GEOMETRIC METHOD FOR GLOBAL STABILITY OF DISCRETE POPULATION MODELS

. A class of autonomous discrete dynamical systems as population models for competing species are considered when each nullcline surface is a hyperplane. Criteria are established for global attraction of an interior or a boundary ﬁxed point by a geometric method utilising the relative position of these nullcline planes only, independent of the growth rate function. These criteria are universal for a broad class of systems, so they can be applied directly to some known models appearing in the literature including Ricker competition models, Leslie-Gower models, Atkinson-Allen models, and generalised Atkinson-Allen models. Then global asymptotic stability is obtained by ﬁnding the eigenvalues of the Jacobian matrix at the ﬁxed point. An intriguing question is proposed: Can a globally attracting ﬁxed point induce a homoclinic cycle?

(a2) For x ∈ R N + and each i ∈ I N , ∂T i (x) ∂x i > 0 for 0 ≤ x i < r i .
(a3) For each nonempty J ⊂ I N and for any two points x, y ∈ R N + satisfying y i = x i ∈ [0, r i ] for i ∈ I N \ J, 0 = x j < y j ≤ r j and (Ay) j ≤ r j for j ∈ J and (Ay) i = r i for some i ∈ J, the functions T k (x + t(y − x)), k ∈ J, are increasing for t ∈ [0, 1].
(A simplified version of (a3) will be given in section 2 as (a3) .) The discrete dynamical system (1) defined above can be viewed as a population model of N competing species, where x i (n) denotes the population size of the ith species at time n (e.g. nth generation or end of nth time period). Indeed, if a ij > 0, the existing population of the jth species reduces the growth rate G i of the ith species; if a ij = 0, although the current population of the jth species does not affect the next generation of the ith species T i (x) directly, it may reduce the growth rate G i ((AT n x) i )(n ≥ 1) of later generations. This reflects the nature of mutual competition between any two species.
System (1) with (2)-(3) and (a1)-(a3) is competitive not only in ecological context, but the map T is also mathematically a competitive map under the additional condition that the spectral radius of the matrix M (x) = diag(− Recall that a general map T : S → T (S) for a set S ⊂ R N + is called competitive if x < y whenever T (x) < T (y) for x, y ∈ S, and strongly competitive (or retrotone) if x y whenever T (x) < T (y) with x, y ∈ S and y 0 [35,33,15,14] (see section 2 for detailed definition for "<" and " "). By (a1) and (a2) we have G i ≤ 0 and − x i G i G i < 1. Then (4) implies that DT (x) −1 ≥ 0, so T is a competitive map. But since DT (x) −1 may have zero entries, (4) does not imply strong competitiveness of T . However, we are not sure whether (a1)-(a3) imply (4). Thus, without the condition (4) we are not sure whether T is a competitive map.
System (1) with (2)- (5) and (a1)-(a3) includes many known models as special instances. For example, if G i (u) = e r i −u , 0 < r i ≤ 1, then (2) becomes and systems with such a form for T are known as Ricker competition models. If G i (u) = 1+r i 1+u for i ∈ I N , then and systems with such a form for T are known as Leslie-Gower competition models. If and systems with such a form for T are known as Atkinson-Allen models. If Systems with such a form for T are known as generalised Atkinson-Allen Models. It can be checked that all these models defined by (6)-(9) and any system defined by a combination of (6)-(9) meet the requirement of (a1)-(a3) (see Appendix 1). Thus, these models are typical examples of system (1) with (2)- (5) and (a1)-(a3).
A more general discrete population model, known as Kolmogorov system, is (1) with T defined by where F : R N + → R N + is smooth enough with F i (x) > 0 and ∂F i ∂x j ≤ 0(i = j). Hirsch [15], Herrera [13], Wang and Jiang [36] (see also [22] and the references therein) proved the existence and uniqueness of a carrying simplex Σ ⊂ R N + under certain conditions. Since Σ is a global attractor of (1) with (10) restricted to R N + \ {0}, the global dynamics of the system on R N + \ {0} is essentially described by the dynamics on Σ if it exists. Then many researchers explored the behaviour of the dynamics of (1) with (10) on Σ. For a few examples, Jiang, Niu and Wang [25] investigated existence and local stability of heteroclinic cycles of competitive maps via carrying simplices; Herrera [14] investigated exclusion and dominance under conditions which guarantee the existence of Σ; Balreira, Elaydi and Luis [5] provided criteria for global stability of an interior fixed point under the existence of the carrying simplex.
For system (1) with T defined by (2) and (3) with a ij > 0 for all i, j ∈ I N and each G i decreasing, Franke and Yakubu [8,9] described the competition model and studied the exclusion of some species and proved that k-weakly dominant G plus invariance of a set implies that the k-th species will survive and all other species will die out.
For Ricker models (1) with (6), Smith [34] analysed the dynamics in detail for two competing species (N = 2). Roeger [31] (see also [32]) studied the local dynamics near the interior fixed point and Neimark-Sacker bifurcations for the special 3D maps (6) with r 1 = r 2 = r 3 . Hofbauer et al. [16] studied the long term survival of N species. Hirsch [15] showed that (1) with (6) possesses a carrying simplex under mild conditions. Gyllenberg et al. [12] classified all 3D Ricker maps (6) admitting a carrying simplex Σ and derived a total of 33 stable equivalence classes with a typical phase portrait on Σ given for each class. The authors of [5] applied their stability criteria to 3D Ricker models and derived a sufficient condition for the interior fixed point to be globally asymptotically stable when r 1 = r 2 = r 3 < 1 and a ij = a < 1(i = j).
