On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems
aa r X i v : . [ m a t h . D S ] F e b ON EXISTENCE AND UNIQUENESS OF A MODIFIED CARRYINGSIMPLEX FOR DISCRETE KOLMOGOROV SYSTEMS
Zhanyuan Hou
School of Computing and Digital Media, London Metropolitan University,166-220 Holloway Road, London N7 8DB, UK
Abstract.
For a C map T from C = [0 , + ∞ ) N to C of the form T i ( x ) = x i f i ( x ), thedynamical system x ( n ) = T n ( x ) as a population model is competitive if ∂f i ∂x j ≤ i = j ). Awell know theorem for competitive systems, presented by Hirsch (J. Bio. Dyn. 2 (2008)169–179) and proved by Ruiz-Herrera (J. Differ. Equ. Appl. 19 (2013) 96–113) withvarious versions by others, states that, under certain conditions, the system has a compactinvariant surface Σ ⊂ C that is homeomorphic to ∆ N − = { x ∈ C : x + · · · + x N = 1 } ,attracting all the points of C \ { } , and called carrying simplex. The theorem has beenwell accepted with a large number of citations. In this paper, we point out that one of itsconditions requiring all the N entries of the Jacobian matrix Df = ( ∂f i ∂x j ) to be negativeis unnecessarily strong and too restrictive. We prove the existence and uniqueness of amodified carrying simplex by reducing that condition to requiring every entry of Df tobe nonpositive and each f i is strictly decreasing in x i . As an example of applications ofthe main result, sufficient conditions are provided for vanishing species and dominance ofone species over others. Note.
This paper has been accepted for publication in Journal of Difference Equationsand Applications. 1.
Introduction
In this paper, we are concerned with the global asymptotic behaviour of the discrete dy-namical system(1) x ( n ) = T n ( x ) , x ∈ C, n ∈ N , where C = R N + = [0 , + ∞ ) N , N = { , , , . . . } and the map T : C → C has the form(2) T i ( x ) = x i f i ( x ) , i ∈ I N = { , , . . . , N } and f ∈ C ( C, C ) with f i ( x ) > x ∈ C and i ∈ I N . System (1) is a typicalmathematical model for the population dynamics of a community of N species, where each x i ( n ) represents the population size or density at time n (at the end of n th time period),and the function f i ( x ) denotes the per capita growth rate, of the i th species. If ∂f i ∂x j ≤ Mathematics Subject Classification.
Primary: 37B25; Secondary: 37C70, 34D23, 34D05.
Key words and phrases. discrete competitive models, retrotone maps, carrying simplex, existence anduniqueness, dominant species, vanishing species. for all i, j ∈ I N with i = j , then increase of the j th population reduces the per capitagrowth rate of the i th species, so (1) models the population dynamics of a community ofcompetitive species.System (1) and its various particular instances as models have attracted huge interests fromresearchers in the last two decades. One of the important and influential developments isthe existence of a carrying simplex Σ ⊂ C : a compact invariant hypersurface homeomorphicto ∆ N − = { x ∈ C : x + · · · + x N = 1 } such that every trajectory except the origin isasymptotic with a trajectory in Σ. Since Σ attracts all the points of C \ { } , the dynamicsof (1) on C is essentially described by the dynamics on Σ. The carrying simplex theory wasoriginally established by Hirsch [10] (see [13] for latest update) for competitive Kolmogorovsystems of differential equations. Since then the idea of a carrying simplex for discretesystems gradually appeared in literature (see [19], [20], [14] for example). But a moreaccepted theorem for existence and uniqueness of a carrying simplex for (1) was givenby Hirsch [11] without proof. Then Ruiz-Herrera [9] presented a more general theoremcovering Hirsch’s result with a complete proof.For any x, y ∈ C , we write x ≤ y or y ≥ x if x i ≤ y i for all i ∈ I N ; x < y or y > x if x ≤ y but x = y ; x ≪ y or y ≫ x if x i < y i for all i ∈ I N . The map T given by (2) is said tobe retrotone in a subset X ⊂ C if for any x, y ∈ X , T ( x ) < T ( y ) implies x i < y i for all i ∈ I ( y ) = { j ∈ I N : y j = 0 } . Let [0 , r ] = { x ∈ C : 0 ≤ x ≤ r } . The theorem below isTheorem 6.1 in [9]. Theorem 1.1.
Assume that T with T ([0 , r ]) ⊂ [0 , r ] for some r ≫ satisfies the followingconditions: (i) For each i ∈ I N , the map T restricted to the positive half x i -axis has a fixed point q i e i with q i > , e i the i th standard unit vector and q ≪ r . (ii) T is retrotone and locally one to one in [0 , r ] . (iii) For any x, y ∈ [0 , r ] , if T ( x ) < T ( y ) then, for each j ∈ I N , either x j = 0 or f j ( x ) > f j ( y ) .Then the map admits a carrying simplex Σ . Note that Theorem 1.1 can be only applied to the system restricted to the space [0 , r ] ⊂ C if no condition for T on C \ [0 , r ] is provided. However, if for any compact set S ⊂ C thereis a k ∈ N such that T k ( S ) ⊂ [0 , r ], then Theorem 1.1 can be applied directly to the systemon C .When f on C is a C map, T is also a C map with Jacobian matrix(3) DT ( x ) = diag( f ( x ) , . . . , f N ( x ))( I − M ( x )) , ARRYING SIMPLEX 3 where I is the identity matrix and(4) M ( x ) = ( M ij ( x )) = (cid:18) − x i f i ( x ) ∂f i ∂x j ( x ) (cid:19) N × N . Then, by Lemma 4.1, Corollary 6.1 and Remark 6.4 in [9], Theorem 1.1 has the followingversion with easily checkable conditions.
Theorem 1.2.
Assume that T satisfies the following conditions: (i) For each i ∈ I N , the map T restricted to the positive half x i -axis has a fixed point q i e i with q i > , e i the i th standard unit vector and q ≪ r for some r ∈ C . (ii) All entries of the Jacobian Df are negative. (iii) The spectral radius of M ( x ) satisfies ρ ( M ( x )) < for all x ∈ [0 , q ] \ { } .Then the map admits a carrying simplex Σ . A more user-friendly variation of Theorem 1.2 given by Jiang and Niu [16, Theorem 3.1]is the above theorem with simply a replacement of condition (iii) by (iii) ′ below:(iii) ′ For each x ∈ [0 , q ] \ { } with I ( x ) = { j ∈ I N : x j > } , either f i ( x ) + X j ∈ I ( x ) x j ∂f i ∂x j ( x ) > ∀ i ∈ I ( x )or f i ( x ) + X j ∈ I ( x ) x i ∂f i ∂x j ( x ) > ∀ i ∈ I ( x ) . A carrying simplex Σ has the important and interesting features: compact, invariant,unordered ( p ≤ q implies p = q for p, q ∈ Σ), homeomorphic to ∆ N − by radial projection,and attracting all the points of C \ { } . Therefore, if (1) admits a carrying simplex,the dynamics of the system on the N -dimensional space C is essentially described bythe dynamics on this ( N − ZHANYUAN HOU and geometric method for global stability. Although these methods were not based onthe existence of a carrying simplex, comments and comparisons with those using carryingsimplex were made there.We note that condition (ii) in Theorem 1.2 is very restrictive; it excludes the possibility ofapplying the theorem to systems with some zero entries of Df . But actually, condition (ii)is too strong and unnecessary, a compact invariant set attracting all the points of C \ { } with most of the features of a carrying simplex may still exist even if ∂f i ∂x j = 0 for somedistinct i, j ∈ I N .The aim of this paper is to prove the existence and uniqueness of a modified carryingsimplex under a much weaker condition than (ii): instead of (ii) requiring all N entriesof Df to be negative, we require each entry of Df to be nonpositive, with each f i strictlydecreasing in x i , on a compact set. We shall present the main results in section 2 and leavethe proofs to section 5. In section 3, we present some results on dominant species andvanishing species as an application of the main results. In section 4, we deal with someknown models as examples. We finally conclude the paper in section 6.2. Notation and main results
For C = R N + we let ˙ C = { x ∈ C : ∀ i ∈ I N , x i > } and ∂C = C \ ˙ C . Then ˙ C is the interiorof C and ∂C is the boundary of C . The part of ∂C restricted to the i th coordinate planeand the part restricted to the positive half x i -axis are denoted by π i and X i respectively,i.e. π i = { x ∈ C : x i = 0 } , i ∈ I N ,X i = { x ∈ C : x i > , ∀ j ∈ I N \ { i } , x j = 0 } , i ∈ I N . Denote the i th standard unit vector by e i , i.e. the i th component of e i is 1 and others are0. For any nonempty subset I ⊂ I N , define C I = { x ∈ C : ∀ j ∈ I N \ I, x j = 0 } , ˙ C I = { x ∈ C I : ∀ i ∈ I, x i > } . For any x, y ∈ C I , by writing x ≤ I y we mean x i ≤ y i for all i ∈ I ; we write x < I y if x ≤ I y but x = y ; and we write x ≪ I y if y − x ∈ ˙ C I . We may also use y ≥ I x , y > I x and y ≫ I x for x ≤ I y , x < I y and x ≪ I y respectively. If I = I N , we simply drop the subscript “ I ”from these inequalities. For any a, b ∈ C with a ≤ b , we let [ a, b ] = { x ∈ C : a ≤ x ≤ b } .Then [ a, b ] is a k -dimensional cell if b − a has exactly k positive components. For each x ∈ C , the positive limit set ω ( x ) of T n ( x ) is defined by ω ( x ) = ∞ \ n =1 { T k ( x ) : k ≥ n } , ARRYING SIMPLEX 5 where A denotes the closure of any set A . If T is invertible and T − n ( x ) exist for all n ∈ N ,the negative limit set α ( x ) is defined by α ( x ) = ∞ \ n =1 { T − k ( x ) : k ≥ n } . Also, the whole trajectory of x is denoted by γ ( x ) = { T n ( x ) : n ∈ Z } .Suppose a simply connected closed set S ⊂ C \ { } is an ( N − C into three mutually exclusive subsets S − , S and S + with 0 ∈ S − and C = S − ∪ S ∪ S + . A point p ∈ C is said to be below ( on or above ) S if p ∈ S − ( S or S + ).For any nonempty subset S ⊂ C , S is said to be below ( above ) S if S ⊂ S − ∪ S ( S ∪ S + ); S is said to be strictly below ( strictly above ) S if S ⊂ S − ( S + ).Let B be either C or a positively invariant [0 , r ] for some r ∈ ˙ C . For convenience, we definethe concept of a modified carrying simplex as follows. Definition 2.1.
A nonempty set Σ ⊂ B \ { } is called a modified carrying simplex of (1)if Σ meets the following requirements.(i) Σ is compact, invariant and homeomorphic to ∆ N − by radial projection.(ii) Σ attracts all the points of B \ { } , i.e. ω ( x ) ⊂ Σ for each x ∈ B \ { } .Moreover, if x is below Σ with a nonempty support I ( x ) ⊂ I N , then there is a y ∈ Σ with I ( y ) = I ( x ) such that lim n → + ∞ ( T n ( x ) − T n ( y )) = 0.Note that the “unordered” property of Σ is not mentioned in the above definition. Weshall see in Remark 2.1 (f) below that the unordered property of Σ here is slightly differentfrom that for carrying simplex in Hirsch [11], Ruiz-Herrera [9] and the literature. But themain difference between modified carrying simplex and the carrying simplex in literatureis that the latter requires every trajectory in B \ { } to be asymptotic to one in Σ whereasthe former requires every nontrivial trajectory below Σ to be asymptotic to one in Σ and Σto attract all the points of B \ { } . Obviously, the concept of a modified carrying simplexis more general and it includes carrying simplex as a particular class. Definition 2.2.
The map T : C → C defined by (2) is said to be weakly retrotone in asubset X ⊂ C if for x, y ∈ X with T ( x ) > T ( y ) and T ( x ) − T ( y ) ∈ ˙ C I for some I ⊂ I N ,then x > y and x i > y i for all i ∈ I .Comparing this with the definition of retrotone given in section 1 we see that if T isretrotone then it is weakly retrotone, but not vice versa. Theorem 2.3.
