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Competitive-exclusion versus competitive-coexistence for systems in the plane. (English) Zbl 1116.37030
Let us consider a map $$T$$ on a set $${\mathcal R}\subset\mathbb{R}^2$$. The South-East partial ordering is considered on the $$\mathbb{R}^2$$, defined by $$(x_1, y_1)\preceq (x_2,y_2)\Leftrightarrow x_1\leq x_2$$ and $$y_1\geq y_2$$. Then the map $$T$$ is monotone if $$(x_1, y_1)\preceq (x_2, y_2)$$ implies $$T(x_1, y_1)\preceq T(x_2, y_2)$$. Throughout this paper, the properties of monotone maps are studied. If $$T= (T_1, T_2)$$ be a map on a set $${\mathcal R}\subset\mathbb{R}^2$$, the curves $$C_1= \{(x, y)\in{\mathcal R}: T_1(x, y)= x\}$$, $$C_2= \{(x, y)\in{\mathcal R}: T_2(x,y)= y\}$$ divide the set $${\mathcal R}$$ on the four “curvilinear quadrants” and if $$\overline e= (\overline x,\overline y)$$ is a fixed point for $$T$$ $$(\overline e\in C_1\cap C_2)$$ then the eigenvalues and eigenvectors of the Jacobian $$J_T(\overline e)$$ have properties that are fundamental for the theory developed in this paper. The following result is typical: Let $$T$$ be a monotone map on a closed and bounded rectangular region $${\mathcal R}\subset\mathbb{R}^2$$. Suppose that $$T$$ has a unique fixed point $$\overline e$$ in $${\mathcal R}$$. Then $$\overline e$$ is a global attractor of $$T$$ on $${\mathcal R}$$.

MSC:
 37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000) 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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