Permanence and universal classification of discrete-time competitive systems via the carrying simplex

We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of 33 stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes 1-25 and 33; there is always a heteroclinic cycle in class 27; Neimark-Sacker bifurcations may occur in classes 26-31 but cannot occur in class 32. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes 29,31,33 and those in class 27 with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.


1.
Introduction. The theory of the carrying simplex was first developed in [39] by Hirsch for continuous-time competitive systems of Kolmogorov ODEs, which states that every strongly competitive and dissipative system for which the origin is a repeller possesses a globally attracting hypersurface Σ of codimension one, called the carrying simplex in [84,87]. Furthermore, Σ is homeomorphic to the (n − 1)-dimensional standard probability simplex ∆ n−1 = {x ∈ R n + : i x i = 1} by radial projection, and has the property that every nontrivial orbit in the nonnegative cone R n + is asymptotic to one in Σ. It has been proved as a powerful tool to investigate global dynamics of competitive systems, especially for the lower dimensional systems. The reader can consult, for instance, [44,80,83,85,86,61,35,7,4,51,45,11], for more results on continuous-time competitive systems via the carrying simplex.
The theory of the carrying simplex has been extended to discrete-time competitive systems due to the early work of de Mottoni and Schiaffino [16], Hale and Somolinos [36] and Smith [75]. By introducing mild conditions, Wang and Jiang proved in [82] the existence of a carrying simplex for competitive mappings, which solved the conjecture on the carrying simplex proposed by Smith in [75] and was improved further by Diekman, Wang and Yan [17]. Further, Hirsch announced a theory on the existence of a carrying simplex in [40] for the continuous Kolmogorov map (not necessarily invertible) where F i are continuous satisfying F i (x) > 0 for all x ∈ R n + , i = 1, . . . , n. The discrete-time dynamical system induced by such T has been much used to describe the interactions of n species with non-overlapping generations; see [63,47,42,20,70,60,53,40]. The statement of Hirsch's Theorem in [40] was rigorously proved by Ruiz-Herrera [74] under similar assumptions to Hirsch's. Other criteria on the existence of a carrying simplex for Kolmogorov mappings were also established in [5,50,49], and we refer the readers to the paper [49] for a review.
The importance of the existence of a carrying simplex stems from the fact that it captures the relevant long-term dynamics. In particular, it contains all nontrivial fixed points, periodic orbits, invariant circles and heteroclinic cycles (see, for example, [48,50,49,34]). Therefore, the common approach in the study of these systems is to focus on the dynamics on the carrying simplex.
In [74], Ruiz-Herrera provided an exclusion criterion for discrete-time competitive models of two or three species via the carrying simplex. Jiang and Niu derived a fixed point index formula on the carrying simplex for three-dimensional maps in [48], which states that the sum of the indices of all fixed points on the carrying simplex is one. Based on this formula, an alternative classification for 3-dimensional Atkinson-Allen models was provided in [48] and an alternative classification for 3dimensional Leslie-Gower models was provided in [49]. Such a classification has also been given for the 3-dimensional generalized Atkinson-Allen models [34] and Ricker models admitting a carrying simplex [33], respectively. Jiang, Niu and Wang [50] studied the occurrence and stability of heteroclinic cycles for competitive maps with a carrying simplex. Recently, Niu and Ruiz-Herrera proved in [69] that every orbit converges to a fixed point for three-dimensional maps with a carrying simplex when there is a unique positive fixed point such that its index is −1. For the geometrical properties of the carrying simplex and their impact on the dynamics, we refer the readers to [5,6,8,66,67].
In population biology, the question of persistence of interacting species is one of the most important. There has been many papers on permanence for discrete-time Kolmogorov systems; see [47,42,62,55,53]. Here we study the permanence and impermanence for the discrete-time Kolmogorov systems (1) admitting a carrying simplex. The main mathematical tools involve average Liapunov functions (see [47,50]) and the theory of the carrying simplex. A successful case of combining these two approaches is the stability criterion for heteroclinic cycles established by Jiang, Niu and Wang in [50]. Our project here is to provide the minimal conditions to ensure the permanence for such systems via the carrying simplex. Criteria for permanence and impermanence which are simple and easy to apply are provided for systems (1) admitting a carrying simplex. In particular, for three-dimensional systems, permanence and impermanence can be determined by boundary fixed points. As a special case, when the boundary of the carrying simplex is a heteroclinic cycle, the system is permanent if the heteroclinic cycle is repelling, while the system is impermanent if it is attracting.
Finally, we restrict attention to the class of maps given by where r i > 0, A is an n × n matrix with entries a ij > 0, f i : R + ×Ṙ + →Ṙ + are C 1 , and τ denotes transpose. In addition, in this article, we always assume that f i satisfies (i) f i (r, r) = 1, ∂f i (z, r) ∂z < 0, ∀(z, r) ∈ R + ×Ṙ + ; (ii) f i (z, r) + z ∂f i (z, r) ∂z > 0, ∀(z, r) ∈ R + ×Ṙ + . ( Note that f i enjoys the properties: f i (z, r) > 1 for z < r, f i (z, r) = 1 for z = r, and f i (z, r) < 1 for z > r. The discrete-time system induced by map (2) is often used in the modeling of n-species in competition; see [38,58,20,21,70,60,73,14,54,46]. The variable x i is the density of species i and f i is its growth function (or fitness function). Per capita growth rate for species i in the absence of competition is given by f i (0, r i ).
The parameter a ij is the competition coefficient which measures the effect of species j relative to i on the function f i .
Note that when we consider T restricted to the i-th coordinate axis, we have g i (x i ) = x i f i (a ii x i , r i ), which describes the dynamics of the i-th species without inter-specific competition. By (3) (i), f i (a ii x i , r i ) is a decreasing function of x i . The biological meaning is that the per capita growth rate of species i is a decreasing function of population density due to negative density dependent mechanism such as intra-specific competition [19,78]. By (3) (ii), g i (x i ) is an increasing function of x i , so the population dynamics of each species is monotone. Biologically, this means that the intra-specific competition is contest due to increasing utilization of available resources, where each successful competitor gets all resources it requires for survival or reproduction (see [81,37,10,79]). Furthermore, it follows from (3) that f i (0, r i ) > 1 and so 0 is a repeller (growth of small populations), and x i = ri aii is an attracting fixed point for g i .
