Global Dynamics and Boundedness of Discrete Population Models
Discrete dynamical systems are widely used in biological and entomological applications to model interacting populations. The manuscripts included in this thesis present global dynamic results for three different population models. Manuscript 1 presents basic concepts and definitions for general systems of difference equations in order to lay the theoretical foundation for the remaining sections.^ Manuscript 2 discusses competitive systems of difference equations with the form xn+1=b1xna 1+xn+c1yn, yn+1=b2yna 2+c2xn+yn n=0,1,2,..., where the parameters b1, b 2 are positive real numbers and α1, α2, c1, c2 and the initial conditions x0, y0 are arbitrary nonnegative numbers. In particular, the special cases when α1 = α 2 = 0 and when α1 ≠ 0 and α2 = 0 are investigated. The global behavior of the system in these cases is fully characterized. Global results are also established for general competitive systems of difference equations that have a particular orientation of equilibria and certain local stability characteristics.^ In Manuscript 3, the system of difference equations xn+1=axn1+b yn,yn+1= gxnynxn+d yn,n=0,1,2,..., is analyzed, where α, β, γ, δ, x 0, y0 are positive real numbers. The system was formulated by P. H. Leslie in 1948 and models a host-parasite type of prey-predator interaction. Manuscript 3 provides the most complete dynamical analysis to date of this classic model. A boundedness and persistence result along with global attractivity results for various parameter regions are established. Numerical evidence of chaotic behavior is also presented for particular solutions of the system.^ Finally, Manuscript 4 discusses structured models of difference equations with the forms: yn+1=Mf1y 1n ,...,fkyk n t,n=0,1,2...,y0 ∈Rk+ ,I and xn+1=AXn+ℓ=1
David T McArdle,
"Global Dynamics and Boundedness of Discrete Population Models"
Dissertations and Master's Theses (Campus Access).