GLOBAL ASYMPTOTICAL STABILITY OF THE COEXISTENCE FIXED POINT OF A RICKER-TYPE COMPETITIVE MODEL

. We shall obtain the parameter region that ensures the global asymptotical stability of the coexistence ﬁxed point of a Ricker-type competitive model. The parameter region can be illustrated graphically and examples of such regions are presented. Our result partially answers an open problem pro- posed by Elaydi and Lu´ıs [3] and complements the very recent work by Balreira, Elaydi and Lu´ıs [1].

1. Introduction. Discrete time population models are favored by many researchers due to the fact that data is collected in discrete time intervals and many species have non-overlapping generations. For example, Luís, Elaydi and Oliveira [10] have studied systematically the following Ricker-type competitive model x n+1 = x n exp(K − x n − ay n ), y n+1 = y n exp(L − bx n − y n ), (1) where K, L, a, b are all positive constants. The results or phenomena, such as the local asymptotical stability of the equilibria, the competitive exclusion principle, the various bifurcation scenarios and also the center manifolds, have been obtained or analyzed. Ricker-type discrete models are one of the most important models in population biology. They are very simple in the form, but very complicated in the dynamics [11]. While system (1) is a competitive model of Ricker-type, there are also predatorprey models and host-parasite models of this type [5,6]. A general form of model (1) can be presented as x n+1 = x n exp(K − a 11 x n − a 12 y n ), y n+1 = y n exp(L − a 21 x n − a 22 y n ), (2) where K, L, a ij , i, j = 1, 2 are all positive constants. The permanence and/or stability of model (2) have been studied in [4,8,12,13,14]. Note that model (2) is autonomous and is for two species. A non-autonomous version of (2) has been discussed in [2] for two species, and in [14,16] for m (m ≥ 2) species. Global asymptotical stability of the positive equilibrium of a theoretical population model is one of the important research aspects possibly due to the characteristic of maintaining the topological patterns under fluctuations of the environment to some extent. The dynamics of the system will tend to the coexistence fixed point when there are small environmental changes provided this fixed point is globally asymptotically stable. It reflects the self restoration ability of the ecosystem to small environmental changes. Hence, the global asymptotical stability of the equilibria of a theoretical population model has received numerous attention in the literature. In [3], an open problem on the global asymptotical stability of (1) is proposed as the following conjecture.
then the coexistence fixed point (x * , y * ) of (1), is globally asymptotically stable for all (K, L) ∈ Int(S 1 ), where S 1 is the region in the K-L parameter space satisfying (3) and bK < L < K/a.
In this paper, we shall investigate the global asymptotical stability of the positive equilibrium (x * , y * ) of the Ricker-type competitive model (1). Specifically we shall obtain a K-L parameter region in which the coexistence fixed point (x * , y * ) of (1) is globally asymptotically stable. Our result partially answers the open problem proposed in Conjecture 1, moreover it complements the very recent result of Balreira, Elaydi and Luís [1] -the details will be presented and illustrated with figures in the main section, a comparison with the related results of [1] will also be included (see Remarks 2 and 3).
To obtain results on global asymptotical stability of the equilibrium, appropriate Lyapunov functions are usually constructed [9]. However, it is not easy to find a feasible Lyapunov function since the methods to construct these functions vary from different models [7]. Another method to establish the global asymptotical stability of the equilibrium is to verify that this equilibrium is both locally stable as well as globally attractive [15]. In the next section we shall employ this method to obtain the global asymptotical stability of the positive equilibrium of (1).

