Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane

and Applied Analysis 3 We now present some basic notions about competitive maps in plane. Define a partial order on R2 so that the positive cone is the fourth quadrant, that is, x1, y1 x2, y2 if and only if x1 ≤ x2 and y1 ≥ y2. For x,y ∈ R2 the order interval x,y is the set of all z such that x z y. A set A is said to be linearly ordered if is a total order on A. If a set A ⊂ R2 is linearly ordered by , then the infimum i infA and supremum s supA of A exist in R 2 −∞,∞ × −∞,∞ . If both i and s belong to R2, then the linearly ordered set A is bounded, and conversely. We note that the ordering may be extended to the extended plane R 2 in a natural way. For example, 0,∞ a, b if a ≥ 0 or a ∞. If x ∈ R2, we denote with Q x , ∈ {1, 2, 3, 4}, the four quadrants in R 2 relative to x, that is, Q1 x, y { u, v ∈ R 2 : u ≥ x, v ≥ y}, Q2 x, y { u, v ∈ R 2 : x ≥ u, v ≥ y}, and so on. A map T on a set B ⊂ R2 is a continuous function T : B → B. The map is smooth on B if the interior of B is nonempty and if T is continuously differentiable on the interior of B. A set A ⊂ B is invariant for the map T if T A ⊂ A. A point x ∈ B is a fixed point of T if T x x, and a minimal period-two point if T2 x x and T x / x. A period-two point is either a fixed point or a minimal period-two point. The orbit of x ∈ B is the sequence {T x }∞ 0. A minimal period two orbit is an orbit {x }∞ 0 for which x0 / x1 and x0 x2. The basin of attraction of a fixed point x is the set of all y such that T y → x. A fixed point x is a global attractor on a set A if A is a subset of the basin of attraction of x. A fixed point x is a saddle point if T is differentiable at x, and the eigenvalues of the Jacobian matrix of T at x are such that one of them lies in the interior of the unit circle in R2, while the other eigenvalue lies in the exterior of the unit circle. If T T1, T2 is a map on R ⊂ R2, define the sets RT −, : { x, y ∈ R : T1 x, y ≤ x, T2 x, y ≥ y} and RT ,− : { x, y ∈ R : T1 x, y ≥ x, T2 x, y ≤ y}. For A ⊂ R2 and x ∈ R2, define the distance from x toA as dist x,A : inf {‖x − y‖ : y ∈ A}. A map T is competitive if T x T y whenever x y, and T is strongly competitive if x y implies that T x − T y ∈ { u, v : u > 0, v < 0}. If T is differentiable, a sufficient condition for T to be strongly competitive is that the Jacobian matrix of T at any x ∈ B has the sign configuration ( − − ) . 2.2 For additional definitions and results e.g., repeller, hyperbolic fixed points, stability, asymptotic stability, stable and unstable manifolds see 8, 9 for competitive maps, and 10, 11 for difference equations. IfA is any subset of R, we shall use the notation clos A to denote the closure ofA in R, and A◦ to denote the interior of A. The next results are stated for order-preserving maps on R and are known but given here for completeness. See 12 for a more general version valid in ordered Banach spaces. Theorem 2.1. For a nonempty set R ⊂ R and a partial order on R, let T : R → R be an orderpreserving map, and let a, b ∈ R be such that a ≺ b and a, b ⊂ R. If a T a and T b b, then a, b is invariant and i there exists a fixed point of T in a, b , ii if T is strongly order preserving, then there exists a fixed point in a, b which is stable relative to a, b , 4 Abstract and Applied Analysis iii if there is only one fixed point in a, b , then it is a global attractor in a, b and therefore asymptotically stable relative to a, b . Corollary 2.2. If the nonnegative cone of is a generalized quadrant in R, and if T has no fixed points in u1, u2 other than u1 and u2, then the interior of u1, u2 is either a subset of the basin of attraction of u1 or a subset of the basin of attraction of u2. Define a rectangular region R in R2 to be the cartesian product of two intervals in R. Remark 2.3. It follows from the Perron-Frobenius theorem and a change of variables 9 that, at each point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively. Also, one can show that if the map is strongly competitive then no eigenvector is aligned with a coordinate axis. Theorem 2.4. Let T be a competitive map on a rectangular region R ⊂ R2. Let x ∈ R be a fixed point of T such that Δ : R ∩ int Q1 x ∪ Q3 x is nonempty (i.e., x is not the NW or SE vertex of R), and T is strongly competitive on Δ. Suppose that the following statements are true. a The map T has a C1 extension to a neighborhood of x. b The Jacobian matrix JT x of T at x has real eigenvalues λ, μ such that 0 < |λ| < μ, where |λ| < 1, and the eigenspace E associated with λ is not a coordinate axes. Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E at x, and C is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of C in the interior of R are either fixed points or minimal period-two points. In the latter case, the set of endpoints of C is a minimal period-two orbit of T . We shall see in Theorem 2.7 and in the examples in 2 that the situation where the endpoints of C are boundary points of R is of interest. The following result gives a sufficient condition for this case. Theorem 2.5. For the curve C of Theorem 2.4 to have endpoints in ∂R, it is sufficient that at least one of the following conditions is satisfied. i The map T has no fixed points nor periodic points of minimal period two in Δ. ii The map T has no fixed points in Δ, det JT x > 0, and T x x has no solutions x ∈ Δ. iii The map T has no points of minimal period two in Δ, det JT x < 0, and T x x has no solutions x ∈ Δ. In many cases, one can expect the curve C to be smooth. Theorem 2.6. Under the hypotheses of Theorem 2.4, suppose that there exists a neighborhood U of x in R2 such that T is of class C on U ∪ Δ for some k ≥ 1, and that the Jacobian matrix of T at each x ∈ Δ is invertible. Then, the curve C in the conclusion of Theorem 2.4 is of class C. In applications, it is common to have rectangular domains R for competitive maps. If a competitive map has several fixed points, often the domain of the map may be split Abstract and Applied Analysis 5and Applied Analysis 5 into rectangular invariant subsets such that Theorem 2.4 could be applied to the restriction of the map to one or more subsets. For maps that are strongly competitive near the fixed point, hypothesis b of Theorem 2.4 reduces just to |λ| < 1. This follows from a change of variables 9 that allows the Perron-Frobenius theorem to be applied to give that at any point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axes. Smith performed a systematic study of competitive and cooperative maps in 9, 13, 14 and in particular introduced invariant manifolds techniques in his analysis 13–15 with some results valid for maps on n-dimensional space. Smith restricted attention mostly to competitive maps T that satisfy additional constraints. In particular, T is required to be a diffeomorphism of a neighborhood of R that satisfies certain conditions this is the case if T is orientation preserving or orientation reversing , and that the coordinate semiaxes are invariant under T . For such class of maps as well as for cooperative maps satisfying similar hypotheses , Smith obtained results on invariant manifolds passing through hyperbolic fixed points and a fairly complete description of the phase-portrait when n 2, especially for those cases having a unique fixed point on each of the open positive semiaxes. In our results, presented here, we removed all these constraints and added the precise analysis of invariant manifolds of nonhyperbolic equilibrium points. The invariance of coordinate semiaxes seems to be serious restriction in the case of competitive models with constant stocking or harvesting, see 16 for stocking. The next result is useful for determining basins of attraction of fixed points of competitive maps. Compare to Theorem4.4 in 13 , where hyperbolicity of the fixed point is assumed, in addition to other hypotheses. Theorem 2.7. A Assume the hypotheses of Theorem 2.4, and let C be the curve whose existence is guaranteed by Theorem 2.4. If the endpoints of C belong to ∂R, then C separates R into two connected components, namely, W− : {x ∈ R \ C : ∃ y ∈ C with x se y}, W : {x ∈ R \ C : ∃ y ∈ C with y se x}, 2.3 such that the following statements are true: i W− is invariant, and dist T x ,Q2 x → 0 as n → ∞ for every x ∈ W−. ii W is invariant, and dist T x ,Q4 x → 0 as n → ∞ for every x ∈ W . B If, in addition to the hypotheses of part (A), x is an interior point of R, and T is C2 and strongly competitive in a neighborhood of x, then T has no periodic points in the boundary of Q1 x ∪ Q3 x except for x, and the following statements are true. iii For every x ∈ W− there exists n0 ∈ N such that T x ∈ intQ2 x for n ≥ n0. iv For every x ∈ W there exists n0 ∈ N such that T x ∈ intQ4 x for n ≥ n0. Basins of attraction of period-two solutions or period-two orbits of certain systems or maps can be effectively treated with Theorems 2.4 and 2.7. See 2, 6, 11 for the hyperbolic case; for the nonhyperbolic case, see examples in 2, 17 . 6 Abstract and Applied Analysis If T is a map on a set R and if x is a fixed point of T , the stable set Ws x of x is the set {x ∈ R : T x → x}, and unstable setWu x of x is the set { x ∈ R : there exists {xn}n −∞ ⊂ R s.t. T xn xn 1, x0 x, lim n→−∞n x } . 2.4 When T is noninvertible, the setWs x may not be connected and made up of infinitely many cur