Stability And Bifurcation Analysis In Shallow Lake Model With Delayed

The eutrophication of shallow lakes is one complicated kind of natural process, which result in the over-growth of phytoplankton because of the over-loaded salts (eg. nitrogen, phosphorus ). The shallow lake system is a powerful tool to deal with the the deterioration problem of the quality of the lake. It also can integrate organically the theory analysis and the research result, and apply them to the shallow lake. It gradually becomes an independent method to research the eutrophication of shallow lakes.In the study of biomathcmatics system of the shallow lake model, former researchers have paid more attention to the controlling tactics and the realization methods, but few explorations were made in terms of the dynamical behavior of the system. Thus, it is an essential thesis to make an analysis of the dynamical behavior of the system.In this thesis, firstly, we introduce one kind of shallow lake system. Considering the influence of the inner loop of the phosphorus, we discuss the shallow lake system with time-delay. Secondly, we analyze the stability of the equilibrium by using linearizing stability method. When the eigenvalues of the linear part are pure imaginary numbers, we obtain the corresponding delay value. The stability of the steady state is lost when the delay passes through the critical value, as well as there will be a family periodic solutions bifurcate from the steady states, and we find that the stability switch occurs when delay varies. Then, we derive the explicit formulae for determining the direction of the Hopf bifurcation and the stability of these periodic solutions bifurcating from the steady states, by using the normal form method and center manifold theorem. Finally, some numerical simulations are carried out by using MATLAB.