American Institute of Mathematical Sciences

$x_{n+1} = T(x_{n}),\quad n=0,1,2,...$ (E)

where $T:\mathcal{ R}\rightarrow \mathcal{ R}$ is a competitive (monotone with respect to the south-east ordering) map on a set $\mathcal{R}\subset \mathbb{R}^2$ with nonempty interior. We assume the existence of a unique fixed point $\overline{e}$ in the interior of $\mathcal{ R}$. We give very general conditions which are easily verifiable for (E) to exhibit either competitive-exclusion or competitive-coexistence. More specifically, we obtain sufficient conditions for the interior fixed point $\overline{ e}$ to be a global attractor when $\mathcal{ R}$ is a rectangular region. We also show that when $T$ is strongly monotone in $\mathcal{ R}^{\circ}$ (interior of $\mathcal{ R}$), $\mathcal{ R}$ is convex, the unique interior equilibrium $\overline{ e}$ is a saddle, and a technical condition is satisfied, the corresponding global stable and unstable manifolds are the graphs of monotonic functions, and the global stable manifold splits the domain into two connected regions, which under additional conditions on $\mathcal{R}$ and on $T$ are shown to be basins of attraction of fixed points on the boundary of $\mathcal{R}$. Applications of the main results to specific difference equations are given.

$\upsilon_{\tau}=[D(\upsilon)\upsilon_{x}]_{x}+f(\upsilon) \tau\ge 0, x\in R,$

where $f$ is a monostable (i.e. Fisher-type) nonlinear reaction term and $D(\upsilon)$ changes its sign once, from positive to negative values, in the interval $\upsilon \in [0, 1]$ where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density $\upsilon$ at time $\tau$ and position $x$. The existence of infinitely many traveling wave solutions is proven. These fronts are parameterized by their wave speed and monotonically connect the stationary states $\upsilon \equiv 0$ and $\upsilon \equiv 1$. In the degenerate case, i.e. when $D(0) = 0$ and/or $D(1) = 0$, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.

$u'(t)= Au(t)+f(t,u(t),\int_0^tk(t,s)u(s)ds), t>0, t\ne t_i,$
$u(0)= u_0,$
$\Delta u(t_i)= I_i(u(t_i)). i= 1,2,....,p.$

in a Banach space X, where A is the infinitesimal generator of a strongly continuous semigroup,$ \Delta u(t_i)=u(t^+_i)-u(t^-_i)$ and $I's$ are some operator. We apply the semigroup theory to study the existence and uniqueness of the mild solutions, and then show that the mild solution give rise to classical solution if $f$ is continuously differentiable.