Assume that T defined by (2) with T ([0 , r ]) ⊂ [0 , r ] for some r ∈ ˙ C satisfiesthe following conditions: ZHANYUAN HOU (i)
For each i ∈ I N , the map T restricted to X i has a fixed point q i e i with q i > and q ≪ r . (ii) T is weakly retrotone and locally one to one in [0 , r ] . (iii) For any x, y ∈ [0 , r ] , if T ( x ) < T ( y ) and T ( y ) − T ( x ) ∈ ˙ C I for some I ⊂ I N then,for each j ∈ I , either x j = 0 or f j ( x ) > f j ( y ) .Then 0 is a repellor with the basin of repulsion B (0) ⊂ [0 , r ] , (1) has a unique modifiedcarrying simplex Σ and Σ = B (0) \ ( { } ∪ B (0)) . Moreover, for each p ∈ Σ and every q ∈ [0 , r ] \ { } with q < p , we have α ( q ) ⊂ π i provided q i < p i . Remark 2.1 (a) Condition (i) of Theorem 2.3 is the same as that of Theorem 1.1 but conditions (ii)and (iii) are weaker than those of Theorem 1.1.(b) Condition (ii) and the definition (2) imply that T : [0 , r ] → T ([0 , r ]) is a homeo-morphism. This follows from the local one to one property of T on [0 , r ], T ( x ) = 0if and only if x = 0, and Lemma 4.1 in [9].(c) Condition (ii) implies that, for each i ∈ I N , the function T i ( se i ) is strictly increasingfor s ∈ [0 , r i ]. Indeed, from (b) above we know that T is one to one on [0 , r ]. As T j ( se i ) = 0 and T i ( se i ) > j ∈ I N \ { i } and s ∈ (0 , r i ], the one to oneproperty of T ensures that T i ( s e i ) = T i ( s e i ) for 0 < s < s ≤ r i . By (ii) wemust have T i ( s e i ) < T i ( s e i ) for 0 < s < s ≤ r i . By continuity, T i ( se i ) is strictlyincreasing for s ∈ [0 , r i ].(d) Conditions (ii) and (iii) imply that, for each i ∈ I N , f i ( se i ) is strictly decreasingfor s ∈ [0 , r i ]. Indeed, for 0 < s < s ≤ r i , from (c) above we see that 0
Example
Consider the system (1) with T given by(5) T i ( x ) = x i g i ( x i ) , i ∈ I N , where each g i : R + → R is positive, continuous, 0 < g i ( u ) < u ≥ r i > q i > g i ( q i ) = 1, g i ∈ C ([0 , r i ] , R ), g ′ i ( u ) <
0, and g i ( u ) + ug ′ i ( u ) > u ∈ [0 , r i ]. Then T satisfies all the conditions of Theorem 2.3, so it has a unique modified carrying simplex Σ.Note that system (1) with T defined by (5) is a trivial case of (1) with T defined by (2)when there is no interaction between distinct component equations of the system. Since q i is the globally attracting equilibrium of the i th component equation on the positive x i -axis,Σ is the upper boundary surface of the cell [0 , q ], i.e.Σ = { x ∈ [0 , q ] : x i = q i for some i ∈ I N } . Clearly, q ∈ Σ and for each p ∈ Σ \ { q } , we have p < q . Thus, ordered points in the senseof < are permitted on Σ.Now utilising DT and Df , we obtain conditions which guarantee conditions (ii) and (iii)and the following version of Theorem 2.3 with easily checkable conditions. Consider thematrix M ( x ) given by (4) and(6) ˜ M ( x ) = ( ˜ M ij ( x )) = (cid:18) − x j f i ( x ) ∂f i ∂x j ( x ) (cid:19) N × N . Theorem 2.4.
Assume that T given by (2) satisfies the following conditions: (i) For each i ∈ I N , the map T restricted to X i has a fixed point q i e i with q i > and q ≪ r for some r ∈ ˙ C . (ii) The entries of the Jacobian Df satisfy (7) ∀ x ∈ [0 , r ] , ∀ i, j ∈ I N , ∂f i ∂x j ( x ) ≤ , and f i is strictly decreasing in x i ∈ [0 , r i ] for x ∈ [0 , r ] . (iii) For each x ∈ [0 , q ] \ { } , either ρ ( M ( x )) < for M ( x ) given by (4) or ρ ( ˜ M ( x )) < for ˜ M ( x ) given by (6).Then 0 is a repellor with the basin of repulsion B (0) ⊂ [0 , r ] , (1) has a unique modifiedcarrying simplex Σ and Σ = B (0) \ ( { } ∪ B (0)) . Moreover, for each p ∈ Σ and every q ∈ [0 , r ] \ { } with q < p , we have α ( q ) ⊂ π i provided q i < p i . Remark 2.2
ZHANYUAN HOU (a) When ∂f i ∂x i ≤
0, a sufficient condition for f i to be strictly decreasing for x i ∈ [0 , r i ], x ∈ [0 , r ] with x j fixed for all j ∈ I N \ { i } , is that the set Z i of zeros of ∂f i ∂x i in[0 , r i ] is either empty or finite or infinite with only a finite number of accumulationpoints. In particular, when each Z i is empty, condition (ii) in Theorem 2.4 can bereplaced by(ii)* For all i, j ∈ I N , the entries of the Jacobian Df satisfy(8) ∀ x ∈ [0 , r ] , ∂f i ∂x i ( x ) < , ∂f i ∂x j ( x ) ≤ . (b) Comparing Theorem 2.4 with Theorem 1.2, we see that condition (i) of Theorem2.4 is the same as (i) of Theorem 1.2 and (iii) of Theorem 2.4 has one more choicethan (iii) of Theorem 1.2, but condition (ii) of Theorem 2.4 only requires eachentry of Df to be nonnegative instead of N entries of Df to be strictly negativein Theorem 1.2, plus the strictly decreasing requirement of each f i in x i . Even if(ii) is replaced by the stronger condition (ii)* above, it only requires N diagonalentries of Df to be negative. From this point of view, with a trade off of havinga modified carrying simplex rather than the well known carrying simplex, we havesignificantly reduced the cost and generalised the existing results.(c) Under condition (ii) of Theorem 2.4, if(9) f i ( x ) + N X j =1 x i ∂f i ∂x j ( x ) > ∀ i ∈ I N , using one type of matrix norm we have k M ( x ) k = max i ∈ I N N X j =1 (cid:12)(cid:12)(cid:12)(cid:12) x i f i ( x ) ∂f i ( x ) ∂x j (cid:12)(cid:12)(cid:12)(cid:12) < . By Theorem 6.1.3 in [18], ρ ( M ( x )) ≤ k M ( x ) k . Thus, (9) is a sufficient conditionfor ρ ( M ( x )) <
1. By the same reason, if(10) f i ( x ) + N X j =1 x j ∂f i ∂x j ( x ) > ∀ i ∈ I N , then ρ ( ˜ M ( x )) ≤ k ˜ M ( x ) k <
1. Therefore, condition (iii) of Theorem 2.4 is met if(9) or (10) holds for each x ∈ [0 , q ] \ { } . Corollary 1.
Under the conditions of Theorem 2.3 or Theorem 2.4, the following conclu-sions hold. (i)
For any periodic orbit γ ⊂ Σ , the points on γ are unordered, i.e. if p, q ∈ γ with p ≤ q then p = q . ARRYING SIMPLEX 9 (ii)
For any x ∈ Σ , if there are two points p, q ∈ γ ( x ) satisfying p < q then α ( x ) consistsof either a single fixed point or a periodic orbit.Proof. (i) Suppose there are two points p, q ∈ γ satisfying p < q . Then there is at least one i ∈ I N such that p i < q i . From Theorem 2.3 we have α ( p ) ⊂ π i , so q α ( p ), a contradictionto q ∈ γ = α ( p ) due to the periodicity of γ . Therefore, γ is unordered.(ii) By x ∈ Σ we have γ ( x ) ⊂ Σ and α ( x ) ⊂ Σ. Since p, q ∈ γ ( x ) with p < q , we have T ( T − ( p )) = p < q = T ( T − ( q )). Then the weakly retrotone property of T implies that T − ( p ) < T − ( q ) and T − n ( p ) < T − n ( q ) for all n ∈ N . For each i ∈ I N , if there is an n ∈ N such that ( T − n ( p )) i < ( T − n ( q )) i , by Theorem 2.3 we have α ( x ) = α ( p ) ⊂ π i ; otherwise,we have ( T − n ( p )) i = ( T − n ( q )) i for all n ∈ N . Thus, there is a proper subset I ⊂ I N suchthat α ( x ) ⊂ π i for each i ∈ I and ( T − n ( p )) j = ( T − n ( q )) j for all n ∈ N and j ∈ I N \ I . As p and q are two distinct points on γ ( x ), there is an n > T n ( p ) = q or T n ( q ) = p . Hence, since the component ( T n ( p )) j is an n -periodic function for n ∈ N foreach j ∈ I N \ I , we obtain α ( x ) = { T k ( y ) : k ∈ { , , . . . , n − } , ( T k ( y )) i = 0 , i ∈ I ;( T k ( y )) j = ( T k − n ( p )) j , j ∈ I N \ I } Therefore, α ( x ) consists of either a single fixed point or a periodic orbit. (cid:3) Remark 2.3
Just as we mentioned after Theorem 1.1, Theorems 2.3 and 2.4 can be onlyapplied to systems on the space [0 , r ] ⊂ C if no condition for T on C \ [0 , r ] is given.However, a simple additional condition(11) ∀ i ∈ I N , ∀ x ∈ C with x i ≥ r i , < f i ( x ) < S ⊂ C there is a k ∈ N such that T k ( S ) ⊂ [0 , r ], sothat Theorems 2.3 and 2.4 can be applied directly to systems on C .In general, for any topological space X , a system x ( n ) = F n ( x ) for x ∈ X, n ∈ N with amap F : X → X , and a compact invariant set A ⊂ X , A is called a global attractor of thesystem if A attracts the points of any bounded set B ⊂ X uniformly. For our system (1)with (2) on C under the conditions of Theorem 2.3 or Theorem 2.4, since 0 is a repellingfixed point, by saying that Σ is a global attractor of the system in [0 , r ] \ { } ( C \ { } ), wemean Σ uniformly attracts the points of any bounded set B ⊂ [0 , r ] \ { } ( B ⊂ C \ { } )that is bounded away from 0, i.e. B ⊂ [0 , r ] \ { } ( B ⊂ C \ { } ). Corollary 2.
Under the conditions of Theorem 2.3 or Theorem 2.4, the modified carryingsimplex Σ is a global attractor in [0 , r ] \ { } . In addition, if (11) holds, then Σ is a globalattractor of the system in C \ { } . Before we prove Theorem 2.3, Theorem 2.4 and Corollary 2 in section 5, we present anapplication of Theorem 2.4 in next section. Criteria for dominance and vanishing species
In this section, we consider (2) and give sufficient conditions for dominance of some speciesunder the assumption that the conditions of Theorem 2.4 are met.Viewing (2) as a population model for N competitive species, we say that the j th speciesis dominated or vanishing if lim n → + ∞ x j ( n ) = 0 for all x ∈ ˙ C ; we say that the j th speciesis dominant if lim inf n →∞ x j ( n ) > δ > x ∈ ˙ C and all other species are vanishing.Let(12) Γ i = { x ∈ C : f i ( x ) = 1 } , i ∈ I N . Under the general assumptions for (2), each Γ i is a closed set and an ( N − i is simply connected and divides C into three mutually exclusive subsets Γ + i , Γ i and Γ − i with 0 ∈ Γ − i . Then the closure of Γ − i is Γ − i = Γ − i ∪ Γ i . But if we consider the restriction of Γ i to [0 , r ], this assumption is met ifthe conditions of Theorem 2.4 hold: each Γ i ∩ [0 , r ] is a simply connected closed set and an( N − − i ∩ [0 , r ] is strictly below Γ i and Γ + i ∩ [0 , r ]is strictly above Γ i .Under the conditions of Theorem 2.4, let Q i = q i e i , the fixed point of T on X i , the positivehalf x i -axis. Theorem 3.1.