We call the map (2) fixed points linearly determined. Jiang and Niu proved in [49] that all maps given by (2) with f i satisfying (3) admit a carrying simplex unconditionally. In this article, we focus on studying the parameter conditions that guarantee permanence and impermanence for the three-dimensional map (2) with given functions f i . By noticing the above linear structure, we can define an equivalence relation on the parameter space for the three-dimensional map (2) as that for the two specific cases: the Atkinson-Allen model [48] and Leslie-Gower model [49]. Two maps (2) are said to be equivalent relative to 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}. Map (2) is said to be stable relative to the boundary of Σ if all the fixed points on the boundary are hyperbolic. Via the index formula on the carrying simplex established in [48], we list the equivalence classes for all stable maps in Table 1. There are always a total of 33 stable equivalence classes which can be described in terms of inequalities on parameters, and the equivalence classification is independent of the choice of functions f i , which presents a clear picture of the essence of the dynamics for this class of maps. Moreover, based on this classification, one can easily get the parameter conditions that guarantee permanence and impermanence for such systems. Specifically, applying the permanence criteria to each class, we obtain that systems in classes 29, 31, 33 and those in class 27 with repelling heteroclinic cycles are permanent, while systems in classes 1 − 26, 28, 30, 32 and class 27 with attracting heteroclinic cycles are impermanent. For systems in class 33, the permanence can guarantee the global stability of the positive fixed point. It is emphasized that the fixed points linearly determined systems (2) contain many classical systems, such as the Atkinson-Allen model [48] and the Leslie-Gower model [49]; see Section 5 for more. Our investigations will stimulate further study on the global behavior of these systems including the higher order bifurcations, multiplicity of closed invariant curves, and so on.
As we mentioned earlier, many of the important ideas used in this work are due to Hutson and Moran [47], Hofbauer, Hutson and Jansen [42], Jiang and Niu [48,49] and Jiang, Niu and Wang [50].
The paper is organized as follows. In Section 2, we present some notions and recall some known results on average Liapunov functions. In Section 3, we provide the criteria on permanence and impermanence for dissipative Kolmogorov systems and maps admitting a carrying simplex. In Section 4, we define the equivalence relation relative to the boundary dynamics on the parameter space for the three-dimensional map (2), and derive the 33 stable equivalence classes. Based on this classification, we obtain the parameter conditions that guarantee permanence and impermanence. In Section 5, we apply our results to some classical discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models. The paper ends with a discussion in Section 6.

Notation and preliminaries.
Suppose that X is a metric space with metric d X (·, ·) and T : X → X is a continuous mapping. Let Z + = {0, 1, 2, . . .}. For any x ∈ X, we define the positive orbit through x as γ + (x) := {T k x : k ∈ Z + }, and denote the tail from the moment m ≥ 1 of γ + (x) by γ + m (x) := {T k x : k ≥ m}. A negative orbit through x is a sequence {x(−k) : k ∈ Z + } such that x(0) = x, T x(−k − 1) = x(−k) for all k ∈ Z + . The omega limit set ω(x) := k≥0 m≥k T m x of x is the set of limit points of the positive orbit γ + (x). The alpha limit set α(x) := k≥0 m≥k x(−m) associated to a negative orbit {x(−k) : k ∈ Z + } through x is the set of limit points of this negative orbit.
For a subset D ⊆ X, D and D c denote the closure of D in X and the complement of D respectively. Given any set D, let Note that Ω(D) is a subset of the omega limit set ω(D) of the orbit through D, i.e.
We denote E(T ) = {x ∈ X : T x = x} to be the set of the fixed points of T .
A set J is said to attract a set D under T if for any ε > 0, there exists a k 0 ≥ 1 such that T k D belongs to the ε-neighborhood O ε (J) of J for k ≥ k 0 , where O ε (J) = {y ∈ X : d X (y, J) < ε}. A nonempty compact invariant set J ⊆ X is said to be a global attractor of T if J attracts each bounded set D ⊆ X. T is said to be dissipative if it admits a global attractor.
We say that a nonempty, compact, positively invariant subset S ⊆ X repels for T if there exists an ε-neighborhood O ε (S) of S such that for all x ∈ X \ S there exists In the following part of this section, we recall two known results on average Liapunov functions. Lemma 2.1 (Lemma 2.1 in [42]). Let T : X → X be a continuous map, where X is a metric space. Let W be open with compact closure, and suppose that U is open and positively invariant, where W ⊆ U ⊆ X. If γ + (x) ∩ W = ∅ for any x ∈ U , then γ + (W ) is compact and positively invariant such that for any x ∈ U , there exists a m = m(x) > 0 satisfying γ + m (x) ⊆ γ + (W ). Lemma 2.2 (Repelling Set [47]). Suppose that M is a compact metric space and T : M → M is a continuous mapping. Assume that S is a compact subset of M with empty interior such that S and M \ S are positively invariant under T . Suppose that there is a continuous function V : Then S repels for T .
Then S attracts for T , that is, there is an The following proposition shows that the presence of an invariant repelling set S implies the existence of a compact set K contained in S c that absorbs the orbits contained in S c . See [77] for more details. Proof. Set U = M \ S. Since S repels, there exists an ε-neighborhood O ε (S) of S such that for any 3. Permanence criteria. From now on we reserve the symbol n for the dimension of the Euclidean space R n and the symbol N for the set {1, . . . , n}. We will denote by {e 1 , . . . , e n } the usual basis for R n , and by d(·, ·) the usual Euclidean distance. 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 coneṘ n + := {x ∈ R n + : x i > 0, ∀i ∈ N } and the boundary of R n + is ∂R n + := R n + \Ṙ n + . 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 The reverse relations are denoted by ≥, >, , respectively. Given an m × m 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). We shall use I to denote the identity matrix.
Consider the map T : R n + → R n + given by with continuous functions F i satisfying F i (x) > 0 for all x ∈ R n + . Note that this implies that T i (x) > 0 if and only if x i > 0, and hence TṘ n + ⊂Ṙ n + and T (∂R n + ) ⊂ ∂R n + . In particular, Definition 3.1 ( [42,47]). The map T is said to be permanent if there exists a compact positively invariant set K ⊆Ṙ n + such that for every x ∈Ṙ n + , there exists a m = m(x) > 0 such that the tail γ + m (x) ⊆ K.
