# Global dynamics of some discrete dynamical systems with applications

#### Abstract

We investigate the global character of the difference equation of the form ^ *x*_{n}_{+1} = * f*(*x _{n}*,

*x*

_{n}_{ –1}),

*n*= 0, 1,… ^ with several period-two solutions, where

*f*is increasing in all its variables. We show that the boundaries of the basins of attractions of different 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 period-two solutions. An application of our results give global dynamics of three feasible models in population dynamics which includes the nonlinearity of Beverton-Holt and sigmoid Beverton-Holt types. Consider the difference equation^ [vector

*x*]

_{n}_{+1}=

*f*(

*n*,[vector

*x*]

*,...,[vector*

_{n}*x*]

*),*

_{n-k}*n*= 0,1,…, ^ where k∈{0, 1, …} and the initial conditions are real vectors. We investigate the asymptotic behavior of the solutions of the considered equation. We give some effective conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation. Our results are based on application of the linearizations technique. We illustrate our results with many examples that include some equations from mathematical biology. In this paper we consider the cooperative system^

*x*

_{n}_{+1}=

*ax*+

_{n}*by*

^{2}

_{n}/1 +

*y*

^{2}

_{n}

*y*

_{ n}_{+1}=

*cx*

^{2}

_{n }/1 +

*x*

^{2}

_{n}+

*dy*,

_{ n}*n*= 0,1, …, ^ where all parameters

*a,b,c,d*are positive numbers and the initial conditions

*x*

_{0},

*y*

_{ 0}are nonnegative numbers. We 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 effect. We 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 we discuss its basic properties such as a convergence of all solutions to the equilibrium, oscillation of solutions about the equilibrium solutions, Allee's effect, Jillson's effect, etc. We consider the effect of the constant and periodic immigration and emigration on the global properties of Beverton-Holt model. We also consider the effect of the periodic environment on the global properties of Beverton-Holt model.^

#### Subject Area

Mathematics

#### Recommended Citation

Arzu Bilgin,
"Global dynamics of some discrete dynamical systems with applications"
(2016).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI10190612.

http://digitalcommons.uri.edu/dissertations/AAI10190612