Assume that (11) and the conditions of Theorem 2.4 hold. (a)
If for some i ∈ I N and all j ∈ I N \ { i } , ∂f i ∂x i ( Q i ) < and Γ i ∩ [0 , r ] is strictly below Γ j , then lim n → + ∞ x i ( n ) = 0 for all x ∈ C \ X i so the i th species is vanishing. (b) If for some i ∈ I N and all j ∈ I N \ { i } , ∂f i ∂x i ( Q i ) < and Γ i ∩ [0 , r ] is strictlyabove Γ j , then the i th species is dominant and the axial fixed point Q i is globallyasymptotically stable.Proof. By Theorem 2.4 the system has a unique modified carrying simplex Σ.(a) Under the assumption that Γ i ∩ [0 , r ] is strictly below Γ j for all j ∈ I N \ { i } , we firstclaim that(13) Γ − i ∩ Σ = ∅ so that Γ − i ∩ [0 , r ] is strictly below Σ and Σ is above Γ i . Indeed, if (13) were not true thenwe would have a point p ∈ (Γ − i ∩ Σ). As 0 Σ, we have p = 0 and a nonempty J ⊂ I N such that p j > j ∈ J . Since p is below Γ i and Γ i ∩ [0 , r ] is strictly below Γ j for all j ∈ I N \ { i } , p is below Γ j for all j ∈ I N . Let u = T ( p ). Then ∀ j ∈ J, u j = T j ( p ) = p j f j ( p ) > p j ; ∀ k ∈ I N \ J, u k = p k = 0 , so p ≪ J u . As Σ is invariant and p ∈ Σ, we have u ∈ Σ. Then, by Theorem 2.4, we have α ( p ) = { } so p ∈ B (0), a contradiction to p ∈ Σ = B (0) \ ( { } ∪ B (0)). This shows ourclaim (13). ARRYING SIMPLEX 11
Since the axial fixed point Q i is below Γ j for all j ∈ I N \ { i } , the Jacobian matrix DT ( Q i )has N − f j ( Q i ) > j ∈ I N \ { i } and one eigenvalue 1 + q i ∂f i ∂x i ( Q i ). Byassumption, (4) and (6), the only nonzero eigenvalue of M ( Q i ) and ˜ M ( Q i ) is − q i ∂f i ∂x i ( Q i ) >
0, so ρ ( M ( Q i )) = ρ ( ˜ M ( Q i )) = − q i ∂f i ∂x i ( Q i ). By condition (iii) of Theorem 2.4, we have0 < q i ∂f i ∂x i ( Q i ) <
1. So Q i is a saddle point in C with X i as its one-dimensional stablemanifold and a repellor on Σ. Thus, to show that lim n → + ∞ x i ( n ) = 0 for all x ∈ C \ X i ,by the definition of modified carrying simplex, we need only show that lim n → + ∞ x i ( n ) = 0for all x ∈ C \ X i on or above Σ, i.e. x ∈ (Σ ∪ Σ + ) \ X i .Now for any x ∈ C with x i > r i , the assumption (11) ensures that x ( n ) ∈ [0 , r ] for largeenough n ∈ N . Without loss of generality, we only consider x ∈ (Σ ∪ (Σ + ∩ [0 , r ])) \ X i .We first show that the set (Σ ∪ (Σ + ∩ [0 , r ])) \ X i is positively invariant. From the proofof Theorem 2.4 given in section 5 we shall see that the conditions of Theorem 2.4 implythe conditions of Theorem 2.3. Thus, [0 , r ] is positively invariant and, by Remark 2.3 (b), T : [0 , r ] → T ([0 , r ]) is a homeomorphism. As 0 is a repellor with the basin of repulsion B (0) ⊂ [0 , r ], we shall see in section 5 (Lemma 5.4) that B (0) is invariant. Thus, T mapsthe set [0 , r ] \ B (0) = [0 , r ] \ (Σ ∪ B (0) ∪ { } ) = [0 , r ] \ (Σ ∪ Σ − ) = [0 , r ] ∩ Σ + into itself. As Σ is invariant, Σ ∪ (Σ + ∩ [0 , r ]) is positively invariant. For each x ∈ Σ ∪ (Σ + ∩ [0 , r ]) \ X i , there is a j ∈ I N \ { i } such that x j >
0, so T j ( x ) = x j f j ( x ) >
0. Thus, T ( x ) ∈ Σ ∪ (Σ + ∩ [0 , r ]) \ X i . This shows the positive invariance of Σ ∪ (Σ + ∩ [0 , r ]) \ X i .By (13), Σ is above Γ i , so Σ ∪ (Σ + ∩ [0 , r ]) \ X i is above Γ i . Thus, for x ∈ (Σ ∪ (Σ + ∩ [0 , r ])) \ X i , x ( n ) = T n ( x ) ∈ (Σ ∪ (Σ + ∩ [0 , r ])) \ X i , so x ( n ) is on or above Γ i for all n ∈ N . Hence, ∀ n ∈ N , x i ( n + 1) = T i ( x ( n )) = x i ( n ) f i ( x ( n )) ≤ x i ( n ) . This shows that { x i ( n ) } is a bounded monotone nonincreasing sequence, so there is an x ≥ n → + ∞ x i ( n ) = x . Suppose x >
0. Then, for each y ∈ ω ( x ) ⊂ Σ,we have y i = x . As T n ( y ) ∈ ω ( x ) for all integer n , we have T i ( y ) = y i f i ( y ) = x = y i so f i ( y ) = 1 and y ∈ Γ i . Therefore, ω ( x ) ⊂ Γ i ∩ Σ. If ω ( x ) = { Q i } , as Q i is below Γ j for all j ∈ I N \ { i } , there is a δ > O ( Q i , δ ) ∩ [0 , r ] of the open ballcentred at Q i with radius δ restricted to [0 , r ], i.e. O ( Q i , δ ) ∩ [0 , r ], is strictly below Γ j forall j ∈ I N \ { i } . Let m = min { f j ( u ) : u ∈ O ( Q i , δ ) ∩ [0 , r ] , j ∈ I N \ { i }} . Then m >
1. Since lim n → + ∞ x ( n ) = Q i , there is n ∈ N such that x ( n ) ∈ O ( Q i , δ ) ∩ [0 , r ]for n ≥ n . As x X i , we have x j > j ∈ I N \ { i } . Then, for this j and all n ≥ x j ( n + n ) = T j ( x ( n − n ) = x j ( n − n ) f j ( x ( n − n ) ≥ m x j ( n − n ) , so x j ( n + n ) ≥ m n x j ( n ) → + ∞ as n → + ∞ , a contradiction to the boundedness of { x ( n ) } . This contradiction shows the existence of a point y ∈ ω ( x ) \ { Q i } . Since Q i is the unique intersection point of Σ with X i and y ∈ Σ \ { Q i } , we have y X i so y j > j ∈ I N \ { i } . Since Γ i ∩ [0 , r ] is strictly below Γ j and ω ( x ) ⊂ Γ i ∩ Σ ⊂ Γ i ∩ [0 , r ], ω ( x )is strictly below Γ j . Let ρ = min u ∈ ω ( x ) f j ( u ) . Then, by the continuity of f and the compactness of ω ( x ), ρ > y j ( n + 1) = T j ( y ( n )) = y j ( n ) f j ( y ( n )) ≥ ρy j ( n ) . Thus, y j ( n ) ≥ ρ n y j → + ∞ ( n → + ∞ ) , a contradiction to the boundedness of ω ( x ). This contradiction shows that we must have x = 0, i.e. lim n → + ∞ x i ( n ) = 0 for all x ∈ C \ X i .(b) Under the condition that Γ i ∩ [0 , r ] is strictly above Γ j for every j ∈ I N \ { i } , we firstshow that Σ is below Γ i by assuming the opposite: there is a point p ∈ Σ ∩ Γ + i . As p = 0,there is a nonempty J ⊂ I N as the support of p . As p is above Γ i and Γ i ∩ [0 , r ] is strictlyabove Γ j for all j ∈ I N \ { i } , p is above Γ j for all j ∈ I N . Thus, ∀ j ∈ J, T j ( p ) = p j f j ( p ) < p j , so T ( p ) ≪ J p . By Theorem 2.4, α ( T ( p )) = { } so T ( p ) ∈ B (0), a contradiction to T ( p ) ∈ Σ = B (0) \ ( { } ∪ B (0)). This shows that Σ must be below Γ i .We need only show that Q i is stable and attracts all the points of C \ π i as the dominanceof the i th species is implied by the global attraction of Q i . As Q i is above Γ j for all j ∈ I N \ { i } , we have f j ( Q i ) ∈ (0 ,
1) for all j ∈ I N \ { i } . By the assumption ∂f i ∂x i ( Q i ) < q i ∂f i ∂x i ( Q i ) ∈ (0 , DT ( Q i )is in (0 ,
1) so Q i is asymptotically stable. To show the global attraction of Q i in C \ π i , bythe assumption (11) and the definition of a modified carrying simplex, we need only showthat lim n → + ∞ x ( n ) = Q i for all x ∈ (Σ ∪ (Σ + ∩ [0 , r ])) \ π i .If x ∈ Σ \ π i , as x ( n ) ∈ Σ for all n ∈ N and Σ is below Γ i , the sequence { x i ( n ) } is boundedand monotone nondecreasing. Thus, there is a β > n → + ∞ x i ( n ) = β . Forany y ∈ ω ( x ), we have T i ( y ) = y i f i ( y ) = β = y i so f i ( y ) = 1. Thus, y ∈ Γ i and ω ( x ) ⊂ Γ i .We claim that ω ( x ) = { Q i } . To verify this claim, as ω ( x ) is compact, ω ( x ) ⊂ Γ i andΓ i ∩ [0 , r ] is strictly above Γ j for all j ∈ I N \ { i } , there is a δ > O ( ω ( x ) , δ ) ∩ [0 , r ] of the open set O ( ω ( x ) , δ ) ∩ [0 , r ] with O ( ω ( x ) , δ ) = { z ∈ C : k z − u k < δ for some u ∈ ω ( x ) } is strictly above Γ j for all j ∈ I N \ { i } . Let µ = max { f j ( u ) : u ∈ O ( ω ( x ) , δ ) ∩ [0 , r ] , j ∈ I N \ { i }} . By the continuity of f and the compactness of O ( ω ( x ) , δ ) ∩ [0 , r ], we have 0 < µ <
1. Bythe definition of ω ( x ), there is an integer N ≥ x ( n ) ∈ O ( ω ( x ) , δ ) ∩ [0 , r ] for all n > N . Let J ⊂ I N be the support of x . Then x j ( n ) > n ∈ N if and only if j ∈ J . ARRYING SIMPLEX 13 If J = { i } then x = Q i and the above claim is obviously true. Now suppose { i } ⊂ J = { i } .Then, ∀ n > N , ∀ j ∈ J \ { i } , x j ( n + 1) = x j ( n ) f j ( x ( n )) ≤ µx j ( n ) , so x j ( n + N ) ≤ µ n x j ( N ) → n → + ∞ . This shows that ω ( x ) = Σ ∩ X i = { Q i } .Now suppose x ∈ (Σ + ∩ [0 , r ]) \ π i . By the asymptotic stability of Q i there is a δ > z ∈ O ( Q i , δ ) ∩ C is attracted to Q i , i.e. lim n → + ∞ z ( n ) = Q i . Thus, aslong as Q i ∈ ω ( x ), there is an m ∈ N such that x ( m ) ∈ O ( Q i , δ ) ∩ C so that ω ( x ) = { Q i } .We now prove Q i ∈ ω ( x ) by contradiction. If Q i ω ( x ) and there is a y ∈ ω ( x ) ⊂ Σ with y i >
0, by the previous paragraph we have lim n → + ∞ y ( n ) = Q i . As ω ( x ) is compact and y ( n ) ∈ ω ( x ) for all n ∈ N , we have Q i ∈ ω ( x ), which contradicts the condition Q i ω ( x ).Thus, if Q i ω ( x ) then ω ( x ) ⊂ Σ ∩ π i . If ω ( x ) is strictly below Γ i , by definition of ω ( x ) there is a K ∈ N such that x ( n ) is below Γ i for all n ≥ K . Thus, { x i ( n ) } is anincreasing sequence for n ≥ K so that x i ( n ) ≥ x i ( K ) > n ≥ K and each y ∈ ω ( x )satisfies y i ≥ x i ( K ) >
0, a contradiction to the assumption ω ( x ) ⊂ π i . If there is a point p ∈ ω ( x ) ⊂ (Σ ∩ π i ) on or above Γ i , there is a nonempty J ⊂ I N \ { i } as the support of p .As Γ i ∩ [0 , r ] is strictly above Γ j for all j ∈ I N \ { i } , p is above Γ j for all j ∈ I N \ { i } so T ( p ) ≪ J p . This leads us to α ( T ( p )) = { } by Theorem 2.4, so T ( p ) ∈ B (0), a contradictionto T ( p ) ∈ Σ = B (0) \ ( { } ∪ B (0)). These contradictions show that we must have Q i ∈ ω ( x )so ω ( x ) = { Q i } . (cid:3) We note that Theorem 2.3 in [9] is consistent with our Theorem 3.1 (b) but under thestronger conditions of Theorem 1.1. While Theorem 3.1 used one surface Γ i ∩ [0 , r ] com-paring with the other N − j to obtain one species vanishing, our next resultrepeat such a condition several times to get multiple species vanishing. Theorem 3.2.
Assume that (11) and the conditions of Theorem 2.4 hold. Assume alsothe existence of an integer k ∈ I N \ { N } such that for all i ∈ { , . . . , k } , ∂f i ∂x i ( Q i ) < and (14) ∀ j ∈ { i + 1 , . . . , N } , ( ∩ i − ℓ =1 π ℓ ) ∩ Γ i ∩ [0 , r ] is strictly below Γ j . Then the i th species is dominated for all i ∈ { , . . . , k } . In addition, if k = N − and ∂f N ∂x N ( Q N ) < , then the N th species is dominant and the N th axial fixed point Q N is globallyasymptotically stable in C . Remark 3.1
Here the symbol ∩ i ∈∅ π i is deemed as C . So, (14) for i = 1 is simplifiedas(15) ∀ j ∈ { , . . . , N } , Γ ∩ [0 , r ] is strictly below Γ j . Proof of Theorem 3.2.