Remark 3.1. Since K ⊆Ṙ n + is compact, the distance d(K, ∂R n + ) of K from the boundary ∂R n + is thus non-zero, and ω(x) ⊆ K for all x ∈Ṙ n + . Equivalently, permanence means that there exist δ, D > 0 such that for all x ∈Ṙ n + ( [43,53,77]). In a permanent system, species can coexist permanently in the sense that when the population densities of all species are positive, after some generations each population density will be bounded away from zero and infinity for all the time. As a consequence extinction and explosion cannot occur. In this paper, the map T is said to be impermanent if it is not permanent.
3.1. Permanence for dissipative systems. Our first criteria on permanence and impermanence for dissipative systems defined on R n + is the following theorem based on the technique of average Liapunov functions. A similar result was given by Garay and Hofbauer for discrete-time replicator dynamics in [22]. See also [77] for a detailed discussion on the technique of average Liapunov functions.
where ∂J = J ∩ ∂R n + , then T is permanent; if instead then T is impermanent.
Proof. We first show that if (10) holds then T is permanent. Let Note that V (x) = 0 if and only if x ∈ ∂R n + . For x ∈Ṙ n + , we let θ(k, Since Note that for any x ∈Ṙ n + and k ≥ 2, So (14) provides a continuous extension of θ(k, ·) to R n + , ∀k ≥ 2. Consider the maps T | J : J → J and V | J : J → R + given by (12), where T | J denotes the restriction of T on J, and similarly for V | J . Note that V | J (x) = 0 if and only if x ∈ ∂J. For x ∈ ∂J, one has Let ψ J (x) := sup k>0 ϑ J (k, x) for x ∈ ∂J. By condition (10), exp{g(x)} > 1 for all x ∈ Ω(∂J), that is Therefore, ψ J (x) > 1 for all x ∈ Ω(∂J), and obviously ψ J (x) ≥ ϑ J (1, x) > 0 for all x ∈ ∂J. In Lemma 2.2 take M = J, S = ∂J and T = T | J . Then by Remark 2.1, we have ψ J (x) > 1 for all x ∈ ∂J. Let W = O ε (J) be an ε-neighborhood of J in R n + . Since J is the global attractor, for all x ∈ R n + , γ + (x) ∩ W = ∅. It then follows from Lemma 2.1 that γ + (W ) is compact and positively invariant such that for any x ∈ R n + , there exists a k = k(x) > 0 satisfying γ + k (x) ⊆ γ + (W ).
Let ψ M (x) := sup k>0 ϑ M (k, x) for x ∈ S. Note that for all x ∈ ∂J, It then follows from Remark 2.1 that ψ M (x) > 1 for all x ∈ S, and hence S repels for T | M by Lemma 2.2. Therefore, there exists a compact set K ⊆ M \S of the subspace M which is positively invariant under T | M , such that for every x ∈ M \ S, there exists a m = m(x) > 0 such that γ + m (x) ⊆ K by Proposition 2.4. Of course, K is also a compact subset of R n + and positively invariant under T . Note that K ⊆ M \ S ⊆Ṙ n + . Recall that for any x ∈Ṙ n + , there exists a k = k(x) > 0 such that T k (x) ∈ M , and hence T k (x) ∈ M \ S because TṘ n + ⊆Ṙ n + . Therefore, there exists a m = m(x) ≥ k such that the tail γ + m (x) ⊆ K, that is T is permanent. Now suppose that (11) holds. Consider the maps T | J : J → J and V | J : J → R + given by (12). In Lemma 2.3 take M = J, S = ∂J and T = T J . Since θ(1, ·) : It follows from (11) that exp{g(x)} < 1 and hence θ(1, x) < 1, for all x ∈ Ω(∂J). Then for all x ∈ Ω(∂J), one has ϕ J (x) ≤ ζ J (1, x) = θ(1, x) < 1, where ϕ J (x) and ζ J (1, x) are defined in Lemma 2.3. Thus, ∂J attracts for T J . Therefore, there exists some x ∈Ṙ n + such that ω(x) ⊆ ∂J ⊆ ∂R n + , and hence T is impermanent.

3.2.
Permanence via carrying simplex. Before presenting the permanence and impermanence criteria for the map T given by (8) admitting a carrying simplex Σ, we first recall the properties of carrying simplex.
A carrying simplex for the map T is a subset Σ of R n + \ {0} with the following properties: (P1) Σ is compact and invariant under T ; (P2) for any x ∈ R n + \ {0}, there exists some z ∈ Σ such that lim , and homeomorphic to the probability simplex ∆ n−1 via radial projection; (P4) T : Σ → Σ is a homeomorphism.
(P1) and (P2) imply that the long-term dynamics of T is accurately reflected by that in Σ, and (P3) means that Σ is topologically simple. We denote the boundary of Σ, i.e. Σ ∩ ∂R n + by ∂Σ, and the interior of Σ, i.e. Σ \ ∂Σ byΣ. We denote by H + i the i-th positive coordinate axis and by π i = {x ∈ R n + : x i = 0} the i-th coordinate plane. Note that each π i is positively invariant under T and ∂Σ ∩ π i is the carrying simplex of T | πi , that is ∂Σ is composed of the carrying simplices of T | πi , i = 1, . . . , n. Σ contains all non-trivial fixed points, periodic orbits and heteroclinic cycles, etc. Every vertex of Σ is a fixed point of T , where Σ and some positive coordinate axis meet, and denote by q {i} = q i e i the fixed point at the vertex where Σ and H + i meet. For one-dimensional case, T admits a carrying simplex if and only if it has a globally attracting positive fixed point in We first recall a readily checked criterion provided by Jiang and Niu [49] on the existence of a carrying simplex for the competitive map T of type (8). Theorem 3.4. Assume that T admits a carrying simplex Σ. If there are real numbers ν 1 , . . . , ν n > 0 such that then T is impermanent.