For Leslie-Gower models (1) with (7), Cushing et al. [7] thoroughly analysed the 2D model and showed convergence of every orbit to a fixed point. For 3D models, Jiang et al. [25] analysed the existence and stability of heteroclinic cycles. For N -dimensional models, Hirsch [15] and Herrera [14] verified the existence of the carrying simplex Σ under some conditions, but Jiang and Niu [24] showed the unconditional existence of Σ. Moreover, the authors of [24] classified all 3D Leslie-Gower models via the boundary dynamics on Σ and derived a total of 33 stable equivalent classes. For a special case of 3D models with b 1 = b 2 = b 3 and a ij = a(i = j), Balreira et al. [5] obtained a condition for global stability of the interior fixed point.
The Atkinson-Allen models (1) with (8) were first built as a plant competition model by Atkinson [3] and Allen et al. [2] (see also a discrete model in [30]). For 3D models, Jiang and Niu [23] proved the index formula for fixed points on the carrying simplex and, based on which, gave a complete classification of 3D models into 33 stable equivalent classes. The generalised Atkinson-Allen models (1) with (9) were proposed by Gyllenberg et al. [10] and a complete analysis for 3D models with (9) was given similar to [23].
In this paper, we are concerned with the global dynamics and, in particular, the global asymptotic stability of a boundary or interior fixed point, of the system (1) with (2)- (5) and (a1)-(a3). There are some available methods and results for stability: the Liapunov function method initiated by LaSalle [27] for general discrete dynamical systems, the Gfunction method for asymptotic stability introduced by Bouyekhf and Gruyitch [6], the method of using convexity of the per-capita growth rate by Kon [26], the split Liapunov function method developed by Baigent and Hou [4] for Kolmogorov systems (1) with (10), the stability criteria for monotone Kolmogorov systems obtained by Balreira et al. [5] and a revised version by Gyllenberg et al. [11], and some stability results for specific systems (see the references in [6], [4], [5] and [11]). No doubt that all of these are precious contributions to the development of stability theory and methods for discrete dynamical systems. However, it is also obvious that the application of each of these methods or criteria has its limitation due to its conditions and requirements. For this reason, more alternative methods will be welcome and expected.
The aim of this paper is to provide a geometric method for global attraction and stability of a fixed point either in the interior or the boundary of R N + for (1) with (2)-(5) and (a1)-(a3). We do not rely on the construction of Liapunov functions, nor on the existence of the carrying simplex. The nature of global attraction will be simply derived by the relative position of the N nullcline hyperplanes defined by the component equations of Ax = r within the N -dimensional cell [0, r]. Superficially, this method can be viewed as an extension to our discrete system (1) of the geometric method for Lotka-Volterra differential systems and Kolmogorov differential systems buried in a large number of publications (e.g. [37,1,28,29,17,18,19,20,21]) since conditions derived for the discrete system here is similar to those for (11). But if we bear in mind the essential difference between an orbit x(n), n ≥ 1, for (1) and an orbit x(t), t ≥ 0, for (11) described below, we shall realise that this is not a simple extension as the similarity of the conditions for both discrete and continuous systems suggests: the former consists of isolated points and the latter is a smooth continuous curve; the former can go from one side to the other side of a plane without actual intersection with the plane, i.e. jumps over the plane, but the latter going from one side to the other of a plane must have a intersection point with the plane. Indeed, from later sections we shall appreciate the subtlety of the techniques employed to tackle some hard obstacles laying in the process of proofs of the main theorems.
The virtue of the method is that the derived criteria for global attraction only uses the matrix A and the point r, irrelevant to the functions G k as long as (a1)-(a3) are met, and the criteria can be applied to a broad class of systems (1) with (2)-(5) and (a1)-(a3), universal to all the above models with T defined by (6)- (9). Then, by finding the eigenvalues of the Jacobian matrix DT (x * ), we know that either the fixed point is globally asymptotically stable or a homoclinic cycle is induced. Conclusion. Appendix 1. Proof of (a1)-(a3) for models (6-(9). Appendix 2. Proof of Proposition 2.3. Appendix 3. Proof of Lemma 5.1.

Notation and preliminaries
Denote the interior of R N + by intR N + and the boundary of R N + by ∂R N + , and define (12) π i = {x ∈ R N + : x i = 0}, i ∈ I N . Then π i is the part of the ith coordinate plane restricted to R N + and is part of Then [u, v] is a k-dimensional cell if v − u has exactly k non-zero components. For any u ∈ R N + and I ⊂ I N , we also introduce the following notation Then Γ i is the ith nullcline plane for T defined by (2)-(5) and (a1)-(a3), i.e. T i (x) = x i on Γ i . We abuse the notation slightly by using 0 to denote scalar number zero, vector zero and the origin in R N . For any plane Γ in R N + with 0 ∈ Γ, R N + is divided into three mutually exclusive connected subsets Γ − , Γ and Γ + with 0 ∈ Γ − such that R N We view ∩ k∈∅ π k as R N + and denote the nonnegative half x i -axis by X i . With the above notation, the condition (a3) given in section 1 can be simplified as follows: (a3) For any nonempty J ⊂ I N and for any point x ∈ [0, r] which is on Γ i for some i ∈ J but on or below Γ j for all j ∈ J, the functions T j (x I N \J + tx J ), j ∈ J, are increasing for t ∈ [0, 1].
