ON THE CLASSIFICATION OF GENERALIZED COMPETITIVE ATKINSON-ALLEN MODELS VIA THE DYNAMICS ON THE BOUNDARY OF THE CARRYING SIMPLEX

. We propose the generalized competitive Atkinson-Allen map which is the classical Atkson-Allen map when r i = 1 and c i = c for all i = 1 ,...,n and a discretized system of the competitive Lotka-Volterra equations. It is proved that every n -dimensional map T of this form admits a carrying simplex Σ which is a globally attracting invariant hypersurface of codimension one. We deﬁne an equivalence relation relative to local stability of ﬁxed points on the boundary of Σ on the space of all such three-dimensional maps. In the three-dimensional case we list a total of 33 stable equivalence classes and draw the corresponding phase portraits on each Σ. The dynamics of the generalized competitive Atkinson-Allen map diﬀers from the dynamics of the standard one in that Neimark-Sacker bifurcations occur in two classes for which no such bifurcations were possible for the standard competitive Atkinson-Allen map. We also found Chenciner bifurcations by numerical examples which implies that two invariant closed curves can coexist for this model, whereas those have not yet been found for all other three-dimensional competitive mappings via the carrying simplex. In one class every map admits a heteroclinic cycle; we provide a stability criterion for heteroclinic cycles. Besides, the generalized Atkinson-Allen model is not dynamically consistent with the Lotka-Volterra system.


(Communicated by Chongchun Zeng)
Abstract. We propose the generalized competitive Atkinson-Allen map which is the classical Atkson-Allen map when r i = 1 and c i = c for all i = 1, ..., n and a discretized system of the competitive Lotka-Volterra equations. It is proved that every n-dimensional map T of this form admits a carrying simplex Σ which is a globally attracting invariant hypersurface of codimension one. We define an equivalence relation relative to local stability of fixed points on the boundary of Σ on the space of all such three-dimensional maps. In the three-dimensional case we list a total of 33 stable equivalence classes and draw the corresponding phase portraits on each Σ. The dynamics of the generalized competitive Atkinson-Allen map differs from the dynamics of the standard one in that Neimark-Sacker bifurcations occur in two classes for which no such bifurcations were possible for the standard competitive Atkinson-Allen map. We also found Chenciner bifurcations by numerical examples which implies that two invariant closed curves can coexist for this model, whereas those have 1. Introduction. By Hirsch's carrying simplex theory [26], it is known that every strongly competitive and dissipative system of Kolmogorov ODEs for which the origin is a repeller possesses a globally attracting invariant hypersurface Σ of codimension one. Furthermore, Σ is homeomorphic to the (n − 1)-dimensional standard probability simplex ∆ n−1 = {x ∈ R n + : i x i = 1}, such that every nontrivial orbit in the nonnegative cone R n + is asymptotic to one in Σ. This result implies that n-dimensional strongly competitive continuous-time systems behave like general (n − 1)-dimensional systems, and hence the Poincaré-Bendixson theorem holds for the 3-dimensional case. Based on this remarkable theory, many researchers have obtained a lot of results on nontrivial dynamics for 3-dimensional continuoustime competitive systems, including the existence and multiplicity of limit cycles [22,23,24,28,34,38,47,48,52]; the existence of centers and heteroclinic cycles [8,34,52]; and ruling out periodic orbits [8,34,45,52]. Moreover, the readers can consult [4,6,7,29,30,39,49,50,51] for the geometrical properties of carrying simplices and their impact on the dynamics.
The research on the existence of carrying simplex for discrete-time systems began with Smith's work [43] on the dynamical behavior of the Poincaré map induced by time-periodic competitive Kolmogorov ODEs. Based on the early work of Hirsch [26] and Smith [43], there have been many results on the existence of carrying simplex for competitive mappings; see [46,12,25,42,5,33,32]. We refer the readers to the most recent article [32] for a review of the development of carrying simplex theory for competitive mappings. In [32], Jiang and Niu provided a readily checked criterion that guarantees the existence of carrying simplex for the continuous map T : R n + → R n + of the type T (x) = (T 1 (x), · · · , T n (x)) = (x 1 G 1 (x), · · · , x n G n (x)), where G i (x) > 0, i = 1, · · · , n, for all x ∈ R n + . They applied this criterion to show that all maps in a large family of competitive maps have a carrying simplex. Their result enriches the existing literature on discrete-time competitive dynamical systems with carrying simplices.
The importance of the existence of carrying simplex Σ stems from the fact that Σ captures the relevant long-term dynamics. It contains all non-trivial fixed points, periodic orbits, invariant closed curves and heteroclinic cycles, etc. In order to analyze the global dynamics of such discrete-time systems, it suffices to investigate the dynamics on Σ. However, compared with the continuous-time competitive systems, the research in discrete-time competitive systems via carrying simplices is much less. In [42] Ruiz-Herrera provided an exclusion criterion for discrete-time competitive models of two or three species via carrying simplices. Jiang and Niu [31] deduced an index formula on the sum of the indices of all fixed points on Σ for the three-dimensional map T of type (1): θ∈Ev I (θ, T ) + 2 θ∈Es I (θ, T ) + 4 θ∈Ep I (θ, T ) = 1. (2) was given in [31] and an alternative classification for 3-dimensional Leslie-Gower models was also given in [32]. Neimark-Sacker bifurcations were investigated within each class of these two types of models. Neimark-Sacker bifurcation is the birth of an invariant closed curve from a fixed point in discrete-time dynamical systems, and either all orbits are periodic, or all orbits are dense on the invariant closed curve. Such an invariant closed curve corresponds to either a subharmonic or a quasiperiodic solution in continuous-time systems. In [33], Jiang et al. studied the occurrence of heteroclinic cycles via carrying simplices for competitive maps (1) and provided their stability criteria.
