Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On competitive discrete systems in the plane. I. Invariant Manifolds   Let $T$ be a $C^{1}$ competitive map on a rectangular region $R\subset
\mathbb{R}^{2}$. The main results of this paper give conditions which guarantee
the existence of an invariant curve $C$, which is the graph of a continuous
increasing function, emanating from a fixed point $\bar{z}$. We show that $C$
is a subset of the basin of attraction of $\bar{z}$ and that the set consisting
of the endpoints of the curve $C$ in the interior of $R$ is forward invariant.
The main results can be used to give an accurate picture of the basins of
attraction for many competitive maps.
  We then apply the main results of this paper along with other techniques to
determine a near complete picture of the qualitative behavior for the following
two rational systems in the plane.
  $$x_{n+1}=\frac{\alpha_{1}}{A_{1}+y_{n}},\quad
y_{n+1}=\frac{\gamma_{2}y_{n}}{x_{n}},\quad n=0,1,...,$$ with
$\alpha_1,A_{1},\gamma_{2}>0$ and arbitrary nonnegative initial conditions so
that the denominator is never zero.
$$x_{n+1}=\frac{\alpha_{1}}{A_{1}+y_{n}},\quad
y_{n+1}=\frac{y_{n}}{A_{2}+x_{n}},\quad n=0,1,...,$$ with
$\alpha_1,A_{1},A_{2}>0$ and arbitrary nonnegative initial conditions.

18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat