Basins of attraction of certain rational anti-competitive system of difference equations in the plane
© Moranjkić and Nurkanović; licensee Springer 2012
Received: 20 March 2012
Accepted: 23 August 2012
Published: 4 September 2012
We investigate the global asymptotic behavior of solutions of the following anti-competitive system of rational difference equations:
where the parameters , , , and are positive numbers and the initial conditions are arbitrary nonnegative numbers. We find the basins of attraction of all attractors of this system, which are the equilibrium points and period-two solutions.
where , are continuous functions is competitive if is non-decreasing in x and non-increasing in y, and is non-increasing in x and non-decreasing in y.
System (1) where the functions f and g have a monotonic character opposite of the monotonic character in competitive system will be called anti-competitive.
where the parameters , , , and are positive numbers and the initial conditions are arbitrary nonnegative numbers. In the classification of all linear fractional systems in , System (2) was mentioned as System (16, 39).
Competitive and cooperative systems of the form (1) were studied by many authors such as Clark and Kulenović , Clark, Kulenović and Selgrade , Hirsch and Smith , Kulenović and Ladas , Kulenović and Merino , Kulenović and Nurkanović [7, 8], Garić-Demirović, Kulenović and Nurkanović [9, 10], Smith [11, 12] and others.
The study of anti-competitive systems started in  and has advanced since then (see [14, 15]). The principal tool of the study of anti-competitive systems is the fact that the second iterate of the map associated with an anti-competitive system is a competitive map, and so the elaborate theory for such maps developed recently in [4, 16, 17] can be applied.
The main result on the global behavior of System (2) is the following theorem.
which satisfy (see Figure 1(b)):
(c) If , then (see Figure 1(c))
for , that belong to the segment of the line (15) in the first quadrant.
(ii) All minimal period-two solutions and the equilibrium are stable but not asymptotically stable.
that emanate from and , for all , such that the curves are pairwise disjoint, the union of all the curves equals . Solutions with initial points in converge to and solutions with an initial point in have even-indexed terms converging to and odd-indexed terms converging to ; solutions with an initial point in have even-indexed terms converging to and odd-indexed terms converging to .
and solutions with an initial point in have even-indexed terms converging to and odd-indexed terms converging to , solutions with an initial point in have even-indexed terms converging to and odd-indexed terms converging to (see Figure 1(d)).
We now give some basic notions about systems and maps in the plane of the form (1).
Consider a map on a set , and let . The point is called a fixed point if . An isolated fixed point is a fixed point that has a neighborhood with no other fixed points in it. A fixed point is an attractor if there exists a neighborhood of E such that as for ; the basin of attraction is the set of all such that as . A fixed point E is a global attractor on a set if E is an attractor and is a subset of the basin of attraction of E. If T is differentiable at a fixed point E, and if the Jacobian has one eigenvalue with modulus less than one and a second eigenvalue with modulus greater than one, E is said to be a saddle. See  for additional definitions.
The mean value theorem and the convexity of may be used to show that T is monotone, as in .
For , define for to be the usual four quadrants based at x and numbered in a counterclockwise direction, for example, .
The following definition is from .
Definition 1 Let be a nonempty subset of . A competitive map is said to satisfy condition (O+) if for every x, y in , implies , and T is said to satisfy condition (O−) if for every x, y in , implies .
The following theorem was proved by de Mottoni-Schiaffino for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps .
Theorem 2 Let be a nonempty subset of . If T is a competitive map for which (O+) holds then for all , is eventually componentwise monotone. If the orbit of x has compact closure, then it converges to a fixed point of T. If instead (O−) holds, then for all , is eventually componentwise monotone. If the orbit of x has compact closure in , then its omega limit set is either a period-two orbit or a fixed point.
The following result is from , with the domain of the map specialized to be the Cartesian product of intervals of real numbers. It gives a sufficient condition for conditions (O+) and (O−).
Theorem 3 Let be the Cartesian product of two intervals in . Let be a competitive map. If T is injective and for all then T satisfies (O+). If T is injective and for all then T satisfies (O−).
The map T has a extension to a neighborhood of .
The Jacobian matrix of T at has real eigenvalues λ, μ such that , where , and the eigenspace associated with λ is not a coordinate axis.
Then there exists a curve through that is invariant and a subset of the basin of attraction of , such that is tangential to the eigenspace at , and is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of in the interior of are either fixed points or minimal period-two points. In the latter case, the set of endpoints of is a minimal period-two orbit of T.
Theorem 5 (Kulenović & Merino)
and T is strongly competitive on .
The point is the only fixed point of T in .
The map T is continuously differentiable in a neighborhood of .
At least one of the following statements is true:
T has no minimal period two orbits in .
and only for .
is a saddle point.
The unstable manifold is connected, and it is the graph of a continuous decreasing curve.
For every , eventually enters the interior of the invariant set , and for every , eventually enters the interior of the invariant set .
Let and be the endpoints of , where . For every and every such that , there exists such that , and for every and every such that , there exists such that .
3 Linearized stability analysis
If , then System (2) has a unique equilibrium point .
If , then System (2) has two equilibrium points and , , .
is satisfied, i.e., . □
If , then is locally asymptotically stable.
If , then is non-hyperbolic.
If , then is a repeller.
If , then , i.e., is locally asymptotically stable.
If , then , which implies that is non-hyperbolic.
If , then , which implies that is a repeller.
- (1)Assume that and(8)
- (2)Assume that(9)
- (3)Assume that(10)
Then the positive equilibrium is locally asymptotically stable.
If , then for all , which implies that is a saddle point. If , then for (, ).
- (2)If , then , hence , i.e.,
- (3)If , then and
holds, so is a locally asymptotically stable. □
4 Periodic character of solutions
In this section, we give the existence and local stability of period-two solutions.
So, periodic solutions are located along line (15) with endpoints given by (14) using conditions (9). It is easy to see that if . □
where , , , .
The points are non-hyperbolic fixed points for the map , and both of them have eigenvalues and .
Eigenvectors corresponding to the eigenvalues and are not parallel to coordinate axes.
(b) Eigenvectors corresponding to the eigenvalues and are and . By condition (23) it is easy to see that these vectors are not parallel to the coordinate axes. □
locally asymptotically stable if and ,
non-hyperbolic if ,
saddle points if .
so it comes to the same conclusion! □
5 Global results
In this section, we present the results on the global dynamics of System (2).
, , .
If , then , .
, where , that is, is an invariant box.
is an attracting box, that is .
so it follows that , if .
Proof of 3. and 4. is an immediate checking. □
Lemma 6 The map is injective and , for all and .
and the Jacobian matrix of is invertible for all and . □
Corollary 1 The competitive map satisfies the condition (O+). Consequently, the sequences , , , of every solution of System (2) are eventually monotone.
Proof It immediately follows from Lemma 6, Theorem 2 and 3. □
- (a)If , , then
- (b)If , , then
Proof Since is injective, then . □
Proof of Theorem 1 Case 1
which is an invariant box. In view of Corollary 1 and Theorem 2, every solution converges to minimal period-two solutions or . System (2) has no minimal period-two solutions (Lemma 2). So, every solution of System (2) converges to .
Case 2 and
The existence of the set with the stated properties follows from Lemmas 6, 2, 7, 8, Corollary 1, Theorems 4 and 5.
Cases (i) and (ii) from (c) in Theorem 1 are the consequence of Lemmas 1, 2, 4 and Theorems 6 and 7.
and Lemma 7 completes the conclusion (d) of Theorem 1. □
The authors are very grateful to Professor M.R.S. Kulenović for his valuable suggestions. They thank also the referees for their useful comments.
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