In this paper, we study the long-term dynamics of the map defined on R n + , which plays a role as a discrete-time Lotka-Volterra system. When r i = 1 and c i = c, the model induced by the map (4) reduces to the Atkinson-Allen model (3). Other generalizations of (3) considered by Roeger and Allen [41], Atkinson [2] and Allen et al. [1] are also special cases of (4). We call the model induced by (4) the generalized Atkinson-Allen model. Smith [44] analyzed a related two-dimensional discrete-time model for competition between populations of cystnematodes, due to Jones and Perry [35]. Similar models are also treated in the monograph [40]. He showed that it generates a monotone map in R 2 + . For the analysis of a special case of three-dimensional model (4), we refer the readers to [41,12,31]. Our principal aim is to investigate whether the discrete-time model (4) admits a carrying simplex such that one can study its long-term dynamics via the carrying simplex as studying the continuous-time competitive systems.
The derivation of discrete-time population models from first principles is notoriously difficult and mistakes are frequently made. In particular, a straightforward discretization of an established continuous-time model almost inevitably leads to equations void of biological content ( [21,16]). We therefore give a mechanistic derivation of the model induced by the map (4).
We assume that the season is divided into three periods: one of competition, one of reproduction and one of survival. During the first period, the individuals of n species do not die, but compete for n different resources S j . Assuming chemostat dynamics of the resources and a Holling type I functional response, we obtain the following equations for the resource dynamics: Assuming the resource dynamics is fast, the resource concentrations will have reached the following steady state by the end of the period of competition: During the period of reproduction, individuals of species i will use only the resource S i given by (6) to produce offspring (with a conversion factor γ i ) born at the beginning of the next period of competition. The adults of species i survive to the next period of competition with probability c i . Putting all these assumptions together, we finally arrive at the following map T taking the state of the community at the beginning of one period of competition to the next one: which is, of course, exactly the map (4), but with the parameters denoted in a different way. We shall keep the parameters of (4) to make comparison with the similar models treated in the papers mentioned above easier. The generalized Atkinson-Allen model (4) can also be derived from the classical continuous-time competitive Lotka-Volterra (LV) system by using discretization scheme. Set By substituting (x i (t+h)−x i (t))/h for dx i (t)/dt and using a mixed implicit-explicit approximation we derive which can be expressed more simply as Set h = 1. Then we obtain the discrete-time generalized Atkinson-Allen system Based on the criterion to guarantee the existence of carrying simplex provided by Jiang and Niu [32], we can prove that any n-dimensional model (4) possesses a carrying simplex. The long-term dynamics of (4) is studied further via the carrying simplex. We define an equivalence relation on the space of all three-dimensional models (4) which is similar to the one for the standard Atkinson-Allen models [31] and Leslie-Gower models [32]. Two models (4) are said to be equivalent relative to ∂Σ (the boundary of Σ) if their boundary fixed points have the same locally dynamical property on Σ after a permutation of the indices {1, 2, 3}. We classify all three-dimensional generalized Atkinson-Allen models (4) by this equivalence relation using the index formula (2), and derive a total of 33 stable equivalence classes in terms of simple inequalities on the parameters c i , r i and b ij . Then one can investigate the qualitative properties of the orbits, bifurcations and the occurrence of heteroclinic cycles within each class.
Eighteen classes (classes 1 to 18) which do not possess a positive fixed point have trivial dynamics, i.e., every nontrivial orbit converges to some fixed point on ∂Σ. The other 15 classes all have a unique positive fixed point which may contain much more complex dynamics. We prove that Neimark-Sacker bifurcations do not occur in classes 19 − 25 and class 32 while they do occur in classes 26 − 31. We construct examples in classes 26 − 29 and 31 which admit supercritical Neimark-Sacker bifurcations, so these classes can have stable invariant closed curves on Σ. Class 30 can admit subcritical Neimark-Sacker bifurcations, so this class can admit unstable invariant closed curves on Σ. However, Neimark-Sacker bifurcations can not occur in classes 28 and 30 for the three-dimensional standard Atkinson-Allen model (3). We also provided numerical examples in classes 26 − 29 which admit Chenciner bifurcations. The Chenciner bifurcation is a two-parameter bifurcation phenomenon of a fixed point, which can bifurcate two invariant closed curves simultaneously. So classes 26 − 29 can possess two invariant closed curves on Σ, which are first found in competitive mappings via carrying simplices. Specifically, we find that a large unstable invariant closed curve surrounding a small stable invariant closed curve can occur on Σ in classes 26, 28 and 29, while a stable fixed point and a stable invariant closed curve, separated by an unstable invariant closed curve can coexist on Σ in class 27. Numerical simulations show that two attracting invariant closed curves can coexist on Σ for some maps in class 29 (see Fig. 8), which is also found in competitive mappings via the carrying simplex for the first time. Via the carrying simplex, we show that each map in class 27 has a heteroclinic cycle, i.e. a cyclic arrangement of saddle fixed points and heteroclinic connections. We further provide the stability criteria on heteroclinic cycles. This cyclical fluctuation phenomenon has also been found in many other models; see Cushing [10], Davydova et al. [11] and Jiang et al. [33]. Moreover, it is shown that the generalized Atkinson-Allen model (4) is not dynamically consistent with the continuous-time competitive LV system (8). Our works will make it possible to study various interesting dynamics within each of classes 19 − 33 further.
The paper is organized as follows. Section 2 presents some notations and preliminaries. In Section 3, it is shown that any n-dimensional generalized Atkinson-Allen model admits a carrying simplex. The formula on the sum of all indices of fixed points on Σ for three-dimensional models is reviewed. In Section 4, we define an equivalence relation on the space of all three-dimensional generalized Atkinson-Allen models and derive a total of 33 stable equivalence classes. The 33 stable equivalence classes with their corresponding phase portraits on Σ in terms of simple inequalities on the parameters are listed in Table 1 in the appendix. Furthermore, the dynamics on Σ of each class is studied. In Section 5, we compare the similarities and differences in the generalized Atkinson-Allen model and the Lotka-Volterra system numerically. The paper ends with a discussion in Section 6, where we list a few open problems for future investigation.
