Qualitative Analysis And Bifurcation Research In Several Biological Dynamic Systems

Qualitative theory, which directly judges the qualitative properties of solutions by ordinary differential equations(ODEs) themselves, is very important in the study of ODEs. The ideas of qualitative theory is influencing other mathematical branches gradually. Plentiful results are obtained from systems of lower dimensions, especially from planar systems. A fundamental task here is to analyze the topological or qualitative structures of all limit sets (or invariant sets), which include (1) equilibrium, (2) periodic solution (limit cycle), or (3) equilibria and the orbits which go towards the equilibria as t→+∞, t→-∞. After these limit sets are discussed thoroughly, the structure of system can be determined approximately. However, when system has some degeneracies, it is obviously not sufficient to discuss them only in a simple way or only discuss any one of them. Thus, with the successive change of system parameters, we need to further consider another typical phenomenon-bifurcation. As parameters of a differential system are varied, changes may occur in the qualitative structure of the solutions for certain parameter values. These changes are called bifurcations. If bifurcations or even a versal unfolding are given, all physical phenomena of this system are clear. The difficult cases are bifurcations of codimension≥2 and non-local bifurcations.Portraying systems in bionomics, iatrology and demotogy by constructing mathematical models has gone through a long history and ODEs are the main instrument to analyze their dynamical behaviours. Many people are focusing on the following issues: how to make a model with experiment result or statistical information to reflect the dynamics of actual system; how to extend or perfect the known models so that their dynamics can satisfy the practice or experiment data more accurately, etc.. These models are related to many branches of ODE's research. Corresponding to qualitative analysis, we pay our attention to the existence of local and global solutions, the existence, stablity and non-existence of equilibrium and periodic solutions, globally asmptotical behaviours, and other dynamical properties such as bifurcation and chaos.In Chapter 2, we first introduce the related content of vector field, degenerate equilibrium and its bifurcations. Cmpared with the investigation of equilibria at finity, the analysis of equilibria at infinity also plays an important role. It reflects the tendency of discussed variables growing in large amount, which is helpful for us to discuss global qualitative properties of the system. Then, some methods for discussing infinite equilibrium are also given in Chapter 2.In Chapter 3, we consider periodic solutions and equilibria at infinity for a generalized Brusselator system. This system is a polynomial differential system with order p + q and portrays a process of multi-molecular reaction. Its qualitative results are already very abundant for some special cases. But for the most general case, i.e., p and q are unspecific integers, its qualitative properties need more consideration. After some literature survey, we perfect the qualitative discussion of the system and prove that the multiplicity is at most one when the unique equilibrium is a weak focus, and therefore correct the previous Hopf result. Moreover, applying the Poincare - Bendixson theorem and Bendixson-Dulac criterion, we further discuss the existence and non-existence of periodic solution. Although the generalized Brusselator system has been discussed detailedly, we further analyze it at infinity to get its global dynamics. In some cases, the degeneracies of infinite equilibria are so high that the common methods including the blowing-up method (which decomposes a complicated equilibrium into several simple ones with the Briot-Bouquet's transformation), the Z-sector method and the normal sector method do not work. Our difficulties are overcome in this paper by using the method of generalized normal sectors(GNS for short), provided by [Nonlinearity 17(2004): 1407-1426], which allows curves and orbits to be part of its boundary, may not be an angular neighborhood of the characteristic direction. Using this method we determine the characteristic directions and the numbers of orbits which go towards or away from the equilibria at infinity.Reference [J. Diff. Equ. 188(2003), 135-163] considered Hopf bifurcation, ogdanovTakens bifurcation and existence and uniqueness of periodic solution for a reduced SIRS model with a nonlinear incidence rate. For some difficulties in computation the possiblities of weak focus of multiplicity more than 1 are not proved in the reference. Moreover, the conditions of uniqueness and co-existence of two periodic solutions are hard to check with practical parameters. In Chapter 4, we first prove that the maximal multiplicity of the weak focus is 2 by technically dealing with complicated multivariate polynomials in the computation of the second order Liapunov value. Then, by transforming the reduced SIRS system into a stardard Lienard form we get a feasible condition expressed by the practical parameters for the uniqueness of periodic solu- tion. Furthermore, we consider a typical bifurcation phenomenon-Bogdanov-Takens bifurcation in this system, which is a bifurcation of codimension 3 for a cusp. Since the discussion on non-degerate Bogdanov-Takens bifurcation in [J. Diff. Equ. 188(2003), 135-163] cannot display all periodic solutions, and therefore neither the bifurcation value for the co-existence of two periodic solutions nor the one for the co-existence of a periodic solution and a homoclinic loop is found. In this chapter, these conditions are obtained by a reduction to a form of universal unfolding for a cusp of codimension 3 and a discussion on bifurcations of periodic solutions and homoclinic loops of order 2.In chapter 5, we move over to an Enzyme-Catalyzed reaction system. This system depends on a cubic polynomial with such a complicated relation between its coefficients s_O,α_O,α,κ,ρand the original parameters that the coordinates of equilibria or even the number of equilibria can hardly be determined in many cases. All found results on its qualitative properties and bifurcations are given indirectly for the artificial parameter s_*, a coordinate of a general equilibirum. In this thesis, not following the common idea of computing eigenvalues at equilibria, we give a complete analysis of equilibria directly for those original parameters by using continuity, monotonicity and some techniques of inequality. Moreover, in order to investigate the global trend of system change when the concentrations of the two chemical species increse excessively, we also discuss infinite equilibria of the system by using the GNS method. As for the Bogdanoov-Takens bifurcation analysis of such a system, since not reducing the corresponding perturbed Bogdanoov-Takens system to a versal unfolding form, the condition of parameters for existence of periodic orbits and homoclinic ones was not obtained, although those orbits were simulated numerically for some specified parameters. Therefore, in order to exhibit all phenomena in its Bogdanov-Takens bifurcation (in particular, those phenomena not displayed before), we reduce the original system to its normal form and give a versal unfolding to the parametersκ,ρ. Finally, bifurcation curves of periodic orbits and homoclinic orbits are presented explicitly with the original parameters.