For k > ∀ i ∈ { , . . . , k } , ( ∩ i − ℓ =1 π ℓ ) ∩ Γ − i ∩ Σ = ∅ . The proof of (16) is similar to that of (13). Suppose (16) is not true. Then, for some i ∈ { , . . . , k } , there exists a point u ∈ ( ∩ i − ℓ =1 π ℓ ) ∩ Γ − i ∩ Σ, so u is below Γ i . By (14), ( ∩ i − ℓ =1 π ℓ ) ∩ [0 , r ] ∩ Γ i is strictly below Γ j for all j ∈ { i + 1 , . . . , N } . Thus, u is below Γ j for all j ∈ { i, . . . , N } . Note that u ∈ ( ∩ i − ℓ =1 π ℓ ) ∩ Σ implies u > u j = 0 for all j ∈ { , . . . , i − } . Thus, there is a nonempty J ⊂ { i, . . . , N } such that u j > j ∈ J . Then, ∀ j ∈ J, T j ( u ) = u j f j ( u ) > u j , so u ≪ J T ( u ). As T ( u ) ∈ Σ, by Theorem 2.4 we obtain u ∈ B (0), a contradiction to u ∈ Σ = B (0) \ ( { } ∪ B (0)). This contradiction shows the truth of (16).For k ≥ i = 1, from Remark 3.1 and Theorem 3.1 we know that the first species isvanishing, i.e. ω ( x ) ⊂ Σ ∩ π for all x ∈ C \ X .Then for k ≥ i = 2, from (14) we see that Γ ∩ π ∩ [0 , r ] is strictly below Γ j for all j ∈ { , . . . , N } . As ω ( x ) ⊂ Σ ∩ π for all x ∈ C \ X , we can prove that ω ( x ) ⊂ Σ ∩ ( π ∩ π )for all x ∈ C \ ( X × X ) (the proof is included in the general case below).Then for k ≥ i = 3, from (14) we see that Γ ∩ ( π ∩ π ) ∩ [0 , r ] is strictly below Γ j for all j ∈ { , . . . , N } . As ω ( x ) ⊂ Σ ∩ ( π ∩ π ) for all x ∈ C \ ( X × X ), we can prove that ω ( x ) ⊂ Σ ∩ ( ∩ ℓ =1 π ℓ ) for all x ∈ C \ ( X × X × X ) (the proof is included in the generalcase below).In general, for k > i = k , from (14) we see that Γ k ∩ ( ∩ k − ℓ =1 π ℓ ) ∩ [0 , r ] is strictlybelow Γ j for all j ∈ { k + 1 , . . . , N } . Suppose we know that ω ( x ) ⊂ Σ ∩ ( ∩ k − ℓ =1 π ℓ ) for all x ∈ C \ ( X × · · · × X k − ). We need to prove that(17) ∀ x ∈ C \ ( X × · · · × X k ) , ω ( x ) ⊂ Σ ∩ ( ∩ kℓ =1 π ℓ ) . From condition (11), Theorem 2.4 and the definition of a modified carrying simplex, insteadof (17) we need only prove that(18) ∀ x ∈ ([0 , r ] \ Σ − ) \ ( X × · · · × X k ) , ω ( x ) ⊂ Σ ∩ ( ∩ kℓ =1 π ℓ ) . The proof of (18) is divided into the following two steps.Step 1. We show that(19) ∀ x ∈ ( ∩ k − ℓ =1 π ℓ ) ∩ ([0 , r ] \ Σ − ) \ X k , ω ( x ) ⊂ Σ ∩ ( ∩ kℓ =1 π ℓ ) . From (16) we know that ( ∩ k − ℓ =1 π ℓ ) ∩ Σ is above Γ k , so ( ∩ k − ℓ =1 π ℓ ) ∩ ([0 , r ] \ Σ − ) \ X k is aboveΓ k . Note that π ℓ and C \ π ℓ are positively invariant for any ℓ ∈ I N . Thus, for any x ∈ ( ∩ k − ℓ =1 π ℓ ) ∩ ([0 , r ] \ Σ − ) \ X k , if x k = 0 then x ∈ ( ∩ kℓ =1 π ℓ ), so x ( n ) ∈ ( ∩ kℓ =1 π ℓ ) for all n ∈ N and ω ( x ) ⊂ Σ ∩ ( ∩ kℓ =1 π ℓ ). If x k >
0, as both ∩ k − ℓ =1 π ℓ and ([0 , r ] \ Σ − ) \ X k = Σ ∪ (Σ + ∩ [0 , r ]) \ X k (from the proof of Theorem 3.1) are positively invariant and ( ∩ k − ℓ =1 π ℓ ) ∩ ([0 , r ] \ Σ − ) \ X k =( ∩ k − ℓ =1 π ℓ ) ∩ (([0 , r ] \ Σ − ) \ X k ), we have x ( n ) ∈ ( ∩ k − ℓ =1 π ℓ ) ∩ ([0 , r ] \ Σ − ) \ X k for all n ∈ N ,so each x ( n ) is on or above Γ k for all n ∈ N . Hence, ∀ n ∈ N , x k ( n + 1) = T k ( x ( n )) = x k ( n ) f k ( x ( n )) ≤ x k ( n ) . As { x k ( n ) } is a positive monotone nonincreasing sequence, there is a µ ≥ n → + ∞ x k ( n ) = µ . Suppose µ >
0. Then, for each y ∈ ω ( x ) ⊂ Σ we have y k = µ . As ARRYING SIMPLEX 15 T ( y ) ∈ ω ( x ), we have µ = T k ( y ) = y k f k ( y ) = µf k ( y ) so f k ( y ) = 1 and y ∈ Γ k . Therefore, ω ( x ) ⊂ Γ k . By the positive invariance of ( ∩ k − ℓ =1 π ℓ ) ∩ [0 , r ], ω ( x ) ⊂ Γ k ∩ ( ∩ k − ℓ =1 π ℓ ) ∩ [0 , r ].Since the set Γ k ∩ ( ∩ k − ℓ =1 π ℓ ) ∩ [0 , r ] is strictly below Γ j for all j ∈ { k + 1 , . . . , N } , ω ( x ) isstrictly below Γ j for all j ∈ { k + 1 , . . . , N } . As ω ( x ) is compact, there is a δ > O ( ω ( x ) , δ ) ∩ [0 , r ] is strictly below Γ j for all j ∈ { k +1 , . . . , N } . For this δ , there is an m ∈ N such that x ( n ) ∈ O ( ω ( x ) , δ ) ∩ [0 , r ] for all n ≥ m . Note that x ∈ ( ∩ k − ℓ =1 π ℓ ) ∩ ([0 , r ] \ Σ − ) \ X k implies x j > j ∈ { k + 1 , . . . , N } . For this j , let η = min { f j ( u ) : u ∈ O ( ω ( x ) , δ ) ∩ [0 , r ] } . Then η > n ≥ x j ( n + m ) = T j ( x ( n + m − x j ( n + m − f j ( x ( n + m − ≥ ηx j ( n + m − . It follows from this that x j ( n + m ) ≥ η n x j ( m ) → + ∞ as n → + ∞ , a contradiction to theboundedness of { x ( n ) } . This contradiction shows that µ = 0 and (19) follows.Step 2. Now we prove (18). For x ∈ ([0 , r ] \ Σ − ) \ ( X × · · · × X k ), we show that ω ( x ) ⊂ Σ ∩ ( ∩ kℓ =1 π ℓ ). From the supposition we know that ω ( x ) ⊂ Σ ∩ ( ∩ k − ℓ =1 π ℓ ). Suppose ω ( x ) Σ ∩ ( ∩ kℓ =1 π ℓ ). Then either ω ( x ) = { Q k } or there is a y ∈ ω ( x ) \ { Q k } with y k > n → + ∞ x ( n ) = Q k . Since Q k is below Γ j for all j ∈ { k + 1 , . . . , N } ,there is an ε > O ( Q k , ε ) ∩ [0 , r ] is strictly below Γ j for all j ∈ { k + 1 , . . . , N } .For this ε >
0, there is an m ∈ N such that x ( n ) ∈ O ( Q k , ε ) ∩ [0 , r ] for all n ≥ m . That x X × · · · × X k ensures the existence of some j ∈ { k + 1 , . . . , N } with x j > x j ( n ) > n ∈ N . For this j , let η = min { f j ( u ) : u ∈ O ( Q k , ε ) ∩ [0 , r ] } . Then η > n ≥ x j ( n + m ) = T j ( x ( n + m − x j ( n + m − f j ( x ( n + m − ≥ η x j ( n + m − . This leads to x j ( n + m ) ≥ η n x j ( m ) → + ∞ as n → + ∞ , a contradiction to the boundednessof { x ( n ) } .In the latter case, from Step 1 we see that lim n → + ∞ y k ( n ) = 0. Without loss of generality,we may assume that 0 < y k < q k , where Q k = q k e k . Since the whole trajectory γ ( y ) iscontained in ω ( x ) and ω ( x ) is compact, from Step 1 we derive that ω ( y ) ⊂ ω ( x ) ∩ ( ∩ k − ℓ =1 π ℓ ) ∩ Σ. Let η = max { f k ( u ) : u ∈ [0 , r ] } and take a small ε ∈ (0 , y k η ). Since the set S = { z ∈ ( ∩ k − ℓ =1 π ℓ ) ∩ Σ : ε ≤ z k ≤ y k } is compact and ω ( z ) ⊂ ( ∩ kℓ =1 π ℓ ) ∩ Σ for all z ∈ S from Step 1, by continuous dependencethere is a δ ∈ (0 , min { ε , y k } ) such that ∀ u ∈ O ( S, δ ) ∩ [0 , r ] , ∃ z ∈ S, ∃ n ( z ) ∈ N , such that z k ( n ) < ε , ∀ n ∈ { , . . . , n } , k u ( n ) − z ( n ) k < ε . (20)For this δ , there is an m ∈ N such that x ( n ) ∈ O (( ∩ k − ℓ =1 π ℓ ) ∩ Σ , δ ) for all n ≥ m . As ∅ 6 = ω ( y ) ⊂ ω ( x ) ∩ ( ∩ kℓ =1 π ℓ ) ∩ Σ, there is an m ≥ m such that 0 < x k ( m ) < ε . Then x k ( m + 1) = T k ( x ( m )) = x k ( m ) f k ( x ( m )) ≤ η ε < y k . Thus, either (i) x k ( m +1) < ε or (ii) x ( m +1) ∈ O ( S, δ ). In case (i), by taking m = m +1we have x k ( m + 1) < y k . In case (ii), by (20) there is a z ∈ S and n ∈ N such that z k ( n ) < ε and k x ( m + 1 + n ) − z ( n ) k < ε for all 0 ≤ n ≤ n . By the positive invarianceof ( ∩ k − ℓ =1 π ℓ ) ∩ Σ, for any z ∈ ( ∩ k − ℓ =1 π ℓ ) ∩ Σ, we have z ( n ) ∈ ( ∩ k − ℓ =1 π ℓ ) ∩ Σ for all n ∈ N .As (16) implies that ( ∩ k − ℓ =1 π ℓ ) ∩ Σ is above Γ k and S ⊂ ( ∩ k − ℓ =1 π ℓ ) ∩ Σ, z k ( n ) is monotonenonincreasing in n for each z ∈ S . Then, x k ( m + 1 + n ) < z k ( n ) + 12 ε ≤ z k + 12 ε < x k ( m + 1) + ε < y k , ≤ n ≤ n and x k ( m + 1 + n ) < z k ( n ) + ε < ε . Take m = m + 1 + n . In either (i) or (ii),we see that x k ( n ) < y k for all m ≤ n ≤ m . Repeating the above process we obtain x k ( n ) < y k for all n ≥ m , a contradiction to y ∈ ω ( x ).The contradictions in both cases above show that ω ( x ) ⊂ ( ∩ kℓ =1 π ℓ ) ∩ Σ. Then (18) follows.Finally, if k = N −
1, we have lim n → + ∞ x ( n ) = Q N for all x ∈ C with x N >
0. As Q N is above Γ j for all j ∈ { , . . . , N − } and ∂f N ∂x N ( Q N ) <
0, every eigenvalue of the Jacobianmatrix DT ( Q N ) is in the interval (0 , Q N is globally asymptotically stable. (cid:3) Note that the statement of Theorem 3.2 used the natural ascending order of numbers forthe species. Obviously, the statement is still true after a permutation from ascending orderof numbers.
Corollary 3.