2. Notation and preliminaries. Throughout this paper, we reserve the symbol n for the dimension of the euclidean space R n and the symbol N for the set {1, · · · , n}. We denote the standard basis for R n by {e {1} , · · · , e {n} }. We use R n + to denote the nonnegative cone {x ∈ R n : x i ≥ 0, ∀i ∈ N }. The interior of R n + is the open conė R n + := {x ∈ R n + : x i > 0, ∀i ∈ N } and the boundary of R n + is ∂R n + := R n + \Ṙ n + . We write Z + for the set of nonnegative integers. We denote by H + {i} the ith positive coordinate axis and by π i = {x ∈ R n + : x i = 0} the ith coordinate plane. The symbol 0 stands for both the origin of R n and the real number 0.
Given two points x, z in R n , we write x ≤ z if z−x ∈ R n + , x < z if z−x ∈ R n + \{0}, and x z if z −x ∈Ṙ n + . The reverse relations are denoted by ≥, >, , respectively. Let X ⊂ R n and let T : X → X be a map. The positive orbit (trajectory) emanating from y ∈ X is the set {y(j) : j ∈ Z + }, where y(j) = T j (y) and y(0) = y.
A fixed point y of T is a point y ∈ X such that T (y) = y. We call z ∈ X a kperiodic point of T if there exists some positive integer k > 1, such that T k (z) = z and T m (z) = z for every positive integer m < k. The k-periodic orbit of the k-periodic point z, {z, z(1), z(2), . . . , z(k − 1)}, is often called a periodic orbit for short. A quasiperiodic curve is an invariant simple closed curve with every orbit being dense. For a differentiable map T , we let DT (y) denote the Jacobian matrix of T at the point y.
Given a k × k matrix A, we write A ≥ 0 if A is a nonnegative matrix (i.e., all its entries are nonnegative) and A > 0 if A is a positive matrix (i.e., all its entries are positive). The spectral radius of A, denoted by ρ(A), is defined to be the maximum of the absolute values of its eigenvalues. Given ∅ = J ⊆ N , we denote by A J the submatrix of A with rows and columns from J. We use I to denote both the identity matrix and the identity mapping.
A map T : R n A carrying simplex for the map T is a subset Σ of R n + \ {0} with the following properties: (P1) Σ is compact and unordered; (P2) Σ is homeomorphic via radial projection to the (n − 1)-dimensional standard probability simplex ∆ n−1 = {x ∈ R n + : i x i = 1}; (P3) ∀x ∈ R n + \ {0}, there is some z ∈ Σ such that lim j→∞ |T j x − T j z| = 0; (P4) T (Σ) = Σ, and T : Σ → Σ is a homeomorphism. We denote the boundary of the carrying simplex Σ relative to R n + by ∂Σ = Σ ∩ ∂R n + and the interior of Σ relative to R n + byΣ = Σ \ ∂Σ. We denote the set of all maps taking R n + into itself by T (R n + ) and the set of all generalized competitive Atkinson-Allen maps on R n + by CGAA(n). In symbols: The competitiveness of each map in CGAA(n) will be clear in §3.1. Finally we let B denote the n × n matrix with entries b ij . 3. Carrying simplex and index theory. From now on we assume that T (x) = ( Note that this implies that T i (x) > 0 if and only if x i > 0 and, in particular, that The existence of the carrying simplex. We first restate a criterion provided in [32] on the existence of carrying simplex for the map T .  [32]). Suppose that A1) ∂G i (x)/∂x j < 0 for all x ∈ R n + and i, j ∈ N ; is the support of x and q = q {i} = (q 1 , · · · , q n ).
Then T possesses a carrying simplex Σ.
Conditions A1) and A3) imply that T is competitive and also one-to-one in [0, q]. Specifically, A3) implies that det DT (x) > 0 for all x ∈ [0, q], and together with A1) it guarantees (DT (x) κ(x) ) −1 > 0 for all x ∈ [0, q] \ {0} by the proof of Theorem 3.1 in [32]. Then Proposition 4.1 in [42] ensures that T is competitive and one-to-one in [0, q]. Condition A1) also means that G i (y) < G i (x) for all i ∈ N provided x < y. This follows from so 0 is a hyperbolic repeller for T . All nontrivial fixed points, periodic points and invariant closed curves lie on Σ.
i.e., A2) in Theorem 3.1 holds. Finally, for any x ∈ R n + , A3) in Theorem 3.1 also holds. The result now follows from Theorem 3.1.

Remark 1.
Recall that when r i = 1 and c i = c, i = 1, · · · , n, the map T ∈ CGAA(n) is the standard Atkinson-Allen map (see [41,12,31]) So Proposition 1 also implies that every Atkinson-Allen map admits a carrying simplex Σ, while this result was proved only for the three-dimensional case in [12].
Remark 2. Each map T ∈ CGAA(n) is competitive and one-to-one on R n + . Specifically, to show the injectivity of T we employ Lemma 3.4 in [9, p. 27], which says that if T is a continuous, locally homeomorphic map on a connected metric space for which the inverse image of every compact set is compact, then the cardinal number of the inverse image of every point is finite and the same for all points. Since, as noticed above, in our case T −1 ({0}) = {0}, this constant is one and hence T is one-to-one if T satisfies the above mentioned properties. Recall that A3) in Theorem 3.1 holds at every x ∈ R n + \ {0} for each T ∈ CGAA(n), which implies that det DT (x) > 0 for all x ∈ R n + (see the proof of Theorem 3.1 in [32]), so T is locally homeomorphic on R n + . The injectivity of T will now follow once we have showed that the inverse image of every compact set is compact. To end this, let W ⊆ T (R n + ) be a compact set in T (R n + ). Because T is continuous, T −1 (W ) is a closed set in R n + . Next we show that T −1 (W ) is bounded. If this was not the case, there would exist a sequence x k ∈ T −1 (W ), such that x k i → +∞ as k → +∞ for at least one i ∈ N . Then by (4), one would have T i (x k ) → +∞ as k → +∞, which would contradict the compactness of W . On the other hand, together with A1) in Theorem 3.1, A3) implies that (DT (x) κ(x) ) −1 > 0 holds at every x ∈ R n + \ {0}. It then follows from Proposition 4.1 in [42] that T is competitive on R n + . 3.2. The index formula on the carrying simplex. Let Q = I − T and F = −Q = T − I. Let x be a fixed point of T , that is, a zero of Q and F . The index of T at x is denoted by I (x, T ) and the index of Q and F at x is denoted by X (x, Q) and Assume n = 3. We call the fixed point x of the map T : R 3 + → R 3 + an axial fixed point if it lies on some coordinate axis; a planar fixed point if it lies in the interior of some coordinate plane; and a positive fixed point if it lies inṘ 3 + . We denote the set of all nontrivial axial, planar, and positive fixed points by E v , E s , and E p , respectively.