Proof. We first show that if (18) holds then T is permanent. Since Σ is invariant and compact such that By the property (P2) of Σ, one has ω(x) ⊆ Ω(∂Σ) for any x ∈ S, and hence Ω(S) ⊆ Ω(∂Σ). So, if (18) holds, then g(x) > 0 for all x ∈ Ω(S). Now consider the map T | M : Note that V (x) = 0 if and only if x ∈ S. By the above analysis, we know exp{g(x)} > 1 for all x ∈ Ω(S), that is, F ν1 1 (x) · · · F νn n (x) > 1 for all x ∈ Ω(S). On the other hand, for x ∈ S, one has If (19) holds, then ∂Σ attracts for T | Σ by [50,Theorem 2], and hence there exists some x ∈Ṙ n + such that ω(x) ⊆ ∂Σ ⊆ ∂R n + , which implies that T is impermanent. For the two-dimensinal (i.e. n = 2) map T given by (8) with a carrying simplex Σ, we know that T | Σ is topologically conjugate to a strictly increasing homeomorphism h taking [0, 1] onto [0, 1] (similarly for (T | Σ ) −1 ) because Σ is homeomorphic to [0, 1] and T | Σ is a homeomorphism taking Σ onto Σ, and hence every nontrivial orbit of T converges to some fixed point on Σ (see also [75,76]). In particular, it follows from Theorem 3.4 that T is permanent if F i (q {j} ) > 1 for i = j, i, j = 1, 2, where q {j} is the fixed point on H + j . By the above arguments, we know that for the two-dimensional map T which admits a carrying simplex, the dynamics is relatively simple, that is ω(x) ⊆ E(T ) for all x ∈ R 2 + , and hence Ω(Σ) ⊆ E(T ). For the three-dimensional case, since ∂Σ ∩ π i is the carrying simplex of T | πi , which is a two-dimensional map, the boundary dynamics for T is simple, i.e. Ω(∂Σ) ⊂ E(T ).
Proof. The conclusion follows from the above analysis and Theorem 3.4 immediately.
4. Extensions to competitive systems. In this section, we study the Kolmogorov map T given by (2).
For the convenience of the study, we set a ij = r i µ ij . Let R = diag[r i ], the n × n diagonal matrix with diagonal entries r i , i = 1, . . . , n, and U be the n × n matrix with entries µ ij > 0. Then A = RU , and (2) is written as µ ij x j , r i ), i = 1, · · · , n. (22) PERMANENCE AND CLASSIFICATION FOR DISCRETE-TIME SYSTEMS 13 In this form, the fixed points of T are determined by the linear equations which depend only on the parameters µ ij .
Denote by F the collection of all C 1 functions f : the set of all maps on R n + of the from (22) with the given function f ∈ F n , where T (R n + ) denotes the set of all maps taking R n + into itself. f ∈ F n is called a generating function for the map (22).
Let T ∈ DCS(n, f ). The entries of the Jacobian matrix DT (x) at x are given by where δ ij = 1 for i = j and δ ij = 0 for i = j, i, j = 1, . . . , n. DT (x) "splits" into two blocks: the square matrix DT (x) κ(x) defines the "internal" block which corresponds to the Jacobian matrix of the restriction of T to the subspace R is the submatrix of DT (x) with rows and columns from κ(x); the square matrix DT (x) N \κ(x) is the "external" block which is a diagonal matrix with diagonal entries F j (x) > 0, where j / ∈ κ(x) and DT (x) N \κ(x) is the submatrix of DT (x) with rows and columns from N \ κ(x). Letx = (x 1 , . . . ,x n ) be a fixed point of T . Then F i (x) = 1 for any i ∈ κ(x), and The external block DT (x) N \κ(x) of DT (x) is a diagonal matrix whose entries are the external eigenvalues F j (x) > 0 (we call it the external eigenvalue in direction . . , n, so the trivial fixed point 0 is a hyperbolic repeller. Lemma 4.1 (Gerschgorin Circle Theorem [65]). Let B be an n × n matrix with entries b ij . Define the i-th Gerschgorin disc D i in the complex plane to be the closed disc centered at b ii with radii j =i |b ij |. Each D i contains an eigenvalue of B and, moreover, for any distinct i 1 , · · · , i m , there are at least m eigenvalues  Proof. We first show that T is a local diffeomorphism. According to the inverse function theorem, it suffices to prove that det DT (x) > 0 for all x ∈ R n + . Recall that DT (x) splits into two blocks: the internal block DT (x) κ(x) and the external block DT (x) N \κ(x) which is a diagonal matrix with positive diagonal entries, so we only need to show that det DT (x) κ(x) > 0. Therefore, without loss of generality, we assume that x ∈Ṙ n + , and show det DT (x) > 0. By (23), DT (x) can be written as Note that the (i, j)-th entry of W diag[x k ] is given by Then by (3) (ii) one has It follows from (25) and (26) ∂xj lies in the right half-plane. Then by Lemma 4.1 all the eigenvalues of B have positive real parts, and hence det DT (x) = det B > 0. At this moment we have proved that T is a local diffeomorphism. Now we show that T is one-to-one. By a contradiction argument assume that there exist x = y such that T x = T y. Then one can choose some l > 0 such that Since, as noticed above, The competitiveness of T will now follow once we have proved (DT (x) κ(x) ) −1 > 0 for all x ∈ R n + \ {0} by Proposition 4.1 in [74]. Recall that ∂Fi(x) ∂xj < 0 for all x ∈ R n + and i, j ∈ N , so the (i, j)-th entry of DT (x) κ(x) is negative for i = j. Then it follows from the proof of Theorem 3.1 in [49] that (26) This completes the proof. By (26), we know that each map T ∈ DCS(n, f ) satisfies the condition Υ3) in Lemma 3.3. Since each map T ∈ DCS(n, f ) also satisfies the conditions Υ1) and Υ2), it has a carrying simplex by Lemma 3.3, which had been proved in [49,Corollary 3.3].   (22) is one-to-one and competitive on R n + , and it has a carrying simplex unconditionally if each f i satisfies (3) (i) and (3) (ii). However, if (3) (ii) does not hold for some f i , then T may not be one-to-one or competitive on R n + . For example, the (Ricker) map T with f i (z, r) = exp(r − z), i = 1, . . . , n, is not one-to-one or competitive on R n + and in particular, it has a carrying simplex only under certain additional conditions; see [33] for details.
