researchers have taken various measures to control cyanobacteria,such as physical and chemical methods.Considering the relationship between nutrition and phytoplankton interaction,this paper establishes the relevant mathematical models,and analyzes the dynamic properties of the corresponding model by using pulse differential equation and Filippov dynamics system.By means of Matlab software,numerical simulation is carried out to verify the theoretical results.The main results of this paper are as follows:The first chapter introduces the present situation of freshwater ecosystem,the background knowledge of water resources,the research status of mathematical models and the basic knowledge that will be involved in this paper.In the second chapter,a continuous nutrition-phytoplankton model with variable consumption term is constructed for variable consumption problem.The positivity of the model,the existence and stability of boundary equilibrium and positive equilibrium,and the existence of limit cycles are analyzed in detail,and the corresponding numerical simulation is given.In the third chapter,considering the specific measures of algae control,a mathematical model with pulse state feedback control is established.By using the method of successor function of semi-continuous system,the existence of order-1 periodic solution is proved.Combined with the class Poincare criterion,the orbital asymptotic stability of order-1 periodic solution of the corresponding system is obtained,and the theoretical analysis is verified by numerical simulation.In the fourth chapter,based on the qualitative analysis of continuous models in Chapter 2,we construct a class of nutrition-phytoplankton Filippov model with switching effect.Based on the rich dynamic properties of the two subsystems under state switching,sliding mode dynamics of the Filippov system are studied,it mainly includes sliding mode region and sliding mode branch.The theoretical results show that algae growth can be better controlled by reasonable selection of parameters such as threshold level.