Assume further that T possesses a carrying simplex Σ and that the continuous-time systemẋ = F (x) = T (x) − x is dissipative with the origin 0 being a repeller. If T has only finitely many fixed points on Σ and 1 is not an eigenvalue of any of their Jacobian matrices, then T may also have a positive fixed point p inṘ 3 We It is easy to check that for T ∈ CGAA(3), the conditions in Theorem 3.2 hold, so the following corollary is immediate from the above analysis and Theorem 3.2.
Corollary 1. Assume that T ∈ CGAA(3) and 1 is not an eigenvalue of any of the Jacobian matrices at the fixed points on Σ. Then we have has a unique positive solution, then 1 is not an eigenvalue of Then by Perron-Frobenius theorem, 0 < ρ(M ) < 1 is an eigenvalue of M and the magnitudes of the other eigenvalues are all less than 1. Set λ * : 1+ri ]B and M have the same eigenvalues, 0 < λ * < 1 is a real eigenvalue of DT (p) whose associated eigenvector is strictly positive and all the other eigenvalues have real parts greater than 0 and less than 2.

The dynamics of the 3-dimensional generalized Atkinson-Allen model.
In this section, we analyze the long-term behavior of the map T ∈ CGAA (3): It follows from Proposition 1 that T admits a 2-dimensional carrying simplex Σ homeomorphic to ∆ 2 . Each coordinate plane π i is invariant under T , and the restriction of T to π i is a 2-dimensional map T | πi ∈ CGAA(2), which has a onedimensional carrying simplex, so ∂Σ is composed of the one-dimensional carrying simplices of T | πi . Therefore, it is convenient for us to study the two-dimensional generalized Atkinson-Allen model first. We show that there are only four dynamical outcomes for two-dimensional cases. For T ∈ CGAA(1), i.e., T (x) = (1+r)(1−c)x 1+bx + cx, the fixed point p = r/b is the carrying simplex, i.e., p is globally asymptotically stable inṘ + . This can be seen immediately but it also follows as a very special case of Proposition 1.

4.1.
Classification of the 2-dimensional maps. In this subsection, we study the model T ∈ CGAA(2): By Proposition 1, T admits a one-dimensional carrying simplex Σ homeomorphic to the line segment joining the two points (0, 1) and (1, 0). Our first result (Proposition 2 below) says that every nontrivial trajectory of a two-dimensional generalized Atkinson-Allen map converges to a fixed point on the carrying simplex. To prove this, we need the following lemma which is a direct consequence of Corollary 4.4 in [44] adapted to maps on R 2 + .
Lemma 4.1. Assume that the map T : R 2 + → R 2 + satisfies the following conditions: the diagonal entries of DT (x) are nonnegative and its off-diagonal entries are nonpositive, (iv) T is injective.
Then every orbit with compact closure in R 2 + converges to a fixed point of T .
Proposition 2. Every nontrivial trajectory of T ∈ CGAA(2) in R 2 + converges to a fixed point on Σ.
Proof. The Jacobian matrix obviously has nonnegative diagonal entries and nonpositive off-diagonal entries, i.e. (iii) in Lemma 4.1 holds. By Remark 2, one has det DT (x) > 0 for all x ∈ R 2 + and T is injective, so (ii) and (iv) in Lemma 4.1 hold. Now the conclusion follows from Lemma 4.1.
Besides the trivial fixed point 0, the map T has two axial fixed points q  If γ ij > 0 (resp. < 0), then q {i} is a saddle (resp. an asymptotically stable node), and hence repels (resp. attracts) on Σ. Moreover, q {i} is hyperbolic if and only if γ ij = 0.
Remark 4. Recall that γ ij > 0 (resp. < 0) if and only if q {i} ∈ B j (resp. U j ). So the nature of the fixed point q {i} can be determined by the position of q {i} relative to the line S j , i = j. Moreover, if γ 12 γ 21 > 0 (resp. < 0), then S 1 and S 2 intersect (resp. do not intersect) inṘ 2 + , i.e., there exists (resp. does not exist) a positive fixed point p.
The following theorem is an immediate consequence of Proposition 2, Lemma 4.2 and Remark 4. The analysis in [31] carries over to the present situation in a straightforward way. (d) If γ 12 , γ 21 < 0, then T has a positive fixed point p which is a hyperbolic saddle. Moreover, every nontrivial orbit tends to one of the asymptotically stable nodes q {1} or q {2} or to the saddle p. one of the species will oust the other. The surviving species depends on the initial conditions. (Convergence to the positive saddle happens only for initial conditions in a set of measure zero and is hence impossible in nature). The situations mentioned above are of particular interest when the two populations 1 and 2 are not different species, but different traits (resident and mutant) of the same species. To begin with, the resident (i = 1) is at the fixed point q {1} and then the mutant q {2} is introduced in small quantities. Case (i) γ 12 > 0 gives the condition for successful invasion. Case (ii) describes trait substitution. Case (iii) is an example of protected dimorphism. For a discussion of these notions and their consequences for evolutionary dynamics we refer the reader to [14,15,17,18].