Note from (5) and (16) that the plane Γ i intersects the half axis X i at a point Q i , i.e. Γ i ∩ X i = {Q i }, with r i as its ith component and 0 as other components. Thus, T has N axial fixed points Q i , i ∈ I N . Clearly, 0 is a repelling fixed point since the Jacobian matrix of T at the origin is DT (0) = diag{G 1 (0), . . . , G N (0)} with G i (0) > 1 for all i ∈ I N by (a1). We shall see that the point r = (r 1 , . . . , r N ) T 0 plays a very important role in this paper.
If T is invertible, then the negative orbit γ − (x) and the orbit γ(x) can be defined as and γ(x) = γ + (x) ∪ γ − (x). The positive limit set ω(x) and the negative limit set α(x) are defined as usual: Denote the open ball centred at x ∈ R N with radius δ > 0 by B(x, δ). A fixed point For a nonempty subset J ⊂ I N and a fixed point x * is said to be globally attracting if for any x ∈ R N + \ (∪ i∈J π i ) we have ω(x) = {x * }, and globally asymptotically stable if it is stable and globally attracting. Therefore, for a globally asymptotically stable fixed point x * > 0, if x * ∈ intR N + then it attracts all the interior points; if x * ∈ ∂R N + it attracts not only the interior points but also the boundary points with x i > 0 for all i ∈ J.
The principal idea of proving global attraction of a fixed point x * ∈ R N + \ {0} is as follows: (iii) Then repetition of (i) and (ii) leads to ω(x) = {x * } for all x ∈ R N + \ (∪ i∈J π i ) under the assumptions.
The proofs of these are technical, so we leave them to later sections except the proof for (i) (Proposition 2.1) with the following reason: At this stage we need to observe that the cell [0, r] is positive invariant and globally attracting. Instead of giving a direct proof for this observation here, we prove Proposition 2.1 from which the observation immediately follows.
Let J ⊂ I N be a nonempty set and let u ∈ R N + such that u i = 0 for each i ∈ I N \ J and u ∈ Γ − j for all j ∈ J. Define v ≥ u by (19) we see that if u = 0 then v = r. Proposition 2.1. Assume the existence of a nonempty set J ⊂ I N and a point u ∈ R N + satisfying u i = 0 for all i ∈ I N \ J and u ∈ Γ − j for all j ∈ J such that ω(x) ⊂ R N + (u) for all x ∈ R N + \ (∪ i∈J π i ). Then, for (1) with (2)-(5) and (a1)-(a3) and v defined by (19), Proof. Take an arbitrary i ∈ I N and an arbitrary x ∈ R N + \(∪ j∈J π j ). Then ω(x) ⊂ R N + (u) by assumption. Thus, for any small enough δ > 0, there is Let v ≥ u be defined by (19) with the replacement of u, v by u and v . If (16) and (19), so T N 1 +1 i (x) < T N 1 i (x) and x i (n) is decreasing in n as long as x(n) ∈ Γ + i for n > N 1 . Thus, either there is an integer K > N 1 such that In the latter case, x(n) ∈ Γ + i so the sequence {x i (n)} is decreasing and bounded below by v i . We claim that lim n→+∞ for all n > N 1 and as k → +∞, a contradiction to x i (n) > v i for all n > N 1 . This shows the above claim. In the former case, we show that Repeating the above process, we obtain Then ω(x) ⊂ [u , v ] follows from the arbitrariness of i ∈ I N . By letting δ → 0 we From Proposition 2.1 with u = 0 we see that [0, r] is positive invariant and globally attracting on R N + . Since (a1) implies that 0 a repelling fixed point, we also observe that system Then S 2 is closed and for all x ∈ S 1 , by the positive invariance of [0, r] the set S 2 is positive invariant. As [0, r] is globally attracting on R N + and 0 is repelling, we can easily check that S 2 is globally attracting on R N Note that under conditions which guarantee the existence of a carrying simplex Σ, we must have S 0 = Σ.
For a fixed i ∈ I N , consider the following geometric condition (20) ∀k ∈ I N \ {i}, Γ k ∩ [0, r] ∩ π i is strictly below Γ i and the algebraic inequality Our next proposition establishes the relationship between (20) and (21). (20) and (21) the following statements hold.
(ii) Conversely, if (21) holds for a fixed i ∈ I N and all j ∈ I N \ {i}, then (20) holds.
(iii) Hence, condition (20) holds for all i ∈ I N if and only if (21) holds for all i, j ∈ I N with i = j.
Although the proof of this proposition is similar to that of Lemma 2.4 in [17], to ease the pain of juggling between different contexts under different notation, a self-contained proof of Proposition 2.3 is provided in Appendix 2 at the end of this paper.