Assume that (11) and the conditions of Theorem 2.4 hold. Assume alsothe existence of a permutation p : I N → I N and an integer k ∈ I N \ { N } such that ∂f p ( i ) ∂x p ( i ) ( Q p ( i ) ) < for all i ∈ { , . . . , k } and (21) ∀ j ∈ { i + 1 , . . . , N } , ( ∩ i − ℓ =1 π p ( ℓ ) ) ∩ Γ p ( i ) ∩ [0 , r ] is strictly below Γ p ( j ) . Then the p ( i ) th species is dominated for all i ∈ { , . . . , k } . In addition, if k = N − and ∂f p ( N ) ∂x p ( N ) ( Q p ( N ) ) < then the p ( N ) th species is dominant and the p ( N ) th axial fixed point Q p ( N ) is globally asymptotically stable in C . ARRYING SIMPLEX 17 Some Examples
In this section, we apply our results obtained in sections 2 and 3 to some known models asexamples. All these models fit well into our system (1) for maps T : C → C of the form(2), where the sign of each entry ∂f i ( x ) ∂x j of the Jacobian Df is completely determined bythe corresponding entry a ij of a constant matrix A = ( a ij ) N × N : ∀ i, j ∈ I N , ∀ x ∈ C, ∂f i ( x ) ∂x j = − σ ij ( x ) a ij , σ ij ( x ) > . All entries of the matrix A are assumed positive in most of the references cited here eitherdue to their particular meaning in the original model or due to convenience of theoreticalanalysis by using available results such as Theorem 1.1, Theorem 1.2 and their variationsmentioned in section 1. Under such an assumption, each system models the populationdynamics of a community of N competing species where the population of the j th speciesdirectly affects the growth rate of the population of the i th species in a negative way as a ij > j th species affects the growthrate of the population of the i th species in a negative way, directly or indirectly, due to a ij > a ij = 0. With the help of our Theorem 2.3 and Theorem 2.4, we are now ableto deal with these models under the relaxed assumption: ∀ i, j ∈ I N , a ij ≥ , a ii > . Since our results obtained in section 3 and this section below are all based on the assump-tions of Theorem 2.4, if ∂f i ( x ) ∂x j = 0 ( a ij = 0 for the models below) for at least one pair ofindices i, j at some point x ∈ [0 , r ], then these results are not achievable by using Theorem1.2 and its variations as their conditions are not fully met. This demonstrates that theclass of systems to which Theorem 2.4 is applicable is broader than that for Theorem 1.2and its variations. Hence, our main results are a significant improvement of those availablein literature.4.1. The competitive Leslie-Gower models.
The competitive Leslie-Gower models aresystem (1) for maps T : C → C of the form (2): T i ( x ) = x i f i ( x ), where(22) ∀ i, j ∈ I N , f i ( x ) = c i P Nk =1 a ik x k , c i > , a ij ≥ , a ii > . Under the condition that a ij > i, j ∈ I N , Jiang and Niu [16] have shown that eachLeslie-Gower model admits a carrying simplex.I. Following the same lines as those in [16], we check that each Leslie-Gower model with a ij ≥ a ii > x ∈ X i , f i ( x ) = 1 if and only if x i = c i − a ii = q i , so T restricted to X i has a unique fixed point Q i = q i e i . For i, j ∈ I N , ∂f i ∂x j = − c i a ij [1 + P Nk =1 a ik x k ] , so ∂f i ∂x j ≤ ∂f i ∂x i <
0. Also, for all x ∈ C , ∀ i ∈ I N , f i ( x ) + N X j =1 x j ∂f i ∂x j = c i [1 + P Nk =1 a ik x k ] > . By Remark 2.2 (c), conditions (i)–(iii) of Theorem 2.4 and (11) are all met for any r ≫ q .The surfaces Γ i are now ( N − C : ∀ i ∈ I N , Γ i = { x ∈ C : a i x + · · · + a iN x N = c i − } . II. If for some i ∈ I N , the following inequalities hold:(23) ∀ j ∈ I N \ { i } , ∀ k ∈ I N , a jk ( c i − < a ik ( c j − , then the intersection point of Γ i with each positive half axis X k is below Γ j for every j ∈ I N \ { i } . So Γ i is strictly below Γ j in C for all j ∈ I N \ { i } . By Theorem 3.1 (a), the i th species is dominated.III. If for some i ∈ I N ,(24) ∀ j ∈ I N \ { i } , ∀ k ∈ I N , either a ik = a jk = 0 or a ik ( c j − < a jk ( c i − , then either X k is parallel to both Γ i and Γ j or the intersection point of Γ j with X k is belowΓ i , so Γ i is strictly above Γ j for all j ∈ I N \ { i } . By Theorem 3.1 (b), the i th species isdominant and the fixed point Q i is globally asymptotically stable.IV. Note that (24) is a sufficient condition for Γ i to be strictly above Γ j for all j ∈ I N \ { i } in C . But the condition in Theorem 3.1 (b) only requires the relationship of such planesrestricted to [0 , r ]. So (24) is much stronger than the requirement of Theorem 3.1 (b). Forexample, let us consider the three-dimensional Leslie-Gower model with(25) f ( x ) = 21 + x + 0 . x , f ( x ) = 21 + 2 x + x + 0 . x , f ( x ) = 21 + 2 x + x . Clearly, e , e and e are the axial fixed points. Take r = (1 . , . , . ≫ (1 , ,
1) = q .The intersection points of Γ with the X and X are (0 . , ,
0) and e respectively. As f (0 . , ,
0) = . > f ( e ) = . >
1, both (0 . , ,
0) and e are below Γ . Since X is parallel to both Γ and Γ , Γ is strictly above Γ in R . The intersection points of Γ with the axes are (0 . , , e and (0 , , . , ,
0) is below Γ already.As f ( e ) = 2 > f (0 , ,
5) = . < e is below Γ but (0 , ,
5) is above Γ . So Γ is not above Γ on R and (24) is not met. However, restricted to [0 , r ], Γ intersects oneof the edges of [0 , r ] at (0 . , , .
1) and f (0 . , , .
1) = . >
1. So Γ ∩ π ∩ [0 , r ] isstrictly below Γ . This, together with e below Γ , implies that Γ ∩ [0 , r ] is strictly above ARRYING SIMPLEX 19 Γ . By Theorem 3.1 (b), the first species is dominant and the fixed point e is globallyasymptotically stable.V. Now suppose the following inequalities hold: ∀ i ∈ I N \ { N } , ∀ j, k ∈ { i + 1 , . . . , N } , (26) a ji ( c i − < a ii ( c j − , a ( i +1) k ( c i − < a ik ( c i +1 − . Then, for each i ∈ I N \ { N } , the intersection point of Γ i with X i is below Γ j for all j ∈{ i + 1 , . . . , N } and Γ i ∩ ( X i +1 × · · · × X N ) is strictly below Γ i +1 . Thus, Γ N − ∩ ( X N − × X N )is strictly below Γ N , Γ N − ∩ ( X N − × X N − × X N ) is strictly below Γ N − and Γ N , . . . ,Γ i ∩ ( X i × · · · × X N ) is strictly below Γ j for all j ∈ { i + 1 , . . . , N } . By Theorem 3.2, the N th species is dominant and Q N = c N − a NN e N is globally asymptotically stable.4.2. The generalised competitive Atkinson-Allen models.
The generalised com-petitive Atkinson-Allen models are systems (1) for maps T : C → C of the form (2): T i ( x ) = x i f i ( x ), where(27) ∀ i, j ∈ I N , f i ( x ) = c i + (1 + u i )(1 − c i )1 + P Nk =1 a ik x k , < c i < , u i > , a ij ≥ , a ii > . Under the condition that a ij > i, j ∈ I N , Gyllenberg et al [6] have shown thateach such model admits a carrying simplex.I. Following the same lines as those in [6], we check that each generalised Atkinson-Allenmodel with a ij ≥ a ii > x ∈ X i , f i ( x ) = 1 if and only if x i = u i a ii = q i , so T restricted to X i has a uniquefixed point Q i = q i e i . For i, j ∈ I N , ∂f i ∂x j = − (1 + u i )(1 − c i ) a ij [1 + P Nk =1 a ik x k ] , so ∂f i ∂x j ≤ ∂f i ∂x i <
0. Also, for all x ∈ C , ∀ i ∈ I N , f i ( x ) + N X j =1 x j ∂f i ∂x j = c i + (1 + u i )(1 − c i )[1 + P Nk =1 a ik x k ] > . By Remark 2.2 (c), conditions (i)–(iii) of Theorem 2.4 and (11) are all met for any r ≫ q .The surfaces Γ i are now ( N − C :(28) ∀ i ∈ I N , Γ i = { x ∈ C : a i x + · · · + a iN x N = u i } . II. If for some i ∈ I N , the following inequalities hold:(29) ∀ j ∈ I N \ { i } , ∀ k ∈ I N , a jk u i < a ik u j , then the intersection point of Γ i with each positive half axis X k is below Γ j for every j ∈ I N \ { i } . So Γ i is strictly below Γ j in C for all j ∈ I N \ { i } . By Theorem 3.1 (a), the i th species is dominated.III. If for some i ∈ I N ,(30) ∀ j ∈ I N \ { i } , ∀ k ∈ I N , either a ik = a jk = 0 or a ik u j < a jk u i , then either X k is parallel to both Γ i and Γ j or the intersection point of Γ j with X k is belowΓ i , so Γ i is strictly above Γ j for all j ∈ I N \ { i } . By Theorem 3.1 (b), the i th species isdominant and the fixed point Q i is globally asymptotically stable.IV. Note that (30) is much stronger than the requirement of Theorem 3.1 (b). Similar to(25), we can easily construct a three-dimensional generalised Atkinson-Allen model as anexample which fails (30) but satisfies the condition of Theorem 3.1 (b).V. Now suppose the following inequalities hold:(31) ∀ i ∈ I N \ { N } , ∀ j, k ∈ { i + 1 , . . . , N } , a ji u i < a ii u j , a ( i +1) k u i < a ik u i +1 . Then, for each i ∈ I N \ { N } , the intersection point of Γ i with X i is below Γ j for all j ∈{ i + 1 , . . . , N } and Γ i ∩ ( X i +1 × · · · × X N ) is strictly below Γ i +1 . Thus, Γ N − ∩ ( X N − × X N )is strictly below Γ N , Γ N − ∩ ( X N − × X N − × X N ) is strictly below Γ N − and Γ N , . . . ,Γ i ∩ ( X i × · · · × X N ) is strictly below Γ j for all j ∈ { i + 1 , . . . , N } . By Theorem 3.2, the N th species is dominant and Q N = u N a NN e N is globally asymptotically stable.VI. The standard Atkinson-Allen models are systems (1) for maps T : C → C of the form(2): T i ( x ) = x i f i ( x ), where(32) ∀ i, j ∈ I N , f i ( x ) = c + 2(1 − c )1 + P Nk =1 a ik x k , < c < , a ij ≥ , a ii > . Note that f defined by (32) is a special case of (25) with c i = c and u i = 1 for all i ∈ I N .Thus, the results obtained above for generalised Atkinson-Allen models can be appliedto the standard Atkinson-Allen models with simplified conditions ( u i , u j replaced by 1 in(29)–(31)). For these models with a ij > i, j ∈ I N , see [5], [15] and the referencestherein for further results.4.3. The competitive Ricker models.
The competitive Ricker models are systems (1)for maps T : C → C of the form (2): T i ( x ) = x i f i ( x ), where(33) ∀ i, j ∈ I N , f i ( x ) = exp " u i − N X k =1 a ik x k ! , u i > , a ij ≥ , a ii > . Under the conditions that a ij > i, j ∈ I N and(34) ∀ i ∈ I N , u i < a ii / N X j =1 a ij , or ∀ i ∈ I N , u i < / N X j =1 a ij a jj , Gyllenberg et al [8] have shown that each such model admits a carrying simplex.
ARRYING SIMPLEX 21
I. We check that, under (34), each Ricker model with a ij ≥ a ii > x ∈ X i , f i ( x ) = 1 if and onlyif x i = a ii = q i , so T restricted to X i has a unique fixed point Q i = q i e i . (ii) For i, j ∈ I N , ∂f i ∂x j = − u i a ij f i ( x ) , so ∂f i ∂x j ≤ ∂f i ∂x i <
0. (iii) For all x ∈ [0 , q ], we have ∀ i ∈ I N , f i ( x ) + N X j =1 x j ∂f i ∂x j = f i ( x )[1 − u i N X j =1 a ij x j ] ≥ f i ( x )[1 − u i N X j =1 a ij a jj ] > , or ∀ i ∈ I N , f i ( x ) + x i N X j =1 ∂f i ∂x j = f i ( x )[1 − u i x i N X j =1 a ij ] ≥ f i ( x )[1 − u i a ii N X j =1 a ij ] > . By Remark 2.2 (c), conditions (i)–(iii) of Theorem 2.4 and (11) are all met for any r ≫ q .Then, by Theorem 2.4, each Ricker model with (34) has a modified carrying simplexΣ.The surfaces Γ i are now ( N − C : ∀ i ∈ I N , Γ i = { x ∈ C : a i x + · · · + a iN x N = 1 } . II. If for some i ∈ I N , the following inequalities hold:(35) ∀ j ∈ I N \ { i } , ∀ k ∈ I N , a jk < a ik , then the intersection point of Γ i with each positive half axis X k is below Γ j for every j ∈ I N \ { i } . So Γ i is strictly below Γ j in C for all j ∈ I N \ { i } . By Theorem 3.1 (a), the i th species is dominated.III. If for some i ∈ I N ,(36) ∀ j ∈ I N \ { i } , ∀ k ∈ I N , either a ik = a jk = 0 or a ik < a jk , then either X k is parallel to both Γ i and Γ j or the intersection point of Γ j with X k is belowΓ i , so Γ i is strictly above Γ j for all j ∈ I N \ { i } . By Theorem 3.1 (b), the i th species isdominant and the fixed point Q i is globally asymptotically stable.IV. Note that (36) is much stronger than the requirement of Theorem 3.1 (b). Similar to(25), we can easily construct a three-dimensional Ricker model as an example which fails(36) but satisfies the condition of Theorem 3.1 (b).V. Now suppose the following inequalities hold:(37) ∀ i ∈ I N \ { N } , ∀ j, k ∈ { i + 1 , . . . , N } , a ji < a ii , a ( i +1) k < a ik . Then, for each i ∈ I N \ { N } , the intersection point of Γ i with X i is below Γ j for all j ∈{ i + 1 , . . . , N } and Γ i ∩ ( X i +1 × · · · × X N ) is strictly below Γ i +1 . Thus, Γ N − ∩ ( X N − × X N ) is strictly below Γ N , Γ N − ∩ ( X N − × X N − × X N ) is strictly below Γ N − and Γ N , . . . ,Γ i ∩ ( X i × · · · × X N ) is strictly below Γ j for all j ∈ { i + 1 , . . . , N } . By Theorem 3.2, the N th species is dominant and Q N = a NN e N is globally asymptotically stable.4.4. General competitive models with plane nullclines.