The following definition of equivalence appears to be unnecessarily pompous, but it prepares the way for the analogous definition in higher dimensions. Let T,T ∈ CGAA(2). T andT are said to be equivalent relative to ∂Σ if there exists a permutation σ of {1, 2} such that T has a fixed point q {i} if and only ifT has a fixed pointq {σ(i)} , and further q {i} has the the same hyperbolicity and local dynamics asq {σ(i)} . A model T ∈ CGAA(2) is said to be stable relative to ∂Σ if all the fixed points on ∂Σ are hyperbolic. We say that an equivalence class is stable if each mapping in it is stable relative to ∂Σ.
There are a total of 3 stable equivalence classes in CGAA(2). The three dynamical scenarios are presented in Fig. 1.  Biologically, det B > 0 means that both species can invade, while det B < 0 means that none of them can (Remark 5 (i)).

4.2.
Classification of the 3-dimensional maps. We are now ready to analyze the three-dimensional model (14). We let S i be the plane {x ∈ R 3   (3) is said to be stable relative to ∂Σ if all the fixed points on ∂Σ are hyperbolic. We call an equivalence class stable if each map in it is stable relative to ∂Σ.
By the invariance of π i and the analysis of the 2-dimensional case in §4.1, the classification program, statements, proofs in [31] carry over to CGAA(3) in a straightforward way, so we do not need to re-do it.
The biological meaning of the condition γ ij > 0 (resp. < 0) in Lemmas 4.4-4.5 is that species j can (resp. not) invade species i in the absence of species k; here i, j, k are distinct.

Remark 7.
It is easy to check that (Bv τ {k} ) k < r k (resp. > r k ) if and only if b ki β ij + b kj β ji < r k (resp. > r k ). A model T ∈ CGAA(3) is stable relative to ∂Σ if and only if γ ij = 0 and b ki β ij + b kj β ji = r k , i.e., (Bv τ {k} ) k = r k (if v {k} exists). Suppose that T is stable relative to ∂Σ. If T admits a positive fixed point p satisfying (12), then p is the unique positive fixed point. Otherwise, assume that T has two different positive fixed points p andp. Now p s := sp + (1 − s)p is a solution of (12) for any s ≥ 0. Lets := sup{s > 0 : p s ∈ Σ}. Then ps ∈ ∂Σ is a fixed point, which is not hyperbolic, contradicting that T is stable relative to ∂Σ. Therefore, B −1 exists so 1 is not an eigenvalue of DT (p).  Proof. It is a straightforward combinatorial task to classify the stable equivalence classes, which is based on the index formula (17), Remark 7 and a geometric analysis of the positions of the three planes S j .
Step 1 There are a total of 2 6 possibilities for the non-zero values of sgn(r ij ) which reduce to 16 possibilities modulo permutation of the indices.

4.3.
Dynamics on the carrying simplex. As befits the context, we shall consider the families of models given in Table 1 by permutation of the indices, i.e., we assume the parameters b ij , r i , c i of the corresponding class satisfy the conditions listed in the table.
Recalling §3.1, each map T ∈ CGAA(3) satisfies A1)-A2) in Theorem 3.1, which implies that G i (y) < G i (x) for all i ∈ N provided x < y, and T is competitive and one-to-one on R 3 + , so conditions C1)-C3) of Theorem 2.2 in [42] hold for T . By Remark 7, each map T ∈ CGAA(3) which is stable relative to ∂Σ has only finitely many fixed points, so it satisfies C4) of Theorem 2.2 in [42]. Then by Theorem 2.2 in [42] we conclude the following result. According to the index formula (17) and Proposition 4, one has I (p, T ) = 0 for each map T in the stable classes 1 − 18, so there is no positive fixed point for these maps. The following proposition follows from Lemma 4.8 immediately.
Proposition 5. For each map T in classes 1 − 18, every nontrivial orbit converges to some fixed point on ∂Σ.
From a biological point of view, Proposition 5 means that if there is no coexistence, then some of the species will be extinct. Now we only need to study classes 19 − 33. Note that each model in them has a unique positive fixed point p = (p 1 , p 2 , p 3 ). We will focus on analyzing whether these classes can admit Neimark-Sacker bifurcations and Chenciner bifurcations (for a textbook treatment of these two bifurcations, see [36]). These bifurcations can bifurcate invariant closed curves, which either consist of periodic points or are quasiperiodic curves. Moreover, at Chenciner bifurcations two isolated invariant closed curves can be created. We prove that classes 19 − 25 can not admit these two bifurcations, while classes 26 − 31 can admit Neimark-Sacker bifurcations. We also construct examples to show that classes 26 − 29 can admit Chenciner bifurcations numerically by using the methods provided in [19], so in these classes, there may exist two invariant closed curves. Proof. For classes 19 − 25 (resp. classes 26 − 33), it follows from the local dynamics of fixed points on ∂Σ in Table 1 and formula (17)  Proof. By Lemma 4.9 one has that I (p, T ) = −1 and all the three eigenvalues of DT (p) are positive real numbers with one eigenvalue greater than 1 and the other two less than 1. Since Σ is invariant and transverse to all strictly positive vectors, the local dynamics of p on Σ is reflected by the other two eigenvalues except λ * , where 0 < λ * < 1 is defined in Remark 3. Thus, p is a saddle on Σ. Since one necessary condition for Neimark-Sacker bifurcations and Chenciner bifurcations occurring at p is that DT (p) has a pair of complex conjugate eigenvalues of modulus 1, these bifurcations can not occur in these classes. Proof. Note that each map T in class 32 admits a fixed point v {k} repelling along ∂Σ ∩ π k for any k = 1, 2, 3 (see Table 1 (32)), so v {k} is a saddle for T | π k . Thus det B {i,j} < 0 (i < j) by Remark 6, and hence det M {i,j} < 0, where M = diag[p i 1−ci 1+ri ]B. By det B > 0, one has det M > 0. Then it follows from Lemma 4.10 and det M > 0 that M has two eigenvalues with negative real parts. Therefore, DT (p) = I − M has two eigenvalues with real parts greater than 1, i.e, DT (p) has two eigenvalues with magnitudes greater than 1 except λ * . Recall that the local dynamics of p on Σ is reflected by the other two eigenvalues except λ * , so p is a repeller on Σ and hence Neimark-Sacker bifurcations and Chenciner bifurcations do not occur within this class.  By Proposition 8, we may assume that the fixed point p of T from any of the stable classes 26 − 33 is at (1, 1, 1). Then the parameters b ij , r i of T satisfy that j b ij = r i , i = 1, 2, 3, i.e., the sum of the ith row of B = (b ij ) 3×3 is r i . Besides, Hereafter, we always assume that n = 3, and b ij , r i > 0 satisfying det B > 0 and j b ij = r i , i = 1, 2, 3. Consider the map T ∈ CGAA(3) with the parameters b ij , r i , c i , where 0 < c i < 1. Let A := I − diag[ 1−ci 1+ri ]B. Lemma 4.11. Under the above assumptions, we have (a) if det B {i,j} < 0, then for 0 < c k < 1 sufficiently close to 1, the matrix A has two eigenvalues with magnitudes greater than 1, where i, j, k are distinct; (b) if det B {i,j} > 0, then for 0 < c k < 1 sufficiently close to 1, the matrix A has two eigenvalues with magnitudes less than 1, where i, j, k are distinct.
