#### Title

#### Date of Award

2019

#### Degree Type

Dissertation

#### Degree Name

Physician of Philosophy in Mathematics

#### Department

Mathematics

#### First Consultant

Mustafa Kulenović

#### Abstract

This thesis will be provided in manuscript format. The very first chapter will introduce preliminary definitions and theorems of difference equations that’ll be found in chapters 2, 3, and 4.

The 2nd chapter will investigate global behavior of two difference equations with exponential nonlinearities

*x _{n}*

_{+1}=

*be*+

^{-cx}_{n}*px*

_{n}_{-1},

*n*= , 1,…

in which the parameters b c are positive real figures and p 2 ( 1) and

*x _{n+1}* =

*a*+

*bx*

_{n}_{-1}

*e*,

^{-x}_{n}*n*= , 1,…

in which the parameters a b – are positive figures. The first conditions *x*_{-1}, *x*_{} are arbitrary nonnegative figures. The 2 equations are very well known mathematical models in biology, which behavior was studied by other authors and led to partial global dynamics behavior. Within this manuscript, we complete the outcomes of other authors and provide the worldwide dynamics of both equations. To be able to obtain our results we’ll prove several results on global attractivity and boundedness and unboundedness for general second order difference equations

*x _{n}*

_{+1}=

*f*(

*x*,

_{n}*x*

_{n}_{-1}),

*n*= , 1,…

that are of great interest by themselves.

The 3rd chapter will investigate global behavior from the cooperative system

*x _{t}*

_{+1}= min,

*y*

_{t}_{+1}= min,

*t*= , 1,…

in which the initial conditions x0, y0 are arbitrary nonnegative figures. This technique models a population made up of two subpopulations on several patches of land. The model views the minimum between your maximum transporting capacity of every patch (*K*_{1} or *K*_{2} resp.) and also the straight line combination of people from patch i in the before step with individuals who migrated to patch *i* for *i* = 1, 2. We break the behaviour from the system into several cases according to if the straight line combination of people or maximum transporting capacity is larger. We could conclude that each one fixed point is a global attractor from the interior region of ℝ^{2}* _{+}* or there’ll exist a type of fixed points using the stable manifolds because the basins of attractions. Then we extend a few of these leads to the

*n*-dimensional situation using similar techniques. We investigate global behavior from the general cooperative system

*x ^{i}_{t}*

_{+1}= min ,

for *i *= 1, 2, …, *n*, and *t*=, 1, … in which the initial conditions of *xi* are arbitrary nonnegative figures *I* = 1, 2, …, *n*. We could conclude in some instances that certain fixed point is a global attractor from the interior region of ℝ^{2}* _{+}*.

Finally, within the 4th chapter we’ll prove general results concerning the global stability of monotone systems without minimal period two solutions on the rectangular region *R*. We’ll illustrate the overall leads to two types of well-known systems utilized in mathematical biology. The very first from the systems that’ll be investigated is really a modified Leslie-Gower system from the form

[Make reference to PDF for mathematical equation]

in which the parameters *a*, *b*, *c*, *d* are positive figures, α and β are positive values under 1, and also the initial conditions *x*_{}, *y*_{} are arbitrary nonnegative figures. Generally for various values of *a*, *b*, *c*, and *d*, there’ll be either one, two, three, or four equilibrium solutions usual to for the most part each interior equilibrium point. Within the situation when *c* = *d* = 1 and *a* = *b*, there’ll exist thousands of interior equilibrium points by which situation we’ll and also the basin of attraction for each one of the equilibrium points.

The 2nd system that’ll be investigated is a number of Lotka-Volterra type of the shape

[Make reference to PDF for mathematical equation]

in which the parameters of *A*, *K*_{1}, and *K*_{2} are positive and also the initial conditions x0 y0 are arbitrary nonnegative figures, that is a semi implicit discretization from the continuous version. Generally, you will see between one and three equilibrium points with solutions converging to among the points. In a single situation whenever a > *K*_{1} = *K*_{2}, however, there’ll exist thousands of equilibrium points. Within this situation for every equilibrium point, you will see a reliable manifold since it’s basin of attraction.

Resourse:https://digitalcommons.uri.edu/oa_diss/875/

Key:Global Dynamics of Some Discrete Dynamical Systems in Mathematical Bio by Sarah Van Beaver