has a unique positive solution, then 1 is not an eigenvalue of Otherwise, 0 is an eigenvalue of the matrix DT (p) − I, and hence det U = 0. Then (27) has either no solution, or infinitely many solutions, a contradiction. Therefore, the index of p which is given by (−1) m is either 1 or −1, where m is the sum of the multiplicities of all the eigenvalues of DT (p) which are greater than one (see [2,30]). Let Then DT (p) = I − A, and A is similar to B. By the property (3) (i) of f i , we know that A, B are positive matrices. Note that (U p τ ) i = 1, so the sum of the i-th row of B is −r i ∂fi ∂z (r i , r i ) < 1 (see (4)). It then follows from Perron-Frobenius theorem that ρ(B), the spectral radius of B, is an eigenvalue of B satisfying 0 < ρ(B) < 1 and the magnitudes of the other eigenvalues of B are all less than 1. Set λ * := 1 − ρ(B). Since A and B 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 possess real parts greater than 0 and less than 2. In particular, for the two-dimensional case, i.e. n = 2, both of the two eigenvalues of DT (p) are positive real numbers with one less than 1, and p is hyperbolic.
In the remainder of this article, we will focus on analyzing the map T ∈ DCS(3, f ) modeling three mutually competing species. We define an equivalence relation relative to local stability of fixed points on the boundary of Σ for the set DCS(3, f ) as that for all the three dimensional Leslie-Gower maps [49] T : We show that the classification via this equivalence relation for three dimensional Leslie-Gower maps is valid for any DCS(3, f ), and independent of the choice of generating function f ∈ F 3 . Furthermore, according to the equivalence classification, one can easily derive the permanence conditions in terms of simple inequalities on the parameters for T ∈ DCS(3, f ).

Classification via boundary dynamics.
In this subsection, we study the map T ∈ DCS(3, f ): It follows from Lemma 4.4 that T admits a 2-dimensional carrying simplex Σ homeomorphic to ∆ 2 . Each coordinate plane π i is positively invariant under T , and the restriction of T to π i is a 2-dimensional map T | πi ∈ DCS(2, so ∂Σ is composed of the one-dimensional carrying simplices of T | πi . Therefore, before studying the three-dimensional map T ∈ DCS(3, f ), we first study the two-dimensional case.
By Lemma 4.4, T admits a one-dimensional carrying simplex Σ which is homeomorphic to the line segment joining the two points (0, 1) and (1, 0). By Lemma 4.4 and the arguments in Section 3.2, we conclude the following proposition. Set R 2 + \ S i = U i ∪ B i , where U i and B i are the unbounded and bounded disjoint components of R 2 + \ S i , respectively. Let γ ij := µ ii − µ ji for i, j = 1, 2 and i = j. Then q {i} ∈ U j (resp. B j ) if and only if γ ij < 0 (resp. > 0). Lemma 4.6. If γ ij > 0 (resp. < 0), then q {i} is a saddle (resp. an asymptotically stable node), and hence repels (resp. attracts) along Σ. Moreover, q {i} is hyperbolic if and only if γ ij = 0.
The following definition of equivalence appears to be unnecessarily pompous, but it prepares the way for the analogous definition in higher dimensions. for j = i, that is (see (32)) sgn(γ ij ) = sgn(γ σ(i)σ(j) ) for j = i. Definition 4.9. A map T ∈ DCS(2, f ) is said to be stable relative to ∂Σ if all the fixed points on ∂Σ are hyperbolic. An equivalence class is said to be stable if each map in it is stable relative to ∂Σ.
By the positivity of p, γ 12 and γ 21 have the same sign as det U . Therefore, it follows from Proposition 4.7 (c) and (d) that p attracts (resp. repels) on Σ if and only if det U > 0 (resp. det U < 0). Moreover, it follows from Remark 4.2 that if p attracts on Σ then its two positive eigenvalues are less than 1 while it has one eigenvalue greater than 1 if it repels on Σ.
By Proposition 4.7 we conclude the following result immediately.  Proof. Note that γ ij > 0 (resp. < 0) if and only if f j ( µjirj µii , r j ) > 1 (resp. < 1), i.e., F j (q {i} ) > 1 (resp. < 1). So, it follows from the arguments in Section 3.2 that T is permanent if γ 12 , γ 21 > 0, and in this case there is a globally attracting positive fixed point. If some γ ij < 0, then q {i} is an attractor, so T is impermanent in classes 1 and 3.
In this case, v {k} is just the positive fixed point of the map T | π k . T may also admit a positive fixed point p inṘ 3 + which satisfies Hereafter, denote by . Therefore, the sign of r j − (RU q τ {i} ) j is just the sign of F j (q {i} ) − 1, and that the sign of r k − (RU v τ {k} ) k is just the sign of F k (v {k} ) − 1 (see the comments below (3)). Specifically, By the positive invariance of π i and the analysis of the 2-dimensional case, the statements, proofs and classification program in [49] carry over to DCS(3, f ) in a straightforward way, so we do not re-do it unless the need for special details and we only state the corresponding conclusions.
The biological meaning of the condition γ ij > 0 (resp. < 0) in Propositions 4.12 and 4.13 is that species j can (resp. not) invade species i in the absence of species k; here i, j, k are distinct.
Therefore, for a map T ∈ DCS(3, f ) which is stable relative to ∂Σ, if v {k} exists then µ ii µ jj − µ ij µ ji = 0 (here i, j, k are distinct), and it is easy to check that where Thus a map T ∈ DCS(3, f ) is stable relative to ∂Σ if and only if γ ij = 0 and µ ki β ij + µ kj β ji = 1, i.e., (U v τ {k} ) k = 1 (if v {k} exists). Suppose that T is stable relative to ∂Σ. It follows from Propositions 4.12-4.14 and (36) that the existence and local dynamics of boundary fixed points on ∂Σ for T are completely determined by the parameters µ ij , i.e. the values γ ij and µ ki β ij +µ kj β ji , which are independent of the generating function f . Moreover, if T admits a positive fixed point p which satisfies (34), 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 (34) for any s ≥ 0. Let s := 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. Thus, 1 is not an eigenvalue of DT (p) by Remark 4.2. Therefore, T has only finitely many fixed points on Σ, i.e. three axial fixed points q {i} , at most three planar fixed points v {i} and at most one positive fixed point p, and 1 is not an eigenvalue of any of their Jacobian matrices.