In [12], the competitivemodels given by system (1) for maps T : C → C of the form (2) T i ( x ) = x i f i ( x ) areconsidered, where(38) ∀ i ∈ I N , f i ( x ) = G i (( Ax ) i )with(39) A = a ii a · · · a N a a · · · a N · · · · · · · · · · · · a N a N · · · a NN satisfying a ii > a ij ≥
0, ( Ax ) i denoting the i th component of Ax . Assume that thefunctions G i ∈ C ( R + , R + ) satisfy the following conditions:(a1) Each G i is positive and strictly decreasing with G i ( u i ) = 1 and G ′ i ( u i ) < u i > x ∈ C and each i ∈ I N , ∂T i ∂x i > ≤ x i ≤ u i a ii = q i .Then each nullcline surface Γ i is a hyperplane given by (28). Under (a1), (a2) and anothercondition, criteria are established in [12] for global stability of a fixed point by geometricmethod of using the relative positions of the nullcline planes in [0 , q ]. The matrix M ( x )defined by (4) is M ( x ) = − diag( x i G ′ i (( Ax ) i ) G i (( Ax ) i ) ) A .I. Assume that(40) ∀ x ∈ [0 , q ] , ρ ( M ( x )) < . We check that each such model has a unique modified carrying simplex Σ by Theorem 2.4.(i) For x ∈ X i , G i (( Ax ) i ) = 1 if and only if ( Ax ) i = u i , i.e. x i = u i a ii = q i , so T restrictedto X i has a unique fixed point Q i = q i e i . (ii) For i, j ∈ I N , ∂G i (( Ax ) i ) ∂x j = a ij G ′ i (( Ax ) i ) . By a ij ≥ a ii > ∂G i (( Ax ) i ) ∂x j ≤ G i (( Ax ) i ) is strictly decreasing in x i .Condition (iii) of Theorem 2.4 follows from (40). Then, by Theorem 2.4, each model with(38) has a modified carrying simplex Σ.By the same reasoning as that given for generalised Atkinson-Allen models, we obtain thefollowing conclusions.II. If (29) holds, then the i th species is dominated. ARRYING SIMPLEX 23
III. If (30) holds, then the i th species is dominant and Q i is globally asymptotically sta-ble.IV. If (31) holds, then the N th species is dominant and Q N is asymptotically stable.5. Proof of the main theorems
In this section, we aim at providing complete proofs for Theorem 2.3, Theorem 2.4 andCorollary 2. Although some of the ideas used here are credited to [9] and [19, 20], forreaders’ convenience we present an independent proof rather than citing some lemmas andtheorems and modifying their proofs bit by bit. However, this does not mean that theproofs are trivial modifications from those in the references. Actually, the author’s maincontribution in this paper is the sharp observation that the system permits a modifiescarrying simplex if the retrotone property for T is relaxed to weakly retrotone, which leadsto the dramatic relaxation of the conditions of Theorem 1.2 to those of Theorem 2.4. Toprove these results, in addition to inheriting some techniques shown in Lemma 5.2, Lemma5.3, the main part of the proof of Theorem 2.3 and a small part of the proof of Theorem2.4, the author’s own methods and techniques are reflected in Lemma 5.1, Lemma 5.4,Lemma 5.5, the main part of the proof of Theorem 2.4 and Corollary 2. Lemma 5.1.
Assume that T satisfies the conditions of Theorem 2.3. Then, for any x ∈ [0 , r ] , [0 , T ( x )] ⊂ T ([0 , r ]) .Proof. By Remark 2.1 (b) we know that T is a homeomorphism from [0 , r ] to T ([0 , r ]).Thus, T maps an open set of [0 , r ] to an open set of T ([0 , r ]). Clearly, the set[0 , r ) = { x ∈ [0 , r ] : 0 ≤ x ≪ r } is open in [0 , r ], so T ([0 , r )) is also open in T ([0 , r ]). We first show that(41) ∀ x ∈ [0 , r ) , [0 , T ( x )] ⊂ T ([0 , r )) . Suppose (41) is not true. Then, for some x ∈ [0 , r ) \{ } , there is a y satisfying 0 < y < T ( x )but y T ([0 , r )). Since T ([0 , r )) is open and T ( x ) ∈ T ([0 , r )), there is an s ∈ [0 ,
1) suchthat y ( s ) = y + s ( T ( x ) − y ) ∈ T ([0 , r )) for s ∈ ( s ,
1] but y ( s ) = y + s ( T ( x ) − y ) T ([0 , r )).As T is weakly retrotone, we have 0 < z ( s ) = T − ( y ( s )) < x for s ∈ ( s ,
1) and z ( s ) 1. Thus, lim s → s + z ( s ) exists. Define z ( s ) = lim s → s + z ( s ).Then z ( s ) ∈ [0 , x ] ⊂ [0 , r ) so T ( z ( s )) ∈ T ([0 , r )). By continuity of T , T ( z ( s )) = lim s → s + T ( z ( s )) = lim s → s + y ( s ) = y ( s ) , a contradiction to y ( s ) T ([0 , r )). This contradiction shows the truth of (41).Now we show that [0 , T ( x )] ⊂ T ([0 , r ]) for all x ∈ [0 , r ]. This is true by (41) if x ∈ [0 , r ),so we suppose x ∈ [0 , r ] \ [0 , r ). Then T ( x ) ∈ T ([0 , r ] \ [0 , r )) and x ( s ) = sx ∈ [0 , r ) for all s ∈ [0 , 1) with lim s → − x ( s ) = x . Moreover, by (41), [0 , T ( x ( s ))] ⊂ T ([0 , r )) ⊂ T ([0 , r ]) forall s ∈ [0 , y ∈ [0 , T ( x )], if y ∈ [0 , T ( x ( s ))] for some s ∈ [0 , 1) then y ∈ T ([0 , r ]);if y [0 , T ( x ( s ))] for any s ∈ [0 , 1) then there is an increasing sequence { s n } ⊂ [0 , 1) with s n ↑ { y n } with y n ∈ [0 , T ( x ( s n ))] ⊂ T ([0 , r )) such that lim n →∞ y n = y .So y ∈ T ([0 , r )). But since T ([0 , r )) ⊂ T ([0 , r ]), T ([0 , r )) is open and T ([0 , r ]) is closed,we have T ([0 , r )) ⊂ T ([0 , r ]) so y ∈ T ([0 , r ]). This shows that [0 , T ( x )] ⊂ T ([0 , r ]) for all x ∈ [0 , r ]. (cid:3) For any x ∈ [0 , r ], we denote the image of x under ( T − ) k by x ( − k ) if ( T − ) k ( x ) = T − k ( x )exists. Lemma 5.2. Assume that the conditions of Theorem 2.3 hold. Suppose x ∈ [0 , r ] \ { } such that ( T − ) k ( x ) exists and x ( − k ) ∈ [0 , r ] for all k ∈ N . Then, for any y ∈ [0 , r ] with y < x and x − y ∈ ˙ C I for some nonempty I ⊂ I N , y ( − k ) exists in [0 , r ] for all k ∈ N and (42) ∀ i ∈ I, lim k →∞ y i ( − k ) = 0 . Proof. By the existence of x ( − ∈ [0 , r ] we have x = T ( T − ( x )) = T ( x ( − ∈ T ([0 , r ]).Thus, by Lemma 5.1, [0 , x ] ⊂ T ([0 , r ]). As y ∈ [0 , r ] and y < x , we have y ∈ T ([0 , r ])so y ( − 1) exists and y ( − ∈ [0 , r ]. It then follows from the weak retrotone property of T that y ( − < x ( − 1) and y i ( − < x i ( − 1) for all i ∈ I . If y ( − k ) = ( T − ) k ( y ) exists, y ( − k ) < x ( − k ) and y i ( − k ) < x i ( − k ) for all i ∈ I and some k ∈ N , by the same reasoningas above we obtain the existence of y ( − k − 1) = ( T − ) k +1 ( y ), y ( − k − < x ( − k − 1) and y i ( − k − < x i ( − k − 1) for all i ∈ I . By induction, we see the existence of y ( − k ) ∈ [0 , r ]with y ( − k ) < x ( − k ) and y i ( − k ) < x i ( − k ) for all i ∈ I and all k ∈ N .To prove (42) by contradiction, we suppose the existence of i ∈ I such that 0 < y i < x i and y i ( − k ) k → ∞ . As x ( − k ) , y ( − k ) ∈ [0 , r ] for all k ∈ N and [0 , r ] is compact, wecan select a subsequence { σ ( k ) } ⊂ { k } such thatlim k →∞ x ( − σ ( k )) = ¯ x, lim k →∞ y ( − σ ( k )) = ¯ y, ¯ y i > . By 0 < y i < x i we have 0 < y i ( − k ) < x i ( − k ) for all k ∈ N . Now define∆( k ) = y i ( − k ) x i ( − k ) , k ∈ N . Then1 > ∆( k ) = T i ( y ( − k − T i ( x ( − k − y i ( − k − f i ( y ( − k − x i ( − k − f i ( x ( − k − k + 1) f i ( y ( − k − f i ( x ( − k − . By condition (iii) of Theorem 2.3, f i ( y ( − k − f i ( x ( − k − > 1. So ∀ k ∈ N , < ∆( k + 1) < ∆( k ) < . This shows the existence of a β ∈ [0 , 1) such that lim k →∞ ∆( k ) = β . In particular,1 > β = lim k →∞ ∆( σ ( k )) = lim k →∞ y i ( − σ ( k )) x i ( − σ ( k )) = ¯ y i ¯ x i > , ARRYING SIMPLEX 25 so 0 < ¯ y i = β ¯ x i < ¯ x i . By continuity of T , T ( x ( − σ ( k ))) → T (¯ x ) and T ( y ( − σ ( k ))) → T (¯ y )as k → ∞ . Thus, β = lim k →∞ ∆( σ ( k ) − 1) = lim k →∞ y i ( − σ ( k ) + 1) x i ( − σ ( k ) + 1) = lim k →∞ T i ( y ( − σ ( k ))) T i ( x ( − σ ( k ))) = T i (¯ y ) T i (¯ x ) . From this we obtain T i (¯ y ) = βT i (¯ x ) < T i (¯ x ). As y ( − k ) < x ( − k ) for all k ∈ N , we have T ( y ( − σ ( k ))) = y ( − σ ( k ) + 1) < x ( − σ ( k ) + 1) = T ( x ( − σ ( k ))) and, as k → ∞ , T (¯ y ) < T (¯ x ).By condition (iii) of Theorem 2.3, we have f i (¯ y ) f i (¯ x ) > 1. As lim k →∞ f i ( y ( − σ ( k ))) f i ( x ( − σ ( k ))) = f i (¯ y ) f i (¯ x ) and f i ( y ( − k )) f i ( x ( − k )) > k ∈ N , there is an η > f i ( y ( − σ ( k ))) f i ( x ( − σ ( k ))) ≥ η for all k ∈ N . Thus,∆( σ ( k ) − 1) = T i ( y ( − σ ( k ))) T i ( x ( − σ ( k ))) = ∆( σ ( k )) f i ( y ( − σ ( k ))) f i ( x ( − σ ( k ))) ≥ η ∆( σ ( k )) . From this and σ ( k ) − ≥ σ ( k − 1) for k > ∀ k > , ∆( σ ( k )) ≤ η ∆( σ ( k ) − ≤ η ∆( σ ( k − . This implies that ∀ k ∈ N , β < ∆( σ ( k + 1)) ≤ η k ∆( σ (1)) . Letting k → ∞ , we obtain β = 0, a contradiction to β = ¯ y i ¯ x i > 0. This contradiction showsthe truth of (42). (cid:3) Lemma 5.3. Suppose the existence of x, y ∈ [0 , r ] \ { } with support I ( x ) = I ( y ) ⊂ I N satisfying x ( k ) ≤ y ( k ) for all k ∈ N . Then, under the conditions of Theorem 2.3, (43) lim k →∞ ( y ( k ) − x ( k )) = 0 . Proof. If x ( k ) = y ( k ) for some k ∈ N then y ( k ) = x ( k ) for all k ≥ k so (43) holds.Now assume that x ( k ) < y ( k ) for all k ∈ N . For each i ∈ I ( x ), if there is a k ∈ N such that x i ( k ) = y i ( k ), then we must have x i ( k + 1) = y i ( k + 1), for the inequality T i ( x ( k )) = x i ( k + 1) < y i ( k + 1) = T i ( y ( k )) and condition (ii) of Theorem 2.3 wouldimply x i ( k ) < y i ( k ). Thus, x i ( k ) = y i ( k ) for all k ≥ k so lim k →∞ ( y i ( k ) − x i ( k )) = 0.Now suppose for a fixed i ∈ I ( x ), ∀ k ∈ N , < x i ( k ) < y i ( k ) . Define δ ( k ) = x i ( k ) y i ( k ) for all k ∈ N . Then1 > δ ( k + 1) = T i ( x ( k )) T i ( y ( k )) = δ ( k ) f i ( x ( k )) f i ( y ( k )) . As f i ( x ( k )) f i ( y ( k )) > { δ ( k ) } is a positive increasing sequencebounded above by 1. If lim k →∞ δ ( k ) = 1 then y i ( k ) − x i ( k ) = y i ( k )[1 − δ ( k )] → k → ∞ ) . Suppose lim k →∞ δ ( k ) = β for some β ∈ (0 , k →∞ ( y i ( k ) − x i ( k )) = 0, there mustbe a subsequence { σ ( k ) } ⊂ { k } such thatlim k →∞ x ( σ ( k )) = ¯ x, lim k →∞ y ( σ ( k )) = ¯ y, ¯ x < ¯ y, ¯ x i < ¯ y i . Then,(44) 1 > δ ( σ ( k + 1)) ≥ δ ( σ ( k ) + 1) = T i ( x ( σ ( k ))) T i ( y ( σ ( k ))) = δ ( σ ( k )) f i ( x ( σ ( k ))) f i ( y ( σ ( k ))) . By condition (iii) of Theorem 2.3 again, f i ( x ( σ ( k ))) f i ( y ( σ ( k ))) > 1. As T ( x ( σ ( k ))) = x ( σ ( k ) + 1) < y ( σ ( k ) + 1) = T ( y ( σ ( k ))) , lim k →∞ T ( x ( σ ( k ))) = T (¯ x ) and lim k →∞ T ( y ( σ ( k ))) = T (¯ y ), we have T (¯ x ) ≤ T (¯ y ). If T i (¯ x ) < T i (¯ y ) then T (¯ x ) < T (¯ y ). By condition (iii) of Theorem 2.3, we obtain f i (¯ x ) f i (¯ y ) > 1. If T i (¯ x ) = T i (¯ y ), then 1 = T i (¯ x ) T i (¯ y ) = ¯ x i ¯ y i f i (¯ x ) f i (¯ y ) , f i (¯ x ) f i (¯ y ) = ¯ y i ¯ x i > . Therefore, there is an η > f i ( x ( σ ( k ))) f i ( y ( σ ( k ))) ≥ η for all k ∈ N . Then, from (44) weobtain δ ( σ ( k + 1)) ≥ ηδ ( σ ( k )) ≥ η k δ ( σ (1)) → + ∞ ( k → ∞ ) , a contradiction to δ ( k ) < 1. This contradiction shows the conclusion (43). (cid:3) Under the assumptions of Theorem 2.3, T ([0 , r ]) ⊂ [0 , r ]. By Remark 2.1 (b), T : [0 , r ] → T ([0 , r ]) is a homeomorphism, so T maps open sets to open sets and closed sets to closedsets. As ∀ n ∈ N , T n +1 ([0 , r ]) ⊂ T n ([0 , r ])and [0 , r ] is compact, T n ([0 , r ]) is compact for all n ∈ N . From Remark 2.1 (e) we knowthat 0 is a repellor with basin of repulsion B (0) ⊂ [0 , r ]. Lemma 5.4. Assume that the conditions of Theorem 2.3 hold. Let (45) A = ∩ ∞ n =0 T n ([0 , r ]) . Then A is nonempty, compact, invariant and A = B (0) .Proof. That A = ∅ is obvious as 0 , q i e i and all fixed points of T are in A . As each T n ([0 , r ]) is compact and any nonempty intersection of compact sets is compact, by (45) A is compact. The invariance of A follows from (45) and T ([0 , r ]) ⊂ [0 , r ]. Clearly, by(45) we see that A is the largest invariant set of T in [0 , r ]. As B (0) is an open subset of[0 , r ] and invariant, we have B (0) ⊂ A . To show that A = B (0), we take an arbitrarypoint x ∈ A \ B (0) and show that x ∈ B (0). This is trivial if x = 0 as 0 ∈ B (0). If x = 0 then there is a nonempty I ⊂ I N such that x ∈ ˙ C I . Moreover, u s = sx ≪ I x forall s ∈ (0 , A , x ( − k ) exists in A for all k ∈ N . Then, by Lemma ARRYING SIMPLEX 27 u s ( − k ) exists in [0 , r ] for all s ∈ (0 , 1) and all k ∈ N and lim k →∞ u s ( − k ) = 0. Thus, u s ∈ B (0) for all s ∈ (0 , x = lim s → − u s , we have x ∈ B (0). (cid:3) With the help of Lemmas 5.1–5.4 we are now in a position to prove Theorem 2.3. Proof of Theorem 2.3. Let Σ = B (0) \ ( { } ∪ B (0)). We verify that Σ is a modified carryingsimplex. Clearly, Σ ⊂ [0 , r ] \ { } and Σ = ∅ as all the nontrivial fixed points are in Σ.From Lemma 5.4 we see that B (0) is compact and invariant. As { } ∪ B (0) is open andinvariant, Σ is compact and invariant. That T : Σ → Σ is a homeomorphism follows from T being a homeomorphism from [0 , r ] to T ([0 , r ]).To show that Σ is homeomorphic to ∆ N − by radial projection, we define a map m :∆ N − → Σ as follows. For each x ∈ ∆ N − , as λx ∈ B (0) for sufficiently small λ > B (0) is open, there is a unique λ = λ ( x ) > λx ∈ B (0) for all 0 < λ < λ but λ x 6∈ B (0). Since lim λ → λ λx = λ x = 0, we have λ x ∈ B (0) so λ x ∈ Σ. We claim that λx A for λ > λ , where A is given by (45). Indeed, if there is a λ > λ such that u = λ x ∈ A , then u ( − k ) ∈ A for all k ∈ N . By Lemma 5.2, we would have λ x ∈ B (0),a contradiction to λ x 6∈ B (0). Thus, ∀ x ∈ ∆ N − , Σ ∩ { λx : λ > } = { λ ( x ) x } . Then the map m : ∆ N − → Σ defined by m ( x ) = λ ( x ) x is a bijection. The map m is ahomeomorphism if m and m − are continuous.To show that m is continuous, we need only show that λ : ∆ N − → R + is continuous.Suppose λ is not continuous at a point x ∈ ∆ N − , i.e. lim x → x λ ( x ) = λ ( x ). Since λ is obviously bounded, there is a sequence { x k } ⊂ ∆ N − such that x k → x and λ ( x k ) → µ = λ ( x ) as k → ∞ . Then { m ( x k ) } ⊂ Σ and m ( x k ) = λ ( x k ) x k → µx as k → ∞ . Since Σ is compact, we have µx ∈ Σ. This contradicts Σ ∩ { λx : λ > } = { λ ( x ) x } 6 = { µx } . This contradictionshows the continuity of m on ∆ N − .To show that m − : Σ → ∆ N − is continuous, since the continuity of m implies that Σ is acontinuous surface, for each y ∈ Σ, there is a unique µ = µ ( y ) > µ ( y ) y ∈ ∆ N − so that m − ( y ) = µ ( y ) y . Then the continuity of m − follows from showing the continuityof µ : Σ → R + by the same technique as above. Therefore, Σ is homeomorphic to ∆ N − by radial projection.Next, we show that for each x ∈ [0 , r ] \ { } , if x is above Σ then ω ( x ) ⊂ Σ; if x is below Σthen there is a y ∈ Σ with support I ( y ) = I ( x ) such that(46) lim k → + ∞ [ x ( k ) − y ( k )] = 0 . Now suppose x is above Σ. By Lemma 5.4 we have ω ( x ) ⊂ A . As B (0) does not containany positive limit point and 0 ω ( x ), we must have ω ( x ) ⊂ Σ. Next, suppose x is below Σ with support I ( x ) ⊂ I N . As B (0) = { } ∪ B (0) ∪ Σ, we haveΣ − = { } ∪ B (0). By x > 0, we must have x ∈ B (0). Define sets ∀ k ∈ N , U ( k, x ) = { y ∈ Σ : I ( y ) = I ( x ) , x ( k ) < y ( k ) } . Note that T ( B (0)) = B (0) and T (Σ) = Σ. For each fixed k ∈ N , x ( k ) is below Σ so thereis µ > µ x ( k ) ∈ Σ but µx ( k ) ∈ B (0) for 1 ≤ µ < µ . Taking y = T − k ( µ x ( k ))we have y ∈ Σ, I ( y ) = I ( x ) and y ( k ) = µ x ( k ) > x ( k ). Thus, y ∈ U ( k, x ) so U ( k, x ) = ∅ .For each z ∈ U ( k + 1 , x ) we have T ( x ( k )) = x ( k + 1) < z ( k + 1) = T ( z ( k )) . As T is weakly retrotone, we must have x ( k ) < z ( k ) so z ∈ U ( k, x ). This shows that ∀ k ∈ N , U ( k + 1 , x ) ⊂ U ( k, x ) . From the definition we see that each U ( k, x ) is compact. Then ∅ 6 = ∩ ∞ k =0 U ( k, x ) ⊂ Σ.Taking any y ∈ ∩ ∞ k =0 U ( k, x ) we obtain x ( k ) < y ( k ) for all k ∈ N so (46) follows fromLemma 5.3.So far we have proved that Σ = B (0) \ ( { } ∪ B (0)) is a modified carrying simplex. Now foreach p ∈ Σ and every q ∈ [0 , r ] \ { } with q < p , by Lemma 5.2 we know that q ( − k ) ∈ [0 , r ]exists for all k ∈ N and lim k → + ∞ q i ( − k ) = 0, so α ( q ) ⊂ π i , for any i ∈ I N with q i < p i .Finally, we show the uniqueness of the modified carrying simplex Σ. Suppose we haveanother modified carrying simplex Σ = Σ. Then, on a half line starting from the originwe have two distinct points p ∈ Σ and q ∈ Σ so there is a positive number λ = 1 such that p = λq . Clearly p and q have the same support I ⊂ I N so we have either p ≪ I q or q ≪ I p .In the first case, by Lemma 5.2 we would have α ( p ) = { } , a contradiction to α ( p ) ⊂ Σ as0 Σ. In the second case, by Lemma 5.2 again we would have α ( q ) = { } , a contradictionto α ( q ) ⊂ Σ as 0 Σ . This shows that Σ is the unique modified carrying simplex. (cid:3) To prove Theorem 2.4, we need the following lemma. Lemma 5.5. Let U be a small neighbourhood of [0 , r ] and T ∈ C ( U, U ) . Assume that theJacobian matrix DT ( x ) is invertible on [0 , r ] with ( DT ( x )) − = ( t ij ) . If ∀ x ∈ U, ∀ i, j ∈ I N , t ii ( x ) > t ij ( x ) ≥ , then T from U to T ( U ) is one-to-one and is weakly retrotone on [0 , r ] .Proof. Since DT ( x ) is continuous on U and invertible on [0 , r ], there is a small neighbour-hood U ⊂ U of [0 , r ] such that DT ( x ) is invertible on U . Without loss of generality, weassume that U = U . By the inverse function theorem, T from U to T ( U ) is one-to-oneand invertible. Moreover, T − on T ( U ) is differentiable. As g ( u ) = u = T − ( T ( u )) for u ∈ U , by the chain rule of differentiation we have I = Dg ( u ) = [ D ( T − )( T ( u ))][ DT ( u )] , ARRYING SIMPLEX 29 so(47) D ( T − )( T ( u )) = ( DT ( u )) − = ( t ij ( u )) . Now for any x, y ∈ [0 , r ] with T ( x ) < T ( y ) and T ( y ) − T ( x ) ∈ ˙ C I for some nonempty I ⊂ I N , we have y − x = T − ( T ( y )) − T − ( T ( x ))= T − ( T ( x ) + s ( T ( y ) − T ( x ))) | = Z dds T − ( T ( x ) + s ( T ( y ) − T ( x ))) ds = Z D ( T − )( T ( x ) + s ( T ( y ) − T ( x ))) ds ( T ( y ) − T ( x )) . By the assumption on the entries of ( DT ( u )) − and (47), the the diagonal entries of thematrix D ( T − )( T ( x ) + s ( T ( y ) − T ( x ))) are positive and other entries are nonnegative. Asthe matrix Z D ( T − )( T ( x ) + s ( T ( y ) − T ( x ))) ds maintains the same feature as D ( T − )( T ( x )+ s ( T ( y ) − T ( x ))), for each i ∈ I , T ( y ) − T ( x ) > T i ( y ) − T i ( x ) > y − x > y i − x i > 0. Thus, T on [0 , r ] is weaklyretrotone. (cid:3) Proof of Theorem 2.4. We need only show that conditions (ii) and (iii) of Theorem 2.4imply conditions of (ii) and (iii) of Theorem 2.3. Since DT ( x ) = diag( f ( x ) , . . . , f N ( x ))( I − M ( x )) , where I is the identity matrix and M ( x ) is given by (4), if ρ ( M ( x )) < DT ( x ) is invertible with( DT ( x )) − = ( I − M ( x )) − diag (cid:0) f ( x ) , . . . , f N ( x ) (cid:1) = (cid:0) I + ∞ X k =1 M k ( x ) (cid:1) diag (cid:0) f ( x ) , . . . , f N ( x ) (cid:1) . From this it is clear that each diagonal entry of ( DT ( x )) − is positive and other entriesare nonnegative. Then condition (ii) of Theorem 2.3 follows from Lemma 5.5.Now suppose ρ ( ˜ M ( x )) < ρ ( M ( x )) < 1, where ˜ M ( x ) is given by (6). If x ≫ 0, thendiag( 1 x , . . . , x N ) DT ( x )diag( x , . . . , x N ) = diag( f ( x ) , . . . , f N ( x ))( I − ˜ M ( x )) , so DT ( x ) is invertible with( DT ( x )) − = diag( x , . . . , x N )( I − ˜ M ( x )) − diag (cid:0) x f ( x ) , . . . , x N f N ( x ) (cid:1) = diag( x , . . . , x N ) (cid:0) I + ∞ X k =1 ˜ M k ( x ) (cid:1) diag (cid:0) x f ( x ) , . . . , x N f N ( x ) (cid:1) . From this we see that each diagonal entry of ( DT ( x )) − is positive and other entries arenonnegative. Then condition (ii) of Theorem 2.3 follows from Lemma 5.5.If ρ ( ˜ M ( x )) < x 0, then there is a proper subset J ⊂ I N as the supportof x . Without loss of generality, we assume that J = { , . . . , k } for