2.
Global asymptotical stability. In this section, we shall establish the global asymptotical stability of the equilibrium of model (1) by showing that this equilibrium is both locally stable as well as globally attractive [15]. To begin, we consider (1) with initial values x 0 > 0, y 0 > 0. (5) The first lemma is to ensure the permanence of (1). To be specific, model (1) with initial values (5) is said to be permanent if there exist positive constants α, α * , β and β * such that Lemma 2.1. If then model (1) with (5) is permanent.
Proof. Noting that x n+1 ≤ x n exp(K − x n ) and the maximum of the function Similarly, since y n+1 ≤ y n exp(L − y n ), we also have lim sup n→∞ y n ≤ exp(L − 1).
Next, we shall prove that there exists positive constant l, which will be determined later, such that lim inf n→∞ x n ≥ l.
Noting (8), we can choose N 0 sufficiently large such that It then follows from the first equation of (1) that There are two cases to consider.
Case 2. Suppose x n+1 > x n for all n ≥ N 0 . In this case, it is clear that Now, let n → ∞ in the first equation of (1), we get 3258 CHUNQING WU AND PATRICIA J. Y. WONG Using (14) and (10), we find lim inf n→∞ (K − x n − ay n ) ≥ K − l 0 − a(exp(L − 1) + ), and in view of (15) it follows that Consequently, noting the definition of l (see (12)) we find In both cases above (see (13) and (16)), if we denote lim →0 l = l, then it is clear that (9) holds and l > 0.
By a similar argument, we can show that It now follows from (7)- (9) and (17) that model (1) with (5) is permanent. To confirm further, it is possible to verify that l ≤ exp(K − 1) and m ≤ exp(L − 1).
Likewise, it can be shown that m ≤ exp(L − 1). This completes the proof.
From [10], it is known that if bK < L < K/a, ab < 1, is satisfied, then model (1) has a unique positive equilibrium (coexistence fixed point) (x * , y * ) given by (4). Further, it is shown in [10] that this positive equilibrium is a saddle if (19) holds, and the following result. (18) and (3) hold, then the positive equilibrium (x * , y * ) of (1) is locally asymptotically stable.
In the following, we denote It is obvious that f (x, y) is bounded in G 0 . The maximum M 1 is obtained by direct computation, the details are given in the Appendix.
The next result establishes the global attractiveness of the positive equilibrium (x * , y * ). (6) and (18) hold. If and for (x n , y n ) that satisfies (1), then the positive equilibrium (x * , y * ) of (1) is globally attractive in the interior of the first quadrant.
Proof. Under assumptions (6) and (18), model (1) with (5) is permanent (Lemma 2.1) and has a positive equilibrium (x * , y * ) [10]. Hence, there exists positive constants l 1 , l 2 , L 1 and L 2 such that Noting that and likewise l 2 ≥ l 2 exp(L − bL 1 − L 2 ), it follows that and hence the point (L 1 , L 2 ) lies in G 0 . Now we define a region G δ in the first quadrant as follows: G δ ⊃ G 0 and where the lines e 1 : K δ − x − ay = 0 and e 2 : L δ − x − by = 0 are parallel to the lines s 1 : K − x − ay = 0 and s 2 : L − bx − y = 0, respectively, such that the distances of e 1 to s 1 and e 2 to s 2 are both δ > 0. Moreover, e 1 and e 2 are closer to the origin than the lines s 1 and s 2 . The regions G 0 and G δ are illustrated in Figure 1.
In the sequel, we proceed to prove Under the assumption (22), there are four cases to consider.
In all the four cases, we have shown that (30) holds. Coupling with (29), we have lim n→∞ x n = x * and lim n→∞ y n = y * . Hence, (x * , y * ) is globally attractive in the interior of the first quadrant. Proof. Since the conditions of Lemma 2.2 and 2.5 are satisfied, the positive equilibrium (x * , y * ) of model (1) exists, which is both locally asymptotically stable and globally attractive in the interior of the first quadrant. Therefore, (x * , y * ) is globally asymptotically stable in the interior of the first quadrant. Remark 2. Let D be the K-L parameter region obtained in Theorem 2.6, and S 1 be the K-L parameter region associated with Lemma 2.2 [10] (note that S 1 is also the parameter region proposed in Conjecture 1 [3]). To be specific, D is the intersection of conditions (3), (6), (18), (21) and (22), and the global asymptotical stability of (x * , y * ) is guaranteed for (K, L) ∈ D. On the other hand, S 1 is described by (3) and (18), and by Lemma 2.2 the positive equilibrium (x * , y * ) is locally asymptotically stable for (K, L) ∈ S 1 . Logically we should have D ⊂ S 1 . Indeed, by setting a = 0.2 and b = 0.25, we plot S 1 and D in Figure 2 and observe that D ⊂ S 1 .   (1). Suppose that the coexistence fixed point (x * , y * ) is locally asymptotically stable. Further, assume the following conditions: (a) The region R1 is contained in the region Γ1; (b) For all m = n, LC 1 m ∩ LC 1 n = ∅. Then, the positive equilibrium (x * , y * ) of (1) is globally asymptotically stable with respect to the interior of the first quadrant.
then condition (a) of Theorem 6.1 holds.
Let us consider the case when a = 0.2 and b = 0.25. To guarantee the global asymptotical stability of the positive equilibrium (x * , y * ), one computes from (33) the ranges of K and L as 1.0526 < K < 1.4737 and 1.0804 < L < 1.5512 in order to fulfill condition (a) of Theorem 6.1 of [1]. Hence, when the parameters K and L are outside of the above ranges, say (K, L) = (1.5, 2.1), then Theorem 6.1 of [1] may not be applicable to obtain the global asymptotical stability of the positive equilibrium (x * , y * ). On the other hand, it is easy to verify numerically that (K, L) = (1.5, 2.1) is a point in the parameter region D obtained by Theorem 2.6 in this paper (refer to Figure 2(b)). Thus, Theorem 2.6 is able to establish the global asymptotical stability of the positive equilibrium (x * , y * ) when (K, L) = (1.5, 2.1).
One further notes that condition (b) of Theorem 6.1 of [1] is difficult to verify due to the infinite number of curves LC 1 m . Lastly, we remark that the method used in the present work is different from that of [1]. Case (1). x = 0 with y ≥ K/a. In this case, f (x, y) = 0 for all y.
Case (3). The line s 1 : K − x − ay = 0 with 0 ≤ x ≤ x * . We get f (x, y) = x on s 1 . Hence, the maximum of f is x * .
Case (4). The line s 2 : L−bx−y = 0 with x * ≤ x ≤ L/b. In this case, substituting In the following, we shall obtain the maximum of f (x, y) in G 0 from Cases (1)-(4) above. The following statements will be useful in subsequent development.