Let Q = id − T , where id is the identity mapping. Let x be a fixed point of T , that is, a zero of Q. The index of T at x is denoted by Ind(x, T ) and the index of Q at the zero x is denoted by I (x, Q). The index I (x, Q) is defined as the sign of det DQ(x) if det DQ(x) = 0, and the index Ind(x, T ) as I (x, Q); for the general theory see [30].   Proof. Let G(x) = T (x) − x. Consider the continuous-time systeṁ The origin 0 is an equilibrium of system (38), and the eigenvalues of DG(0) are F i (0) − 1 > 0, that is, 0 is a repeller. Note that µ ij x j , r i )r i µ ij < 0, so system (38) is totally competitive. Since for |x| sufficiently large, so system (38) is dissipative. Recall that T ∈ DCS(3, f ) admits a carrying simplex, so the result follows from Remark 4.6 and Lemma 4.17.     Table 1.
Recalling Remark 4.6, the existence and local dynamics of boundary fixed points on ∂Σ for T ∈ DCS(3, f ) are completely determined by the parameters µ ij , i.e. the values γ ij and µ ki β ij + µ kj β ji , which are independent of f , and the same as the Leslie-Gower map (29). Therefore, the classifications are the same for them, which are independent of the choice of the generating function f ∈ F 3 . Any stable map in DCS(3, f ) belongs to one of the 33 classes in Table 1 (modulo permutation of the indices). Moreover, there is no positive fixed point in classes 1 − 18, which have trivial dynamics, i.e. every orbit converges to some fixed point. Each map from classes 19 − 25 admits a unique positive fixed point with index −1, and every orbit also converges to some fixed point for these classes. Each map in classes 26 − 33 has a unique positive fixed point with index 1; and the positive fixed point is globally asymptotically stable in class 33; see Subsection 4.2 for details. Such a classification is also valid for the Ricker models admitting a carrying simplex [33], and we will discuss in Section 5.

Stability and permanence.
As befits the context, we shall consider the families of maps given in Table 1 by permutation of the indices, i.e., we assume the parameters µ ij , r i of the corresponding class satisfy the conditions listed in Table  1. [74] and Theorem 3.1 in [69]). Consider the threedimensional map T given by (8) which satisfies the conditions Υ1), Υ2) and Υ3) in Lemma 3.3. Suppose that T has only a finite number of fixed points. Then the following conclusions hold:

Lemma 4.21 (Theorem 2.2 in
• If T has no positive fixed point, then every nontrivial orbit converges to some fixed point on the boundary of the carrying simplex. • If T has a unique positive fixed point p such that Ind(p, T ) = −1, then p is a saddle on the carrying simplex, and moreover, every nontrivial orbit converges to some fixed point on the boundary of the carrying simplex, except those on the stable manifold of p.  [68] that both the stable manifold and unstable manifold of the saddle p are simple curves, and the phase portrait on the carrying simplex can be described clearly together with the dynamics on the boundary of the carrying simplex.
Recall that each map T ∈ DCS(3, f ) satisfies the conditions Υ1), Υ2) and Υ3) in Lemma 3.3. Moreover, if T is stable relative to ∂Σ, then there is at most one positive fixed point, say p, and Ind(p, T ) = 0 if p exists. Thus, together with Lemma 4.21, Proposition 4.18 and Remark 4.6 imply the following trivial dynamics via boundary fixed points.
Then T has trivial dynamics, i.e. every nontrivial orbit converges to some fixed point on Σ.
Note that, each map T in classes 1 − 18 satisfies (39) in Proposition 4.22, and hence T has no positive fixed point. Therefore, such T has trivial dynamics. That is, we have the following proposition. In the biological sense, Proposition 4.23 means that for three competing species modeled by T , if there is no coexistence state, then some of the species will be extinct.
For each map T in classes 19 − 33, there exits a unique positive fixed point p. Proof. For classes 19 − 25 (resp. classes 26 − 33), it follows from the local dynamics of fixed points on ∂Σ in Table 1, Lemma 4.19 and formula (37) that Ind(p, T ) = −1 (resp. Ind(p, T ) = 1). Moreover, if Ind(p, T ) = −1, then all the three eigenvalues of DT (p) are positive real numbers with one eigenvalue greater than 1 and the other two less than 1 by Remark 4.2. So, two eigenvalues of A are greater than 0 and one is less than 0, which implies that det A < 0, and hence det U < 0. While Ind(p, T ) = 1 ensures that there are zero or two eigenvalues of DT (p) greater than 1 by Remark 4.2. For the former case, also by Remark 4.2 we have one eigenvalue of A is greater than 0 and the other two are either complex numbers or greater than 0. For the latter case, two eigenvalues of A are less than 0 and one is greater than 0. Therefore, one always has det A > 0, and hence det U > 0. Proof. For each map T in class 32, there exists a planar fixed point v {k} in the interior of π k for each k = 1, 2, 3, which is repelling along ∂Σ ∩ π k (see Table 1 (32)), so v {k} is a saddle for T | π k . It then follows from Remark 4.4 that det U {i,j} < 0 Therefore, det A {i,j} < 0 for any i < j. By det U > 0, one also has det A > 0. It follows from Proposition 3.8 in [87] and det A > 0 that A has two eigenvalues with negative real parts. Therefore, DT (p) = I − A has two eigenvalues with real parts greater than 1, i.e, DT (p) has two eigenvalues with magnitudes greater than 1 except λ * , where 0 < λ * < 1 is defined in Remark 4.2. So p is a hyperbolic fixed point and it follows from [68,Theorem 4.6] that the local dynamics of p on Σ is reflected by the other two eigenvalues except λ * , which implies that p is a repeller on Σ and its two-dimensional unstable manifold is contained in Σ (see [68,Corollary 4.5]).
The following lemma is the 3D specialization of Theorem 2.4 in [9] (see also Theorem 1.2 in [32]), which can be used to establish our global stability for class 33.
Lemma 4.27 ([9]). Consider the three-dimensional map T : R 3 + → R 3 + given by (1), where F i are C 1 satisfying F i (x) > 0 for all x ∈ R 3 + , i = 1, 2, 3. Assume that (a) det DT (x) > 0 for all x ∈ R 3 + ; (b) DT (x) −1 > 0 for all x ∈Ṙ 3 + ; (c) for each i = 1, 2, 3, T | πi has a unique interior fixed point v {i} that is globally asymptotically stable in the interior of π i , but a saddle for T ; (d) T admits a carrying simplex; (e) T has a unique positive fixed point p ∈Ṙ 3 + . Then p is globally asymptotically stable inṘ 3 + for T . Figure 3. The phase portrait on Σ for class 33. Every orbit in the interior of Σ converges to p. The fixed point notation is as in Table 1.

PERMANENCE AND CLASSIFICATION FOR DISCRETE-TIME SYSTEMS 25
Theorem 4.28. The positive fixed point p is globally asymptotically stable inṘ 3 + for each map T in class 33, and the phase portrait on Σ is as shown in Fig. 3.
For any map T in class 27, each axial fixed point q {i} is a saddle on Σ, and ∂Σ∩π i is the heteroclinic connection between q {j} and q {k} , where i, j, k are distinct. Therefore, ∂Σ is a heteroclinic cycle of May-Leonard type: q {1} → q {2} → q {3} → q {1} (or the arrows reversed), i.e., any map T in class 27 admits a heteroclinic cycle (see Table 1 (27)).
Proof. The conclusion follows from Lemma 3.6 immediately.
From a biological point of view, these cycles in class 27 may be seen to correspond to the biological environment where in purely pairwise competition species 2 can invade species 1 but not vice versa, species 3 can invade species 2 but not vice versa, and species 1 can invade species 3 but not vice versa. It is this intransitivity in the pairwise competition, which underlies the cycle behavior. G ij > 0 (resp. < 0) means that species j can (resp. not) invade species i; see Remark 4.5.  . The phase portrait on Σ for class 29. The fixed point notation is as in Table 1.
(ii) For each map T in classes 1 − 26, 28, 30 and 32, there always exists a fixed point on ∂Σ which is an attractor on Σ (see Table 1), so it is impermanent. For the map T in class 27 such that < 0, the conclusion follows from Corollary 3.7.

5.
Applications to population models. In this section we apply the previous results in some concrete population models. Throughout this section, A denotes the 3 × 3 matrix with entries a ij > 0, R = diag[r i ] with r i > 0, and U is the 3 × 3 matrix with entries µ ij > 0 such that A = RU , where i, j = 1, 2, 3.
Jiang and Niu [49] have listed the 33 stable equivalence classes in CLG (3); see also Table 1 In [49], it was shown that for CLG(3), Neimark-Sacker bifurcations can occur within each of classes 26 − 31, so these classes can admit invariant closed curves. Here, we provide an example to show that the supercritical Neimark-Sacker bifurcation can occur in class 27 with > 0 for CLG (3). We also provide a numerical example to show that the Chenciner (generalized Neimark-Sacker) bifurcation can occur in class 27 with > 0, which implies that two isolated invariant closed curves can coexist on the carrying simplex in class 27 with a repelling heteroclinic cycle. The Chenciner bifurcation is a two-parameter bifurcation phenomenon of a fixed point, which occurs when there is a pair of complex eigenvalues with modulus one and the first Lyapunov coefficient vanishes; see [56,28] for more details. √ 3i)/2, where i stands for the imaginary unit. By calculating we obtain the first Lyapunov coefficient l 1 ≈ −1.162 × 10 −2 < 0. Since the Lyapunov coefficient is a rather lengthy expression, the approximate value was computed as a rational by using MATLAB [28,57,49]. Therefore, a supercritical Neimark-Sacker ). By numerical calculation [28,29], we find that T [r1,r2] admits a Chenciner bifurcation point at p when r 1 ≈ 0.248332 and r 2 ≈ 0.0633101, where the second Lyapunov coefficient l 2 ≈ −3.574 × 10 −2 < 0. Therefore, a stable fixed point and an attracting (large) invariant closed curve, separated by an unstable invariant closed curve can coexist in class 27 for CLG(3) when the parameters r 1 and r 2 are properly disturbed near 0.248332 and 0.0633101 respectively; see [56,Section 9.4] or [34, pp. 633-636] for details. Furthermore, it follows from (41) that ≈ 0.00011 > 0 for r 1 = 0.248332 and r 2 = 0.0633101, i.e. Chenciner bifurcation can also occur in class 27 with > 0.

Atkinson-Allen model. Consider the generalized Atkinson-Allen model
The model induced by the map (44) is a discretized system of the competitive Lotka-Volterra equations, and see [34] for a mechanistic derivation of this model. A related two-dimensional discrete-time model for competition between populations of cyst-nematodes, due to Jones and Perry [52], was analyzed by Smith [76]. When r i = 1 and c i = c, the map (44) reduces to the standard Atkinson-Allen map PERMANENCE AND CLASSIFICATION FOR DISCRETE-TIME SYSTEMS 29 which is a modified model derived from annual plants competition [3,1,73], and has been analyzed by Jiang and Niu in [48].

5.3.
Mixing growth functions. Consider the following model T = (T 1 , T 2 , T 3 ) on R 3 + , in which the three competing species are assumed to have different types of growth functions: Set . Then the map T can be written as Note that each f i ∈ F , so T admits a carrying simplex Σ. Denote the set of all maps (46) by a ij x j ), r i , a ij > 0, i, j = 1, 2, 3.
The one-dimensional map has been studied in detail by May and Oster [64], where they showed that every orbit converges to the positive fixed point for r ≤ 2, and it will exhibit a scenario of chaotic behavior for large r. The two-dimensional map was analyzed in detail by Smith [76], who showed that it has trivial dynamics provided ν 1 , ν 2 < 1. Roeger [72] studied the local dynamics of the positive fixed point and Neimark-Sacker bifurcations for the map (47) with r 1 = r 2 = r 3 . Hofbauer et al. [42] provided the criteria on permanence for map (47) and also the higher dimensional cases. Set f i (z, r) = exp(r − z), i = 1, 2, 3. Then the Ricker map (47) can be written as Note that (3) (ii) does not hold for f i , that is f i / ∈ F , and unlike the maps in DCS(3, f ) (such as the Leslie-Gower map or the Atkinson-Allen map discussed above), the Ricker map T has a carrying simplex only under certain additional conditions (see [33]). Assume that the parameters satisfy Then one can easily check that the Ricker map (47)  µ ij x j )), r i , µ ij > 0, (49) holds}.