Main results
We shall see that the global attraction of an interior or boundary fixed point x * , even its existence, will be determined purely by the relative position of the N planes Γ 1 , . . . , is given by (22) with the replacement of 1 on the jth row of I by G j ((Ax * ) j ) for all j ∈ J.

Results for global attraction. Our first expected result is for existence and global attraction of an interior fixed point.
Theorem 3.1. Assume that Γ j ∩ [0, r] ∩ π i is strictly below Γ i for each i ∈ I N and every j ∈ I N \ {i}. Then system (1) with (2)-(5) and (a1)-(a3) has a unique interior fixed point This particular case of Theorem 3.1 is stated as a corollary below.
Then the conclusion of Theorem 3.1 holds.
The next result is for existence and global attraction of a general boundary fixed point.
Assume that the following conditions hold.
Then system (1) with (2)-(5) and (a1)-(a3) has a fixed point x * ∈ [0, r] with x * i > 0 if and only if i ∈ J and x * is globally attracting. Remark 1. Note from Theorem 3.1 that condition (i) in Theorem 3.3 ensures that the |J|-dimensional subsystem has a globally attracting fixed point in intR |J| + , so system (1) has a fixed point x * ∈ [0, r] with x * i > 0 if and only if i ∈ J. From the proof given later we shall see that the global attraction of x * in R N + \ (∪ j∈J π j ) requires that (ii) x * is on or above Γ k for every k ∈ I N \ J.
Since x * ∈ ∩ j∈J Γ j , it is clear that condition (ii) in Theorem 3.3 guarantees (ii) above. Thus, if we know x * already, we may replace (ii) by the weaker condition (ii) . However, for global asymptotic stability of x * , we require that every eigenvalue λ of DT (x * ) satisfies |λ| < 1 (see Theorem 3.9 in the next subsection). Since each G k ((Ax * ) k ) for k ∈ I N \ J is an eigenvalue of DT (x * ) and G k ((Ax * ) k ) < 1 if x * is above Γ k , for global asymptotic stability, x * must be above Γ k for all k ∈ I N \ J.
If x * is on or above Γ j for all j ∈ I N \ J, then x * is globally attracting.
So far the above results are stated under geometric conditions in terms of the relative position of the nullcline planes restricted to [0, r] ∩ π i for i ∈ I N . For convenience in checking these geometric conditions, we need to find equivalent algebraic conditions in terms of a ij and r i only. Actually, Proposition 2.3 serves this purpose. By Proposition 2.3 (iii), Theorem 3.1 can be restated as follows.
Note that the point r I N \{i} is below Γ i if and only if (Ar I N \{i} ) i < r i , which is the same as (Ar) i < 2r i . Then Corollary 3.2 can be restated as follows.
Corollary 3.6. Assume that Ar 2r. Then the conclusion of Theorem 3.5 holds.
Assume that the following conditions hold.
If (Ax * ) j ≥ r j for al j ∈ I N \ J, then x * is globally attracting.

3.2.
Results for global asymptotic stability. In this subsection, we first state a general theorem for global asymptotic stability when global attraction is know. Then we give a particular case of Theorem 3.7 when an axial fixed point is globally asymptotically stable.
Theorem 3.9. Assume that x * ∈ R N + \ {0} is a globally attracting fixed point. Then the following conclusions hold: (i) If each eigenvalue λ of DT (x * ) satisfies |λ| < 1, then x * is globally asymptotically stable. (ii) If DT (x * ) is invertible and has an eigenvalue λ satisfying |λ| > 1, then there is a homoclinic cycle.
Proof. (i) If each eigenvalue λ of DT (x * ) satisfies |λ| < 1, then x * is locally asymptotically stable. Then the global asymptotic stability of x * follows from its local stability and global attraction. (ii) If DT (x * ) is invertible and has an eigenvalue λ satisfying |λ| > 1, then there is a point By the global attraction of x * , we also have ω(x) = {x * }. Then γ(x) with x * forms a homoclinic cycle. The next theorem is for global stability of an axial fixed point Q i . Theorem 3.10. Assume that Γ j ∩ [0, r] ∩ π i is strictly below Γ i and Q i is above Γ j for some i ∈ I N and all j ∈ I N \ {i}. Then Q i is globally asymptotically stable.
The following is a particular case of Theorem 3.10 with a simple condition that r I N \{i} is below Γ i . Corollary 3.11. Assume that r I N \{i} is below Γ i and Q i is above Γ j for some i ∈ I N and all j ∈ I N \ {i}. Then Q i is globally asymptotically stable.
The condition that the axial fixed point Q i is above Γ j holds if and only if a ji r i > r j . Then, by Proposition 2.3 (ii), Theorem 3.10 and Corollary 3.11 can be restated as follows.
Theorem 3.12. Assume that r j < a ji r i and for some i ∈ I N and all j ∈ I N \{i}. Then the axial fixed point Q i is globally asymptotically stable.
Corollary 3.13. Assume that (Ar) i < 2r i and r j < a ji r i for some i ∈ I N and all j ∈ I N \ {i}. Then Q i is globally asymptotically stable.

3.3.
Combination of the results for global attraction and Theorem 3.9. After stating the criteria for global attraction of an interior or boundary fixed point and the additional condition required for global asymptotic stability, we are now able to combine Theorems 3.1, 3.3 and 3.9 into a unified version of these results.
