Dynamic Modeling And Theoretical Analysis Of Flocculation Of Unicellular Algae With Size-Structure

In this thesis,several types of unicellular algae models with flocculation effect and size-structure under autotrophic or heterotrophic conditions are constructed and investigated.Using integral semigroups theory,linear stability theory,fixed point theorem,contraction semigroups theory,step method,normal cone,and adjoint system,etc.,the dynamic properties of these models are studied.For example,the well-posedness of the solution,the existence of equilibrium state,the local and global stability of equilibrium state,the minimum principle of optimal control problems,and the existence and uniqueness of optimal control,etc.This thesis specifically includes the following research contents:In the third chapter,based on the effect of cells size on photosynthesis,and the difference in flocculation of cells with different sizes,a size-structured model describing flocculation of unicellular algae under autotrophic condition is constructed.First,the well-posedness of the solution is analyzed.The existence and uniqueness of the solution is demonstrated through the use of the theory of integral semigroups.Then,the equilibrium state of the model is analyzed by some classical methods.It is proved that,if the threshold quantity R₀>1,the model has at least one positive equilibrium state,which means that the algae population will not become to be extinct.If threshold quantity R₀<1,then,when the load of algae cells in the initial state is relatively low,the algae population will become to be extinct;when another threshold quantity V₀<1,the algae population will certainly become to be extinct.Finally,some numerical examples are given,which show the existence of the interesting phenomenon of backward bifurcation when R₀<1<V₀.In the fourth chapter,according to the difference in nutrient absorption of algae cells with different diameters,and the influence of cell diameter on nutrient absorption rate by the cells,an unicellular algae model with heterotrophy,flocculation effect and size-structure under heterotrophic condition is considered,and the wellposedness of the model is proved with the help of the fixed point theorem.Then,the equilibrium state of the model is analyzed,and it is proved that if threshold quantity R₀>1,the model has at least one positive equilibrium state.If threshold quantity V₀<1,there is no positive equilibrium state.By using the theory of contraction semigroups,the sufficient condition for the local stability of the positive equilibrium state of algae is obtained.Sufficient condition for the global stability of the boundary equilibrium state is given by using the characteristic equation.Finally,some numerical simulations are performed to show that the model has the interesting backward bifurcation within the implicit parametric range R₀<1<V₀,and the local stability condition of the positive equilibrium state of algae can be satisfied.In the fifth chapter,based on the time delay of algal reproduction,and during the process of flocculating algae for the purpose of harvesting algae or removing harmful algae from polluted water bodies,it is necessary to simultaneously control algae using methods other than flocculation.In view of this,an optimal control problem of a size-structured unicellular algae model with time delay is proposed.First,the model is dimensionless for the convenience of research and application,the well-posedness is established employing the fixed point theorem and the step method,and stability of equilibrium states are analysed.Then,the minimisation problem is outlined,the existence and uniqueness of optimal control are established,and the optimal feedback control law is provided.Finally,some numerical simulations are performed.It is discovered that the control variable v（t）has a significant effect on algal bloom control,and the functional value J（v*）is different when P0（s,t）or σ is different.These results have implications for the development,utilisation,and management of renewable resources.The main innovations of this thesis are as follows:1.The size of algal cells has a significant impact on photosynthesis and flocculation.Therefore,in the third chapter,a size-structured model describing flocculation of unicellular algae is constructed for the first time.The well-posedness and stability of equilibrium states of the model are analysed by the theory of integral semigroups etc.The biological significance of the results is given,and the existence of the interesting backward bifurcation is observed.2.Heterotrophic culture is widely used in industrial production.On the basis of the research work in the third chapter,a PDE model describing the relationship between the size of algae cells,the nutrient absorption and flocculation effect is proposed in the fourth chapter.By using the fixed point theorem and threshold quantity R₀,the well posedness of the solution and the existence of the positive equilibrium state of algae are obtained.By using the theory of contraction semigroups etc,the sufficient conditions for the global stability of the boundary equilibrium state,and the local stability of the positive equilibrium state of algae are obtained.The appearances of the backward bifurcation of equilibrium states and bistability are observed numerically.3.Other methods of control are typically needed when using flocculation for algal harvesting or sewage treatment.Therefore,an optimal control problem of sizestructured unicellular algae flocculation model is proposed in the fifth chapter.By using the fixed point theorem,the well-posedness of the solution is obtained.For the minimization problem,the optimal feedback control law is derived,and its existence and uniqueness are established.Some numerical simulations are performed.It is discovered that the control variable has a significant effect on algal bloom control.