# Global Dynamics of Some Discrete Dynamical Systems in Mathematical Biology

2019

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Department

Mathematics

Mustafa Kulenović

## Abstract

This thesis will be presented in manuscript format. The first chapter will introduce preliminary definitions and theorems of difference equations that will be utilized in chapters 2, 3, and 4.

The second chapter will investigate the global behavior of two difference equations with exponential nonlinearities

xn+1 = be-cxn + pxn-1, n = 0, 1,...

where the parameters b; c are positive real numbers and p 2 (0; 1) and

xn+1 = a + bxn-1e-xn, n = 0, 1,...

where the parameters a; b are positive numbers. The initial conditions x-1, x0 are arbitrary nonnegative numbers. The two equations are well known mathematical models in biology, which behavior was studied by other authors and resulted in partial global dynamics behavior. In this manuscript, we complete the results of other authors and give the global dynamics of both equations. In order to obtain our results we will prove several results on global attractivity and boundedness and unboundedness for general second order difference equations

xn+1 = f(xn, xn-1), n = 0, 1,...

which are of interest on their own.

The third chapter will investigate the global behavior of the cooperative system

xt+1 = min{r11xt + r12yt, K1}, yt+1 = min{r21xt + r22yt, K2}, t = 0, 1,...

where the initial conditions x0, y0 are arbitrary nonnegative numbers. This system models a population comprised of two subpopulations on different patches of land. The model considers the minimum between the maximum carrying capacity of each patch (K1 or K2 resp.) and the linear combination of the population from patch i from the last time step with those who migrated to patch i for i = 1, 2. We break the behavior of the system into several cases based on whether the linear combination of the population or maximum carrying capacity is greater. We are able to conclude that either one fixed point will be a global attractor of the interior region of ℝ2+ or there will exist a line of fixed points with the stable manifolds as the basins of attractions. We then extend some of these results to the n-dimensional case using similar techniques. We investigate the global behavior of the general cooperative system

xit+1 = min {ri1x1t + ri2x2t + … + riixit + … + rinxnt , Ki},

for i = 1, 2, …, n, and t=0, 1, … where the initial conditions of xi0 are arbitrary nonnegative numbers I = 1, 2, …, n. We are able to conclude in some cases that one fixed point will be a global attractor of the interior region of 2+.

Finally, in the fourth chapter we will prove general results regarding the global stability of monotone systems without minimal period two solutions on a rectangular region R. We will illustrate the general results in two examples of well known systems used in mathematical biology. The first of the systems that will be investigated is a modified Leslie-Gower system of the form

[Refer to PDF for mathematical equation]

where the parameters a, b, c, d are positive numbers, α and β are positive values less than 1, and the initial conditions x0, y0 are arbitrary nonnegative numbers. In most cases for different values of a, b, c, and d, there will either be one, two, three, or four equilibrium solutions present with at most one an interior equilibrium point. In the case when c = d = 1 and a = b, there will exist an infinite number of interior equilibrium points in which case we will and the basin of attraction for each of the equilibrium points.

The second system that will be investigated is a version of a Lotka-Volterra model of the form

[Refer to PDF for mathematical equation]

where the parameters of A, K1, and K2 are all positive and the initial conditions x0; y0 are arbitrary nonnegative numbers, which is a semi implicit discretization of the continuous version. In most cases, there will be between one and three equilibrium points with solutions converging to one of the points. In one case when A > K1 = K2, however, there will exist an infinite number of equilibrium points. In this case for each equilibrium point, there will be a stable manifold as its basin of attraction.

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