Proof. Set M := diag[ 1−ci 1+ri ]B. Then for definiteness, let i = 1, j = 2, k = 3. (a) By det B {1,2} < 0, one has det M {1,2} < 0. For c 3 = 1, the entries in the third row of M are 0, so M has a negative eigenvalue and a positive eigenvalue besides 0. Since the eigenvalues of M depend continuously on c 3 , thus for 0 < c 3 < 1 sufficiently close to 1, M has an eigenvalue with negative real part. Recall that det B > 0, so det M > 0, which implies that M has two eigenvalues with negative real parts. Therefore, A has two eigenvalues with real parts greater than 1, i.e, A has two eigenvalues with magnitudes greater than 1.
(b) By det B {1,2} > 0, one has det M {1,2} > 0. So, M has two positive eigenvalues besides 0 for c 3 = 1 because M {1,2} is a positive matrix. It follows from r i = b i1 + b i2 + b i3 that the sum of each row of M is less than 1. Then the Perron-Frobenius theorem ensures that both of the two positive eigenvalues are less than 1. Thus A has two eigenvalues with magnitudes less than 1 for c 3 = 1, and hence for 0 < c 3 < 1 sufficiently close to 1. Proof. Without loss of generality, assume that det B {1,2} > 0, det B {1,3} < 0. First fix 0 < c 1 < 1, 0 < c 2 = µ 0 < 1. Since det B {1,2} > 0, it follows from Lemma 4.11 that there exists 0 < c 3 = µ 3 < 1 sufficiently close to 1 such that A has two eigenvalues with magnitudes less than 1. Now fix c 1 and c 3 = µ 3 . Since det B {1,3} < 0, Lemma 4.11 ensures that A has two eigenvalues with magnitudes greater than 1 for 0 < c 2 = µ 1 < 1 sufficiently close to 1. Thus, as c 2 varies from µ 0 to µ 1 , at least one of the eigenvalues of A varies continuously from having magnitude less than 1 to magnitude greater than 1, and necessarily crosses the unit circle in the complex plane. Since det B = 0, 1 is not an eigenvalue of A. On the other hand, by Remark 3 one knows that all the eigenvalues of A have positive real parts. So −1, ±i, (−1 ± √ 3i)/2 are not eigenvalues of A; and moreover, there exists a 0 < µ 2 < 1 such that A has a pair of complex conjugate eigenvalues of modulus 1 as c 2 = µ 2 . Now one can choose c 1 , c 2 = µ 2 , c 3 = µ 3 .
Given b ij , r i > 0, 0 < c 1 , c 3 < 1 such that det B > 0 and the proof is a copy of that of Lemma 4.14 in [32] by replacing s c2(s) to be ψ(s).
Theorem 4.14. Given b ij , r i > 0 such that det B > 0 and 3 j=1 b ij = r i , i, j = 1, 2, 3. Consider the map T c ∈ CGAA(3) given by (14) with the parameters b ij , r i > 0 and 0 < c i < 1. If det B {i,j} , i < j, i, j = 1, 2, 3, are not all of the same sign, then there exists someĉ = (ĉ 1 ,ĉ 2 ,ĉ 3 ) with 0 <ĉ i < 1 such that the Jacobian matrix DTĉ(p) has a pair of complex conjugate eigenvalues λĉ 1,2 with modulus 1 which do not equal ±1, ±i, (−1 ± √ 3i)/2, where p = (1, 1, 1) is the positive fixed point. Furthermore, the restriction of T c to the two dimensional center manifold at the critical parameter valueĉ can be transformed to the complex Poincaré normal form where ω is a complex variable and d(β) is a complex function.
It follows from Lemma 4.12 that there exist 0 <ĉ i < 1, i = 1, 2, 3 such that Aĉ has a pair of complex conjugate eigenvalues λĉ 1,2 with modulus 1 which do not equal ±1, ±i, (−1 ± √ 3i)/2. Fix c 1 =ĉ 1 and c 3 =ĉ 3 . Set c 2 = s, and write A s := A c . Let s 0 =ĉ 2 . Then A s admits a pair of complex conjugate eigenvalues with modulus 1 at 0 < s = s 0 < 1. Thus A s has a pair of complex conjugate eigenvalues w(s), w(s) with |w(s 0 )| = 1 for s in a small neighborhood V of s 0 . By Lemma 4.13, one has d|w(s)| ds | s=s0 = 0. Then the conclusion follows from [36,Theorem 4.5], which can be proved in quite the same manner as the Theorem 4.3 in [32] (see also [37]), so we omit it.