Then each Ricker map (48) in CRC(3) admits a carrying simplex. The classification program via the dynamics on ∂Σ and statements for the 3-dimensional maps (2) are also applicable for CRC (3). Specifically, Gyllenberg et al. showed in [33] that there are a total of 33 stable equivalence classes in CRC(3) as shown in Table 1, where the parameters should satisfy the condition (49) in addition to those listed in Table  1 for each class. On the other hand, note that all the criteria on the permanence in Section 3.2 do not depend on the condition (3) (ii) and is applicable to any map T given by (22) which has a carrying simplex. Moreover, by the proof of Proposition 4.30, one can see that the existence of the carrying simplex and conditions (i) and (ii) in Table 1 (29) imply the permanence of the class 29, and similarly for classes 31 and 33, etc. Therefore, for the Ricker map (48), Proposition 4.30 is written in the following manner: It was shown in [33] that for CRC(3), classes 26 and 31 can admit supercritical Neimark-Sacker bifurcations, while classes 27 − 30 can admit subcritical Neimark-Sacker bifurcations. The authors also provided a numerical example to show that the Chenciner bifurcation can occur in class 29. Here, we give an example to show that the supercritical Neimark-Sacker bifurcation can also occur in class 29, and a numerical example to show that the Chenciner bifurcation can also occur in class 26.
Example 5.6. Let r 1 = 1 11 , 0 < r 2 < 1 2 , r 3 = 2 7 . Consider the one-parameter family of maps T [r2] given by (48) with the parameter matrix U given in Example 5.4 Particularly, we define an equivalence relation relative to local dynamics of nontrivial boundary fixed points for the set DCS(3, f ) according to this linear structure. We say that two mappings in DCS(3, f ) are equivalent if all their boundary fixed points have the same local dynamics on the carrying simplices after a permutation of the indices {1, 2, 3}. Via the index formula (37), which states that the sum of the indices of the fixed points on the carrying simplex is one, we list the stable equivalence classes for DCS(3, f ) which are independent of generating functions f ∈ F 3 , and present the phase portraits on Σ. Specifically, • there are always a total of 33 stable equivalence classes, no matter what generating functions are, which are described in terms of inequalities on the parameters, and given in Table 1; • every nontrivial orbit converges to a fixed point on the boundary of Σ in classes 1 − 18; • each map in classes 19 − 25 admits a unique positive fixed point p which is a saddle, such that every nontrivial orbit converges to some fixed point on the boundary of the carrying simplex, except those on the stable manifold of p which is a union of simple curves (see Remark 4.8); • each map in classes 26 − 33 has a unique positive fixed point p with index 1; p is always a hyperbolic repeller in class 32; and p is globally asymptotically stable in class 33; within classes 26 − 31, Neimark-Sacker bifurcations might occur; • there is a heteroclinic cycle in class 27. Applying our permanence and impermanence criteria to each class in DCS(3, f ), we obtain that the systems in classes 29, 31, 33 and class 27 with a repelling heteroclinic cycle are permanent, while those in classes 1 − 26, 28, 30, 32 and class 27 with an attracting heteroclinic cycle are impermanent; for systems in class 33, the permanence can guarantee the global stability of the unique positive fixed point.
However, permanence does not always imply the global asymptotic stability of the unique positive fixed point p, and the local stability of p depends on the generating function f ∈ F 3 by (28). Indeed, Neimark-Sacker bifurcations can happen in permanent classes 29 and 31 for the Leslie-Gower model, the generalized Atkinson-Allen model, the model with different types of growth functions, and the Ricker model. Neimark-Sacker bifurcations can also occur in class 27 with repelling heteroclinic cycles for the Leslie-Gower model and the generalized Atkinson-Allen model. So invariant cycles can occur in these classes, on which all orbits are periodic, or any orbit is dense. Numerical experiments show that Chenciner bifurcations can also happen in class 29 for the generalized Atkinson-Allen model and the Ricker model, and in class 27 with repelling heteroclinic cycles for the Leslie-Gower model, which means that two isolated invariant cycles can coexist on the carrying simplex for such systems. In the impermanent classes, such as classes 26, 28, 30 and class 27 with attracting heteroclinic cycles, Neimark-Sacker bifurcations can also occur; see Section 5 and [48,49,34,33] for more details. By the way, the dynamics in the same class which has a unique positive fixed point might be different for different kinds of generating functions f ∈ F 3 . For example, Neimark-Sacker bifurcations do not happen in classes 28 and 30 for the standard Atkinson-Allen model [48], while they can happen in these two classes for the Leslie-Gower model [49].
Furthermore, the results imply that when all the boundary fixed points are unstable, the system may not be permanent, because impermanence can occur in class 27 with attracting heteroclinic cycles, whose boundary fixed points are all unstable.
When the system admits no heteroclinic cycle, i.e. it is not in class 27, all the boundary fixed points being unstable implies the permanence for T ∈ DCS(3, f ). Biologically, the system is impermanent if one of the following conditions holds: • there exists some species which cannot be invaded by any of the other two species (classes 1 − 3, 7, 8, 13 − 23, 26, 28, 30 and 32); • there exists a two-species steady state which cannot be invaded by the third species (classes 4 − 6, 9 − 12, 24 and 25). The system is permanent if the following conditions hold simultaneously (classes 29, 31 and 33): • each species can be invaded by at least one of the other two species; • there exists one species which can be invaded by both of the other two species; • any coexistence of two species can be invaded by the third species. Such classification also presents a detailed classification for permanence and impermanence. Based on this, one can investigate the further long term dynamical properties within each of classes 26 − 32. Finally, we propose some interesting open problems as follows.
• Give sufficient conditions to guarantee the global asymptotic stability of the positive fixed point for permanent systems in classes 29, 31 and class 27 with repelling heteroclinic cycles.
• Investigate the nontrivial interesting dynamics, such as multiplicity of invariant cycles, in both permanent and impermanent systems.