Combination of Corollary 3.2, Corollary 3.4 and Theorem 3.9 gives the following corollary.
Using Proposition 2.3 and combining Theorems 3.5, 3.7 and 3.9, we obtain the following theorem.
We first apply our results to these four models with N = 2 and derive simple conditions for global asymptotic stability. Then we analyse the global stability for three dimensional systems of these four models with a circulant matrix A and r = r 0 (1, 1, 1) T for r 0 > 0. An example of 4-dimensional system is given to demonstrate the global asymptotic stability of an axial fixed point. A final example is for a system of combination of the four models having a general boundary fixed point.
(iii) If βr 1 < r 2 and αr 2 < r 1 , then there is a globally attracting interior fixed point x * .
In addition, if (25) holds then x * is globally asymptotically stable.
Next, we check that the inequality (25) holds for each of the four models (6)-(9) under the condition βr 1 < r 2 and αr 2 < r 1 . Clearly, αβ < 1. For Ricker models, G i (u) = e r i −u , so G i (r i ) = −1 and Then (25) follows from 0 < r i ≤ 1. For Leslie-Gower models, For generalised Atkinson-Allen models, Therefore, from Theorem 4.2 (iii) we see that if βr 1 < r 2 and αr 2 < r 1 , then there is a globally asymptotically stable interior fixed point x * for the four models (6)- (9). Note that, under the conditions α > 0 and β > 0, the results obtained here for Ricker and Leslie-Gower models is consistent with those given in [5].
Then Ax = r has a solution x * ∈ intR 3 + with x * i = r 0 1+α+β for i ∈ I 3 . We now derive a condition for global attraction of x * by using Theorem 3.5. For i = 3 and j = 1, (21) becomes max{0, i.e.
We have mentioned in section 1 that the stability problem for three-dimensional Ricker models and Leslie-Gower models were dealt with in [4], [5] and [11]. (There might be other references, but these three are the latest.) Now it is time to compare our stability region in Figure 1 and those given in the references.
We first state that the stability region obtained by using the criteria for global asymptotic stability given in [5] and [11] is the open rectangle for each of the four models under the assumption that the carrying simplex exists. For the four models with A given by (26), from Example 4.1 or by the computation given in [5] for general planar systems, we know that (31) is the condition for each axial fixed point to be a repeller on the carrying simplex and for the existence and global asymptotic stability of a fixed point P i ∈ π i in the interior of each boundary plane π i . We now show that each P i is a saddle point. Note that P 3 has coordinates ( r 0 (1−α) 1−αβ , r 0 (1−β) 1−αβ , 0). For Ricker models, . Under the condition (31), we so e r 0 −(AP 3 ) 3 > 1 and P 3 is a saddle point. Similarly, P 1 and P 2 are also saddle points. By Theorem 2.1 in [11] we know that the interior fixed point x * of Ricker models is globally asymptotically stable. The above analysis for each P i to be a saddle point for Ricker models is also applicable to the other three models. By Remark 2.1 in [11], the interior fixed point x * is globally asymptotically stable for all of the four models. Comparing our region in Figure 1 with (31) we see that neither of them is contained in the other. Obviously, our theorems permit α = 0 or β = 0 or both, but the existence of carrying simplex requires α > 0 and β > 0. If α > 0, β > 0 and the carrying simplex exists, then the region given by (31) is larger than our region in Figure 1. In particular, the carrying simplex always exists for Leslie-Gower models [24]. However, for Ricker models with (26) and r i = r 0 > 0, a sufficient condition for existence of a carrying simplex (e.g. Lemma 2.1 in [11]) requires that r 0 satisfies 0 < r 0 < 1 1+α+β . Does a carrying simplex exist for Ricker models when 1 1+α+β ≤ r 0 ≤ 1? In case a carrying simplex does not exist, the stability region (31) is invalid and our region in Figure 1 is safe.

Proof of the main theorems
where the ith row ofÃ is (r i G i (r i )a i1 , . . . , r i G i (r i )a iN ) and each of the other entries ofÃ is 0. Under the conditions of Theorem 3.10, Q i is above Γ j , so (AQ i ) j > r j and the jth Since Q i is globally attracting (to be proved later), by Theorem 3.9, Q i is globally asymptotically stable.
The lemma below is for existence of a fixed point x * in Theorem 3.1.
Then there is a point 0 < x * ≤ r such that ∩ i∈I N Γ i = {x * }.
The proof of Lemma 5.1 is similar to that of Lemma 3.1 in [17] so we omit it here. Considering that the proof in [17] is not easy to follow under different context, for clarity we provide the proof of Lemma 5.1 in Appendix 3.
Let J ⊂ I N be any nonempty subset such that conditions (i) and (ii) of Theorem 3.3 hold.
Since the dynamics on π j is not affected by the position of Γ j on π j , we need only consider the set [u, v] ∩ (∪ j∈J 1 (Γ j ∪ Γ − j )). As J = ∅, if J = J 1 = {j}, then v ∈ Γ j , u is below Γ j but is on or above Γ i for all i ∈ I N \ {j}, [u, v] is a line segment on the x j -axis, so i )) and bounded by the surface planes of [u, v] and possibly a plane Γ such that [u, v] ∩ (∪ i∈J 1 (Γ i ∪ Γ − i )) is below Γ and has as many touching points with Γ as possible. For example, when . When |J 1 | = 2 with J 1 = {i, j}, there are following three cases for the configuration of h([u, v] ∩ (∪ k∈{i,j} (Γ k ∪ Γ − k ))). Case 1: a ij = 0. No matter whether a ji = 0 or a ji > 0, we always have Figure 2. Case 2: a ij > 0, a ji > 0, [u, v] ∩ Γ j is on or below Γ i . In this case, Γ = Γ i and as shown in Figure 3.