Let L 1 (0) := Re(e −iθ(0) d(0)), which is the first Lyapunov coefficient (see [37]). Using Theorem 4.14 and [36, Theorem 4.6], we have the following result. It should be pointed out that the conditions λĉ 1,2 = ±1, ±i, (−1 ± √ 3i)/2, the possible roots of z k = 1 for k = 1, 2, 3, 4, in Theorems 4.14-4.15 are not merely technical; see [36,Chapter 4]. If they are not satisfied, the invariant closed curve may not appear at all, or other complex dynamics might occur; see [36,Chapter 9] and [37] for more details. However, for T ∈ CGAA(3) in the stable classes 26 − 33, Remark 3 ensures that ±1, ±i, (−1 ± √ 3i)/2 can not be the eigenvalues of DT (p). The biological interpretation of the condition det B {i,j} < 0 in Lemmas 4.11-4.12 and Theorems 4.14-4.15 is that at most one of the species i and j can invade the other in the absence of species k, whilst det B {i,j} > 0 means that at least one of the species i and j can invade the other in the absence of species k. Therefore, the biological meaning of the condition det B {i,j} , i < j, being not all of the same sign (say det B {i,j} < 0 and det B {i,k} > 0) is that at most one of the species i and j can invade the other in the absence of species k, whilst at least one of the species i and k can invade the other in the absence of species j.  (p) has a pair of complex conjugate eigenvalues with modulus 1 which do not equal ±1, ±i, (−1 ± √ 3i)/2. Furthermore, by calculating we obtain the first Lyapunov coefficient L 1 (0) = −2.204 × 10 −3 < 0. Since the Lyapunov coefficient is a rather lengthy expression, the approximate value was computed as a rational by using MATLAB [19,37]. Thus, by Theorem 4.15 there is a supercritical Neimark-Sacker bifurcation in class 26, i.e., a stable invariant closed curve bifurcates from the fixed point p.  Figure 2. The orbit emanating from x 0 = (0.7667, 0.7667, 1) for the map T ∈ CGAA(3) with the parameters B [26] , r [26] i and c i , i = 1, 2, 3 tends to an attracting quasiperiodic curve (the blue circle), where B [26] and r [26] i are given in Example 4.1 and c 1 = 0.81, c 2 = 0.5, c 3 = 0.5. (3), it is shown that classes 26 and 27 can admit Neimark-Sacker bifurcations while classes 28, 30 and 32 can not (see [31]). For the 3-dimensional generalized Atkinson-Allen model (4), we have shown that classes 26 − 31 can admit Neimark-Sacker bifurcations. Thus one can see that this is a significant difference between model (3) and model (4), and the generalized Atkinson-Allen model (4) contains much richer dynamics. Furthermore, we will give some numerical examples to show that classes 26 − 29 for model (4) can also admit Chenciner bifurcations, which means that in these classes, two isolated invariant closed curves may coexist.

Remark 8. For the 3-dimensional standard Atkinson-Allen model
Consider a sufficiently smooth map Φ(x, β) : R n × R 2 → R n , where x ∈ R n , β ∈ R 2 . Assume that Φ has a fixed point x = 0 at β = 0 for which the Neimark-Sacker bifurcation conditions hold. Thus DΦ(0, 0) has a pair of conjugate complex eigenvalues lying on the unit circle. Assume further that Φ satisfies some other non-degeneracy conditions such that the restriction of Φ to the two dimensional  (3) with the parameters B [27] , r [27] i and c i , i = 1, 2, 3 tends to an attracting quasiperiodic curve (the blue boundary), where B [27] and r [27] Figure 4. The orbit emanating from x 0 = (0.9333, 1, 0.9333) for the map T ∈ CGAA(3) with the parameters B [29] , r [29] i and c i , i = 1, 2, 3 tends to an attracting quasiperiodic curve (the blue circle), where B [29] , r [29] i , i = 1, 2, 3 are given in Example 4.1 and c 1 = 0.89, c 2 = 0.9995, c 3 = 0.8. center manifold at the critical parameter value β = 0 can be transformed to the normal form in polar coordinates ( , θ) (see [36] for more details):  (3) with the parameters B [31] , r [31] i and c i , i = 1, 2, 3 tends to an attracting quasiperiodic curve (the blue circle), where B [31] , r [31]  where µ = (µ 1 , µ 2 ) and the dots denote terms of higher order in and θ. Truncating the higher order terms gives the map = 0 corresponds to the fixed point of the system and any positive fixed point of the -map in (19) corresponds to an invariant closed curve in phase space. µ 1 = 0 corresponds to the Neimark-Sacker bifurcation curve, for which a pair of conjugate complex eigenvalues lie on the unit circle, and µ 2 is the corresponding first Lyapunov coefficient when µ 1 = 0. For µ 2 < 0, a supercritical Neimark-Sacker bifurcation occurs at µ 1 = 0, whereas for µ 2 > 0 a subcritical Neimark-Sacker bifurcation occurs at µ 1 = 0. For µ 2 = 0 the Neimark-Sacker bifurcation becomes degenerate, which is called the Chenciner bifurcation (see [36,13]). The Chenciner bifurcation occurs at µ = 0 for which a pair of conjugate complex eigenvalues lie on the unit circle and the first Lyapunov coefficient µ 2 = 0. An extra non-degeneracy condition for the Chenciner bifurcation is L 2 (0) = 0. Here we show some details by assuming that L 2 (0) < 0. Without loss of generality, we assume that L 2 (0) = −1.
