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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 2 . The South-East partial ordering is considered on the 2 , defined by (x 1 ,y 1 )(x 2 ,y 2 )x 1 x 2 and y 1 y 2 . Then the map T is monotone if (x 1 ,y 1 )(x 2 ,y 2 ) implies T(x 1 ,y 1 )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 2 , the curves C 1 ={(x,y):T 1 (x,y)=x}, C 2 ={(x,y):T 2 (x,y)=y} divide the set on the four “curvilinear quadrants” and if e ¯=(x ¯,y ¯) is a fixed point for T (e ¯C 1 C 2 ) then the eigenvalues and eigenvectors of the Jacobian J T (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 2 . Suppose that T has a unique fixed point e ¯ in . Then e ¯ is a global attractor of T on .
MSC:
37E30Homeomorphisms and diffeomorphisms of planes and surfaces
39A10Additive difference equations
39A11Stability of difference equations (MSC2000)
37C70Attractors and repellers, topological structure