)), which is bounded by the boundary lines of [u, v] and the line Γ determined by A and B shown in Figure 4.
Let P (i) : R N → R N −1 be the projection such that P (i) simply omit the ith component of each x ∈ R N , i.e. P (i) j (x) = x j for all j ∈ I N \ {i} but P (i) (x) has no ith component.
As the proof of Lemma 5.2 is lengthy but technical, we shall dedicate the next section to it.
We now prove the existence of a fixed point x * with x * i ∈ (0, r i ] if and only if i ∈ J and the global attraction of x * .
Proof of Theorems 3.1-3.3 for global attraction. If J = I N , the existence of a fixed point x * with 0 x * ≤ r follows from condition (i) of Theorem 3.3 and Lemma 5.1. If J = {i} for some i ∈ I N , then x * = Q i is the required fixed point. In general, if J = I N , we can view the |J|-dimensional subsystem on ∩ j∈I N \J π j as a system on R |J| + . Then, by condition (i) and applying Lemma 5.1 to this |J|-dimensional system, we obtain a fixed point . By condition (i), the set π i ∩ [0, r] ∩ (∪ j∈I N \{i} (Γ j ∪ Γ − j )) is strictly below Γ i . By the convexity and definition of the set h(S) for a set S, S i (0) is strictly below Γ i so (Ay) i < r i for all y ∈ S i (0). Since S i (0) is compact and We show that (37) ∀x Suppose (37) is not true. Then, for some x 0 ∈ R N + \ (∪ j∈J π j ), there is y 0 ∈ ω(x 0 ) ∩ ([0, δ i ] × P (i) (S i (0))) and an increasing sequence {n k } such that T n k (x 0 ) → y 0 as k → ∞. As For Since the compact set [0, , a contradiction to the boundedness of {T n (x 0 )}. This shows (37).

Proof of Lemma 5.2
In order to prove Lemma 5.2, we first prove the following two lemmas. Note that the u in the next two lemmas are different from, but less restrictive than, the u in Proposition 2.1 as we do not require u j = 0 for j ∈ I N \ J 1 here. Lemma 6.1. For any u ∈ [0, r], let v ∈ [u, r] be defined by (19). Then, such that y ≥ x and (y − x) I N \{j} = 0. By (a2), T j (x) < T j (y) = y j ≤ w j . Thus, for all x ∈ [u, w], T (x) ≤ w and (39) holds.
Proof. We prove (40) by induction on |J 1 |. When |J 1 | = 0, i.e. J 1 = ∅, [u, w] is above Γ j for all j ∈ I N . Thus, for all x ∈ S, T (x) ≤ x so (40) holds. When |J 1 | = 1, J 1 = {i} for some i ∈ I N . As u is below Γ i but on or above Γ j for all j ∈ I N \ {i}, [u, v] and [u, w] are above Γ j for all j ∈ I N \ {i}. For any x ∈ S, if x ∈ Γ i ∪ Γ + i then x is on or above Γ k for all k ∈ I N so (40) holds with y = x. If x ∈ Γ − i , then there is a y ∈ Γ i ∩ S such that x ≤ y and (y − x) I N \{i} = 0. By (a2), T i (x) < T i (y) = y i . For each j ∈ I N \ {i}, as x is on or above Γ j , we have T j (x) ≤ x j = y j . Hence, T (x) ≤ y and (40) holds when |J 1 | = 1.
We show that (40) holds when |J 1 | = k + 1. For any x ∈ S such that x ∈ Γ i ∪ Γ + i for some i ∈ J 1 , let J 3 ⊂ I N such that x is below Γ j if and only if j ∈ J 3 . Since x is on or above Γ k for all k ∈ {i} ∪ (I N \ J 1 ), we have J 3 ⊂ J 1 \ {i} so 0 ≤ |J 3 | ≤ |J 1 | − 1 = k. As u ≤ x ≤ w ≤ r and u ≤ v ≤ w, defining x ∈ [x, r] by (19), i.e.
we have x j = x j ≤ w j for j ∈ I N \ J 3 and x j ≤ r j − (Au I N \{j} ) j = v j ≤ w j for j ∈ J 3 . Thus, x ≤ x ≤ w ≤ r. Then, by (IH), , ∃y ∈ S such that T (z) ≤ y. From the previous paragraph we know that, for any x ∈ S ∩ Γ i for some i ∈ J 1 but 1]. Otherwise, these points p(t) form a line segment p(0)p(1) = x I N \J 1 x. By (a3), T j (p(t)) for each j ∈ J 1 is increasing for t ∈ [0, 1]. Thus, if p(t) ∈ S, then p(t) is on or above Γ k for all k ∈ I N \ J 1 , so Hence, T (p(t)) ≤ y if p(t) ∈ S. Since the set S ∩ (∩ j∈J 1 (Γ j ∪ Γ − j )) consists of such points p(t) ∈ S, we have proved (40) when |J 1 | = k + 1. By induction, (40) holds for all J 1 with 0 ≤ |J 1 | ≤ N .
