Global dynamics of some discrete dynamical systems with applications
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- Tez No: 403445
- Danışmanlar: Belirtilmemiş.
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2016
- Dil: İngilizce
- Üniversite: Unıversıty Of Rhode Island
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
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Özet (Çeviri)
In my rst manuscript, I investigate the global character of the di erence equation of the form xn+1 = f(xn; xn1); n = 0; 1; : : : (1) with several period-two solutions, where f is increasing in all its variables. I show that the boundaries of the basins of attractions of di erent locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or periodtwo solutions. An application of my results give global dynamics of three feasible models in population dynamics which includes the nonlinearity of Beverton-Holt and sigmoid Beverton-Holt types. In this paper I consider Eq.(1) which has three equilibrium points and up to three minimal period-two solutions which are in North-East ordering. More precisely, I will give su cient conditions for the precise description of the basins of attraction of di erent equilibrium points and period-two solutions. The results can be immediately extended to the case of any number of the equilibrium points and the period-two solutions by replicating my main results.In my second manuscript, I investigate the asymptotic behavior of the solutions of the system of di erence equation ~xn+1 = f(n; ~xn; :::; ~xnk); n = 0; 1; : : : ; where k 2 f0; 1; : : :g and the initial conditions are real vectors. I give some e ective conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation. My results are based on application of the linearization technique. I illustrate my results with many examples that include some transition functions from mathematical biology such as linear (also known as Holling type I functions) [5], Beverton-Holt (also known as Holling type II functions or Holling hyperbolic functions), sigmoid Beverton-Holt (also known as Holling type III functions or sigmoid functions) and exponential functions. In this paper I extend some of the results from [4] to the case of vector equation (21). In my third manuscript, I consider the cooperative system xn+1 = axn + by2n 1 + y2n yn+1 = cx2 n 1 + x2 n + dyn; n = 0; 1; : : : ; where all parameters a; b; c; d are positive numbers and the initial conditions x0; y0 are nonnegative numbers. I describe the global dynamics of this system in number of cases. An interesting feature of this system is that exhibits a coexistence of locally stable equilibrium and locally stable periodic solution as well as the Allee's e ect. All global dynamic results for this system can be extended to the general cooperative discrete system in the plane. In my fourth manuscript, I present some basic discrete models in populations dynamics of single species with several age classes. Starting with the basic Beverton-Holt model that describes the change of single species I discuss its basic properties such as a convergence of all solutions to the equilibrium, oscillation of solutions about the equilibrium solutions, Allee's e ect, etc. I consider the e ect of the constant and periodic immigration and emigration on the global properties of Beverton-Holt model. I also consider the e ect of the periodic environment on the global properties of Beverton-Holt model. In this paper I extend Theorems 34 -39 to the case of several generation model with special emphasis on three generation model. I prove general results about asymptotic stability. both local and global which cover all kind of transition or response functions such as linear , Beverton-Holt, sigmoid Beverton-Holt and exponential functions. In order to do so, I introduce some tools in Section 2 which contains some global attractivity results for monotone systems and some di erence inequalities results which lead to precise global attractivity results for non-autonomous asymptotically autonomous di erence equations. In Sections 3 and 4 I obtain fairly general results for local and global asymptotic stability of k-th generations model that extend all results in this section. In the special case of three generation model I nd the precise basins of attraction of all locally stable equilibrium solutions and locally stable period-two solutions.
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