Electronic Theses and Dissertations

# A-Stability For Two Species Competition Diffusion Systems

2006-08-15

Nguyen, Tung

Dissertation

## Department

Mathematics and Statistics

My dissertation research focuses on establishing the structural stability of the attractor ($\mathcal{A}$-stability) via Morse-Smale property for diffusive two-species competition systems $$\label{abstract} \begin{cases} \p_t u=k_1\Delta u+u f(x,u,v),\quad x\in\Omega,\cr \p_t v=k_2\Delta v+v g(x,u,v),\quad x\in\Omega,\cr Bu=Bv=0,\quad x\in \p\Omega, \end{cases}$$ \noindent on a $C^{\infty}$ bounded domain $\Omega\subset \mathbb{R}^n,\ n\geq 1,$ with either Dirichlet or Neumann boundary conditions. Here $u(t,x)$, $v(t,x)$ are the densities of two competing species, $k_1,\ k_2$ are diffusive constants and $(f,g),\ \ f,g:\bar{\Omega}\times \RR\times \RR \to \RR\ \ C^2\ \mbox{functions}$ satisfying \begin{itemize} \item[](\textbf{H1}) $f(x,0,0)>0,\ g(x,0,0)>0\ \forall x\in \bar{\Omega},$ \item[](\textbf{H2}) $\ \partial_u f(x,u,v),\ \partial_v f(x,u,v),\ \partial_u g(x,u,v),\ \partial_v g(x,u,v)<0,\ \forall\ u, v\geq 0,\ \forall x\in \bar{\Omega},$ \item[](\textbf{H3}) $\sup_{x\in\bar{\Omega},\ v\geq 0}\limsup_{u\rightarrow\infty}f(x,u,v)<0,$ \item[](\textbf{H4}) $\sup_{x\in\bar{\Omega},\ u\geq 0}\limsup_{v\rightarrow\infty}g(x,u,v)<0.$ \end{itemize} These hypotheses describe key features of competition models, and since $u$ and $v$ are the densities of two species, we are only interested in nonnegative solutions $(u,v)$. We therefore consider \eqref{main-eq} in the positive cone of some appropriate phase space. Our main result states for the spatially one-dimensional case that if (\ref{abstract}) is a Morse-Smale system on the positive cone, it is structurally stable. We also establish that the set of functions $(f,g)$ for which \eqref{main-eq} possess the Morse-Smale property, is open in the space of all pairs $(f,g)$ satisfying ({\bf H1})--({\bf H4}) under the topology of $C^2$-convergence on compacta. Moreover, we show as a sufficient condition that if all critical elements of \eqref{main-eq} are hyperbolic with one-dimensional unstable manifolds in case of equilibria (except 0) and two-dimensional unstable manifolds in case of periodic orbits, then system \eqref{main-eq} has the Morse-Smale property. These results will have significant impact on the study of the asymptotic dynamics of various classes of discretizations of (\ref{abstract}).