Equipped with Lemma 6.2, we are now in a position to prove Lemma 5.2.
Proof of Lemma 5.2. By assumption, [u, v] is as described in Proposition 2.1, so u j = 0 for j ∈ I N \ J and u ∈ Γ − i for all i ∈ J, and by (19), v j = u j = 0 for j ∈ I N \ J 1 and v i = r i − (Au I N \{i} ) i for i ∈ J 1 . By Lemma 6.1, By Lemma 6.2 for w = v and We first show the conclusion We now show the truth of (44) by contradiction. Suppose (44) is not true. Then there exist a point x 0 ∈ R N + \ (∪ j∈J π j ) and a point y 0 ∈ ω(x 0 ) ∩ ([u, v] \ S). Since ω(x 0 ) is invariant, there is a y 1 ∈ ω(x 0 ) such that T (y 1 ) = y 0 . By the definition of the set S, T (y 1 ) = y 0 ≤ x for any x ∈ S. Thus, by (42), y 1 ∈ S. As y 1 ∈ ω(x 0 ) ⊂ [u, v], we must have y 1 ∈ ω(x 0 ) ∩ ([u, v] \ S). Note that [u, v] \ S is strictly above Γ j for all j ∈ J 1 . Thus, y 0 j = T j (y 1 ) < y 1 j for all j ∈ J 1 . As y 0 i = y 1 i = 0 for all i ∈ I N \ J 1 , we have y 0 ≤ y 1 ≤ v. For any y ≥ y 0 and any j ∈ J 1 , (Ay) j ≥ (Ay 0 ) j > r j so G j ((Ay) j ) ≤ G j ((Ay 0 ) j ) < G j (r j ) = 1.
Then, by y 1 ∈ [y 0 , v], we have y 0 ≤ Ky 1 . Repeating the above process, we obtain a sequence It then follows that y 0 ≤ K n y n ≤ K n v for all n ≥ 0. By letting n → ∞, we obtain y 0 = 0, a contradiction to y 0 above Γ j for all j ∈ J 1 . This shows the truth of (44). Then (43) follows from (44).
Next, taking a fixed k ∈ J, we show that, if |J 1 | ≥ 2, In order to show (46), we first show that For this purpose, we consider the (N − 1)-dimensional system for all x ∈ R N + with x k = u k . ThenT j (y) = T j (x) for all j ∈ I N \ {k}. We can easily check that the assumptions (a1)-(a3) for T also hold forT . By Lemma 6.1 with w = Then, taking y ∈ [u k , v k ] × P (k) (S k (u)) with P (k) (y) = P (k) (z) and y k = v k , we have T (x) ≤ y. This shows (47).

Conclusion
For a class of discrete dynamical systems (1) with (2)-(5) and (a1)-(a3), we have successfully applied the geometric method of using the relative position of the N nullcline planes Γ i restricted to the cell [0, r] to the study of global dynamics and established criteria for the system to have a globally attracting nontrivial fixed point x * . Then global asymptotic stability of x * or existence of a homoclinic cycle can be determined by the eigenvalues of the Jacobian DT (x * ). We have demonstrated application of the criteria for global asymptotic stability of a fixed point by various concrete examples.
For global asymptotic stability of a fixed point of Kolmogorov type discrete dynamical systems, one typical theorem among the existing criteria in literature was given in [5] and a revised version was given in [11] by assuming the existence of a carrying simplex on a monotone region; another typical result was given in [4] by using split Lyapunov function method. Comparing our results with these two typical results, we reach the following conclusions: (i) While both of the existing typical results deal with global asymptotic stability of an interior fixed point only, our results cover global asymptotic stability of both interior and boundary fixed points.
(ii) The requirement for existence of a carrying simplex in the existing results in [5] and [11] is very restrictive for systems (1) with (2)-(5) and (a1)-(a3). Almost every available sufficient condition for existence of carrying simplex requires a ij > 0 for all i, j ∈ I N . But it is a great relief that we do not require the existence of a carrying simplex and only require a ii > 0, a ij ≥ 0, and permit a ij = 0, for distinct i, j ∈ I N .
(iii) Although the theorems given in [4] are useful in general, and powerful in some particular cases, it is generally difficult in constructing suitable Liapunov functions. Fortunately, we are not bothered here with construction of Liapunov functions and we need only check geometric conditions relating to the relative position of the N nullcline planes restricted to [0, r] ∩ ∂R N + or equivalent algebraic conditions using a ij and r k only.
(iv) From the comparisons given in section 4 at the end of Examples 4.1 and 4.3 we see that none of the three results by three different methods is better than others in general, but each has its advantages for some particular cases. Therefore, all of the results are supplements to each other and enrich the theory and methods of discrete dynamical systems.
Due to the restriction of time, space and the author's knowledge, there are a few problems that the author has not solved to make this paper more attractive and complete. Solutions to the problems below are expected for future investigation. (The author should be grateful to anyone who offers advice by private communications.) 1. For a discrete dynamical system in general, or for system (1) with (2)- (5) and (a1)-(a3) in particular, is a globally attracting fixed point always stable so always globally asymptotically stable? A proof for YES or a counter-example for NO is expected.
