dc.description.abstract | 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
\begin{equation}
\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}
\end{equation}
\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}). | en_US |