* is positive fixed point of the -map in (19) if and only if it is a positive solution to the equation µ 1 + µ 2 2 − 4 = 0, i.e., When µ 1 > 0 there is exactly one positive solution for equation (20). For µ 1 < 0, equation (20) has no solution when  Figure 6. Bifurcation diagram of the Chenciner bifurcation in the (µ 1 , µ 2 )-plane for the case L 2 (0) < 0. The origin is the Chenciner bifurcation point. The vertical dashed line µ 1 = 0 is the Neimark-Sacker bifurcation curve. In the region I below the curve T c , there is only one fixed point which is stable; in the region II (µ 1 > 0), there is a unique invariant closed curve which is stable; in the region III between the curve T c and the positive µ 2 -axis, a stable invariant closed curve (outer) and an unstable invariant closed curve (inner) coexist; on the solid curve T c , these two circles coincide. See Fig. 6 for a sketch of this bifurcation diagram. For L 2 (0) > 0, it can be treated similarly, and in this case, the outer invariant closed curve is unstable, while the inner one is stable.
The Chenciner bifurcation is a two-parameter bifurcation phenomenon of a fixed point. Although the normal form computations for Chenciner bifurcations are straightforward, in practical models they can be very complicated. Here based on the numerical methods provided in [19], we do numerical experiments by using MATLAB [37,20] (3) with the parameters B [29] , r [29] i and c 1 = 0.999655, c 2 = 0.339655, c 3 = 0.2 are asymptotic to the bigger quasiperiodic curve and the smaller one respectively, where B [29] and r [29] i are given in Example 4.1.
near the critical values admits two attracting invariant closed curves on its carrying simplex simultaneously, where the orbits emanating from x 0 = (1.004, 0.9927, 1.48) and x 0 = (1.001, 1.002, 1.001) tend to a bigger attracting quasiperiodic curve and a smaller one respectively.
We now turn to study another interesting phenomenon, the occurrence of heteroclinic cycles. Suppose that the 3-dimensional map T has a carrying simplex Σ, which is homeomorphic to ∆ 2 . Suppose further that q {1} = (q 1 , 0, 0), q {2} = (0, q 2 , 0) and q {3} = (0, 0, q 3 ) are its three axial fixed points lying on the vertices of Σ. If each q {i} is a saddle, and ∂Σ ∩ π i is the heteroclinic connection between q {j} and q {k} , then T admits a heteroclinic cycle of May-Leonard type: q {1} → q {2} → q {3} → q {1} (or the arrows reserved), which is just the boundary of Σ. We refer the readers to [33] for more details and the stability of such heteroclinic cycles. Lemma 4.16 (Theorem 3 in [33]). Suppose that ∂Σ is a heteroclinic cycle above. Then the heteroclinic cycle ∂Σ repels (attracts) if where i ∈ {1, 2, 3} is considered cyclic.
6. Discussion. This paper proves that any n-dimensional generalized Atkinson-Allen map T ∈ CGAA(n) can possess a carrying simplex Σ. Based on the existence of Σ, we define an equivalence on the set CGAA(3), i.e., two mappings in CGAA (3) are said to be equivalent if all the boundary fixed points have the same local dynamics on the carrying simplices after a permutation of the indices {1, 2, 3}. Then using the index formula for fixed points on Σ, we derive a total of 33 stable equivalence classes for CGAA(3) via combinatorial technique. The dynamics of each map from any of classes 1 −18 is trivial, i.e., every nontrivial trajectory converges to some fixed point on ∂Σ and the global dynamics of these maps can be determined by the local dynamics of fixed points on ∂Σ. However, the dynamics of those maps from classes 19 − 33 are relatively complex which may not be determined by the local dynamics of fixed points on ∂Σ only. In classes 19 − 25, each map has a positive fixed point which is a saddle on Σ, so within each of these classes Neimark-Sacker bifurcations and Chenciner bifurcations can not occur. See Table 1  Another interesting phenomenon is that each map in class 27 possesses a heteroclinic cycle, i.e. a cyclic arrangement of saddle fixed points and heteroclinic connections. The competition coefficients in this class can be seen to correspond to the biological environment where in purely pairwise competition 1 beats 2, 2 beats 3, and 3 beats 1. It is this intransitivity in the pairwise competition which leads to such cyclic behavior. We further provide the criteria on the stability of heteroclinic cycles, and also show that this model indeed admits heteroclinic cycle attractors, i.e. under mild conditions the model in class 27 exhibits a general class of orbits which cycle from being composed almost wholly of species 1, to almost wholly 2, to almost wholly 3, back to almost wholly 1 etc. Our classification makes it possible to investigate much more various interesting dynamics within each of classes 19 − 33.
Recall that for the standard 3-dimensional Atkinson-Allen model (3), classes 28 and 30 can not admit Neimark-Sacker bifurcations, while for the generalized Atkinson-Allen model T ∈ CGAA(3), each of classes 26 − 31 can admit Neimark-Sacker bifurcations, and classes 26 − 29 can admit Chenciner bifurcations, so the generalized Atkinson-Allen model contains much richer dynamics. The generalized Atkinson-Allen model is also not dynamically consistent with the continuous-time competitive Lotka-Volterra system. However, it is worth noting that several problems remain open. We propose some as follows.
• Unlike the continuous-time systems, for which the Poincaré-Bendixson theorem holds, how to obtain the global dynamics of classes 19 − 25 for the discrete-time 3-dimensional generalized Atkinson-Allen model is open. • Enlightened by Figs. 7 and 8, it is extremely interesting to provide criteria to guarantee that the 3-dimensional generalized Atkinson-Allen model has multiple invariant closed curves in classes 26 − 33. So far there have been many results on the coexistence of multiple limit cycles for 3-dimensional competitive Lotka-Volterra equations ( [22,23,24,28,34,38,47,48]). • Whether the generalized Atkinson-Allen model can possess a center on Σ or not is also unknown. • Give sufficient conditions to guarantee that the positive fixed point is globally attracting on Σ. Recently, we learn that Baigent provides a sufficient condition to guarantee the global stability of the positive fixed point for the 3-dimensional Leslie-Gower model and Ricker model respectively [3].