2. CONJECTURE. Assume that x * is a globally attracting fixed point and the Jacobian DT (x * ) is invertible. Then either x * is globally asymptotically stable or there is a homoclinic cycle. Is this conjecture correct?
3. If the answer to problem 1 above is YES, then our Theorem 3.9 (ii) is void. Otherwise, find an example that a fixed point x * is globally attracting and DT (x * ) has an eigenvalue with modulus greater than 1 so that x * induces a homoclinic cycle.
From the above detailed check we see that for system (1) with (2)-(5), for each i ∈ I N , if G i (u) is taken to be any one of e r i −u , 1+r i 1+u , b 1+u , then the system as a combination of (6)-(9) still satisfies (a1)-(a3).

Appendix 2. Proof of Proposition 2.3
Proof of Proposition 2.3. (i) Under (20) for all i ∈ J, we suppose (Ar I N \{i,j} ) i ≥ r i for some i, j ∈ J with i = j. Then the point r I N \{i,j} is on or above Γ i . As 0 is below Γ i , we have By (20), [0, r I N \{i} ] ∩ Γ j = Γ j ∩ [0, r] ∩ π i is strictly below Γ i , so [0, r I N \{i} ] ∩ Γ i is strictly above Γ j . Thus, [0, r I N \{i,j} ] ∩ Γ i as a nonempty subset of [0, r I N \{i} ] ∩ Γ i is strictly above Γ j . On the other hand, as (20) also holds with the replacement of i by j, [0, r I N \{j} ] ∩ Γ i = Γ i ∩ [0, r] ∩ π j is strictly below Γ j , so [0, r I N \{i,j} ] ∩ Γ i as a nonempty subset of [0, r I N \{j} ] ∩ Γ i is strictly be low Γ j , a contradiction to [0, r I N \{i,j} ] ∩ Γ i strictly above Γ j . This contradiction shows that (Ar I N \{i,j} ) i < r i , i.e. r I N \{i,j} is below Γ i and Γ j for all i, j ∈ J with i = j. Since (Ar I N \{i} ) j ≥ a jj r j = r j , r I N \{i} is on or above Γ j . Hence, [r I N \{i,j} , r I N \{i} ]∩Γ j = {z}, where (z−r) I N \{i,j} = 0, z i = 0 and z j = r j −(Ar I N \{i,j} ) j > 0. As z ∈ [0, r I N \{i} ] ∩ Γ j , by (20) z is below Γ i so (Az) i = a ij (r j − (Ar I N \{i,j} ) j ) + (Ar I N \{i,j} ) i < r i .
Then (21) follows for all i, j ∈ J with i = j.
(ii) If r I N \{i} is below Γ i then [0, r I N \{i} ] = [0, r] ∩ π i is strictly below Γ i so (20) holds. If r I N \{i} is on or above Γ i , by (21) we have (Ar I N \{i,j} ) i < r i so r I N \{i,j} is below Γ i for all j ∈ I N \ {i}. Thus, [r I N \{i,j} , r I N \{i} ] ∩ Γ i = {P j }. It can be checked that For each fixed k ∈ I N \ {i}, we show that P j is above Γ k for all j ∈ I N \ {i}. In fact, if j = k then (Ar I N \{i,j} ) k ≥ a kk r k = r k , so r I N \{i,j} is on or above Γ k . For j = k, if r I N \{i,k} is above Γ k then P k is above Γ k as P k ≥ r I N \{i,k} ; if r I N \{i,k} is on or below Γ k , as [r I N \{i,k} , r I N \{i} ] ∩ Γ k = {z} and z is below Γ i , [r I N \{i,k} , r I N \{i} ] ∩ Γ k is strictly below Γ i , so [r I N \{i,k} , r I N \{i} ] ∩ Γ i is strictly above Γ k . Hence, P k is above Γ k . As r I N \{i} ≥ P k , r I N \{i} must be above Γ k . Since r I N \{i,j} is on or above Γ k for all j ∈ I N \ {i, k}, the line segment [r I N \{i,j} , r I N \{i} ] \ {r I N \{i,j} }, which contains P j , is strictly above Γ k . Therefore, P j is above Γ k for all j ∈ I N \ {i}. By (49), [0, r I N \{i} ] ∩ Γ i is strictly above Γ k . Hence, [0, r I N \{i} ] ∩ Γ k is strictly below Γ i for all k ∈ I N \ {i}, i.e. (20) holds.
(iii) This is a combination of (i) and (ii).
Appendix 3. Proof of Lemma 5.1 Proof of Lemma 5.1. We prove the statement by induction on N . When N = 2, Γ 1 and Γ 2 are straight line segments in R 2 + , (36) means that the fixed point Q 1 on Γ 1 is below Γ 2 and the fixed point Q 2 on Γ 2 is below Γ 1 . This implies that Γ 1 has a point above Γ 2 and a point Q 1 below Γ 2 , so Γ 1 and Γ 2 has a unique intersection point x * ∈ intR 2 + , i.e. Ax = r has a unique solution x * 0. That x * ≤ r follows from a ii = 1 and a ij ≥ 0.