some positive integer k < N (as we can always rearrange the order of the components). Let J = { k + 1 , . . . , N } and U = diag( x , . . . , x k , , . . . , U − DT ( x ) U = diag( f ( x ) , . . . , f N ( x )) (cid:18) I − M ( x ) − M ( x )0 I (cid:19) , where I and I are k × k and ( N − k ) × ( N − k ) identity matrices respectively, and M ( x ) = (cid:18) − x j f i ( x ) ∂f i ∂x j ( x ) (cid:19) k × k , for i, j ∈ J ,M ( x ) = (cid:18) − f i ( x ) ∂f i ∂x j ( x ) (cid:19) k × ( N − k ) , for i ∈ J , j ∈ J . Note that ˜ M ( x ) = (cid:18) M ( x ) 0 M ( x ) 0 (cid:19) , where M ( x ) is an ( N − k ) × k matrix. Then ρ ( M ( x )) = ρ ( ˜ M ( x )) < 1, so DT ( x ) isinvertible with( DT ( x )) − = U (cid:18) ( I − M ( x )) − ( I − M ( x )) − M ( x )0 I (cid:19) × diag (cid:0) f ( x ) , . . . , f N ( x ) (cid:1) U − . As ( I − M ( x )) − = I + P ∞ n =1 M n ( x ) with positive diagonal entries and nonnegative otherentries, each diagonal entry of ( DT ( x )) − is positive and other entries are nonnegative.Then condition (ii) of Theorem 2.3 follows from Lemma 5.5.For any x, y ∈ [0 , r ], if T ( x ) < T ( y ) and T ( y ) − T ( x ) ∈ ˙ C J for some J ⊂ I N , by the weaklyretrotone property of T we have x < y and x j < y j for all j ∈ J . By condition (ii) ofTheorem 2.4, each f i is nonincreasing in every x j but strictly decreasing in x i for x ∈ [0 , r ].Then we have f ( x ) > f ( y ) and f j ( x ) > f j ( y ) for all j ∈ J , so condition (iii) of Theorem2.3 holds. ARRYING SIMPLEX 31 Finally, we check that [0 , r ] is positively invariant. Note that Remark 2.1 (b) and (c) donot reply on the positive invariance of [0 , r ]. Then, for each x ∈ [0 , r ] and every i ∈ I N , by(ii) of Theorem 2.4 we have T i ( x ) ≤ T i ( x i e i ). By Remark 2.1 (c), T i ( x i e i ) is increasing for x i ∈ [0 , r i ], so T i ( x ) ≤ T i ( x i e i ) ≤ T i ( r i e i ) = r i f i ( r i e i ) < r i . This shows that T ( x ) ≪ r and T ([0 , r ]) ⊂ [0 , r ]. (cid:3) Proof of Corollary 2. For any bounded set B ⊂ [0 , r ] \ { } with B ⊂ [0 , r ] \ { } , there isa small δ > O (0 , δ ) ∩ B = ∅ and O (0 , δ ) ∩ [0 , r ] is strictly below Σ. Since0 is a repellor, B (0) is invariant by Lemma 5.4, [0 , r ] is positively invariant, and T from[0 , r ] to T ([0 , r ]) is a homeomorphism by Remark 2.1 (b), for δ > , r ] \ O (0 , δ ) is positively invariant with B ⊂ [0 , r ] \ O (0 , δ ) and O (0 , δ ) ∩ [0 , r ] is strictlybelow Σ. Then, for each n ∈ N , T n ([0 , r ] \ O (0 , δ )) is compact andΣ ⊂ T n +1 ([0 , r ] \ O (0 , δ )) ⊂ T n ([0 , r ] \ O (0 , δ )) . From this follows Σ ⊂ ∞ \ n =0 T n ([0 , r ] \ O (0 , δ )) . We claim that(48) Σ = ∞ \ n =0 T n ([0 , r ] \ O (0 , δ )) . Indeed, from Lemma 5.4 we know that ∩ ∞ n =0 T n ([0 , r ] \ O (0 , δ )) ⊂ A = B (0) = { } ∪ B (0) ∪ Σ . If (48) is not true, then there is a point p ∈ ( ∩ ∞ n =0 T n ([0 , r ] \ O (0 , δ ))) \ Σ, so T − n ( p ) ∈ [0 , r ] \ O (0 , δ ) for all n ∈ N . This shows that lim n →∞ T − n ( p ) = 0. On the other hand,however, as p 6∈ { } ∪ Σ, we must have p ∈ B (0) so lim n →∞ T − n ( p ) = 0. This contradictionshows the truth of (48).Now from (48) we see that Σ attracts the points of [0 , r ] \ O (0 , δ ) uniformly. As B ⊂ [0 , r ] \ O (0 , δ ), Σ attracts the points of B uniformly. Therefore, Σ is a global attractor in[0 , r ] \ { } under the conditions of Theorem 2.3 or Theorem 2.4.Under the additional condition (11), for any bounded set B ⊂ C \ { } with B ⊂ C \ { } ,from Remark 2.3 we know the existence of an integer k > T k ( B ) ⊂ [0 , r ].By the definition of T , T k ( x ) = 0 if and only if x = 0 on C . As 0 B , 0 T k ( B ) so T k ( B ) ⊂ [0 , r ] \ { } . From the previous paragraph we know that Σ attracts the pointsof T k ( B ) uniformly. Thus, Σ attracts the points of B uniformly. Hence, Σ is a globalattractor in C \ { } . (cid:3) Conclusion We have so far considered the discrete dynamical system (1) with the maps T defined by (2).Recall that the current available carrying simplex theory is about the existence of an ( N − C \ { } .With the existing concept of a carrying simplex for the system and the available criteriaon existence of carrying simplex as the main concern of this paper, we have successfullyachieved our goal of extending this theory to a broader class of systems: We have firstdefined the concept of a modified carrying simplex, which is a slight relaxation from theconcept of a carrying simplex and is still an ( N − C \ { } . We then have established our criteriafor existence and uniqueness of a modified carrying simplex.In comparison with the existing criteria for existence of a carrying simplex, our criteria forexistence and uniqueness of a modified carrying simplex have the following main virtue:Instead of requiring all the entries of the Jacobian Df ( x ) to be negative for all x ∈ [0 , q ], weonly require each entry of Df ( x ) to be nonpositive and each f i ( x ) to be strictly decreasingin x i . Thus, we have significantly reduced the cost of having an ( N − C \{ } . In other words, our criteriacan be applied to a broader class of systems as competitive models.The significance of the carrying simplex theory lies in that the global dynamics of thesystem in C can be described by the dynamics on the modified carrying simplex Σ. Asone application of this theory, we have investigated vanishing species and dominance ofone species over others. Assuming the existence of a modified carrying simplex, we haveobtained sufficient geometric conditions for one or more species to die out. We have alsoobtained conditions for one species to dominate all others and one axial fixed point to beglobally asymptotically stable.Above all, with our theorems for modified carrying simplex Σ, we have laid the foundationfor exploring the global dynamics of the system. We expect future research work will beflourishing based on modified carrying simplex. Open Problem Suppose system (1) with T defined by (2) satisfies the conditions ofTheorem 2.4, so the system permits a modified carrying simplex Σ. Is it possible toconstruct a sequence { T [ k ] } satisfying the following conditions?(i) For each integer k > 0, the map T [ k ] from [0 , r ] to C has the form (2).(ii) Each T [ k ] on [0 , r ] meets the requirements of Theorem 1.2, so system (1) with T replaced by T [ k ] permits a carrying simplex Σ [ k ] .(iii) As k → ∞ , T [ k ] ( x ) → T ( x ) uniformly for x ∈ [0 , r ](iv) As k → ∞ , Σ [ k ] → Σ in the following sense: ∀ ε > , ∃ K > , ∀ k ≥ K, Σ [ k ] ⊂ O (Σ , ε ) . ARRYING SIMPLEX 33 If the answer is YES, then our Theorem 2.4 can be viewed as the result of a limit processfrom Theorem 1.2, i.e. system (1) with (2) satisfying the conditions of Theorem 2.4 canbe approximated by systems satisfying the conditions of Theorem 1.2. Acknowledgements The author consulted Professor Stephen Baigent on this topic and is grateful for his encour-agement of writing up this paper. The author is also grateful to the referees and editorsfor their comments and suggestions adopted in this version of the paper. References [1] S. Baigent, Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems ,J. Differ. Equ. Appl., 22 (5) (2016), 609–620.[2] S. Baigent, Convex geometry of the carrying simple for the May-Leonard map , Discret. Contin. Dyn.Syst. B, 24 (4) (2019), 1697–1723.[3] S. Baigent and Z. Hou, Global stability of discrete-time competitive population models , J. DifferenceEqu. Appl., (2017), 1378–1396.[4] E. Cabral Balreira, S. Elaydi and R. Lu´ıs, Global stability of higher dimensional monotone maps , J.Difference Equ. Appl., (2017), 2037–2071.[5] O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations , Discrete Contin. Dyn. Syst., (2008), 37–52.[6] M. Gyllenberg, J. Jiang, L. Niu and P. Yan, On the classification of generalised competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex , Discret. Contin. Dyn. Syst., (2018), 615–650.[7] M. Gyllenberg, J. Jiang and L. Niu, A note on global stability of three-dimensional Ricker models , J.Difference Equ. Appl., (2019), 142–150.[8] M. Gyllenberg, J. Jiang, L. Niu and P. Yan, On the dynamics of multi-species Ricker models admittinga carrying simplex , J. Difference Equ. Appl., (2019), 1489–1530.[9] A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex , J.Differ. Equ. Appl. (1) (2013), 96–113.[10] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. III: Competingspecies , Nonlinearity (1988), 51-71.[11] M. W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems ,J. Biol. Dyn. (2008), 169–179.[12] Z. Hou, Geometric method for global stability of discrete population models , Discret. Contin. Dyn. Syst.B, 25 (9) (2020) 3305–3334.[13] Z. Hou, On existence and uniqueness of a carrying simplex in Kolmogorov differential systems , Non-linearity (2020) 7067–7087.[14] J. Jiang, J. Mierczy´nski and Y. Wang, Smoothness of the carrying simplex for discrete-time competitivedynamical systems: A characterization of neat embedding , J. Differ. Equ. (4) (2009), 1623–1672.[15] J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Atkinson-Allenmodels relative to the boundary fixed points , Discret. Contin. Dyn. Syst. (1) (2016), 217–244.[16] J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie-Gowermodels via the boundary dynamics on the carrying simplex , J. Math. Biol. (2017), 1223–1261.[17] J. Jiang, L. Niu and Y. Wang, On heteroclinic cycles of competitive maps via carrying simplices , J.Math. Biol. (2016), 939–972.[18] Peter Lancaster, Theory of Matrices , Academic Press, New York and London, 1969.[19] Y. Wang and J. Jiang, The general properties of discrete-time competitive dynamical systems , J. Differ.Equ. (2001), 470–493. [20] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitivedynamical systems , J. Differ. Equ. (2002), 611–632. Email address ::