Existence of the carrying simplex for a retrotone map

We present a dynamical approach to the study of unordered, attracting manifolds of retrotone maps commonly known as carrying simplices. Our approach is novel in that it uses the radial representation of unordered manifolds over the probability simplex coupled with distances between these manifolds measured by way of the Harnack and Hausdorff metrics. We establish Kuratowski convergence of radial representations of unordered manifolds to a unique function which then provides the locally Lipschitz radial representation of the carrying simplex.


Introduction
There is a large class of maps widely used to study discrete-time population dynamics that preserve all faces of the first orthant of Euclidean space that are known as Kolmogorov maps.By varying the functional forms of these maps, all the possible types of species interactions for populations where there is no overlap of generations can be modelled, including predator-prey, mutualism and competition to name a few.Here we are interested in a special subclass of those Kolmogorov maps that feature only competition which, following recent research trends, we call retrotone.While much of the existing research uses competitive map where we use retrotone map -indeed the distinction is not crucial (necessary) for continuous time analogues -we prefer to use 'retrotone' to describe maps that represent competitive interactions that also have the specific feature that they preserve a convex cone backwards in time.The Leslie-Gower map is an example of a competition model that is globally a retrotone map [18], but not all competitive maps are retrotone, even on their global attractor; the planar Ricker map discussed later in Subsection 3.2 is a classic example of a map that represents competitive interactions, but is only retrotone for a limited set of parameter values.
The focus of the present work is the carrying simplex which is a well-studied feature of population models with competitive interactions [7,11,12,23,26,27,31].The carrying simplex is a Lipschitz, codimension-one and compact invariant manifold that attracts all nonzero points, and that is unordered, which means that the carrying simplex is non-increasing in each coordinate direction.As we discuss later there are several ways of characterising the carrying simplex, including as the set of nonzero points with globally defined and bounded forward and backward orbits, or as the relative boundary of the global attractor of bounded sets.The presence of the carrying simplex (which is asymptotically complete) means that the limiting dynamics can be studied on the carrying simplex which gives rise to a system of one fewer degree of freedom.This has been exploited by a number of authors to study the global dynamics of both continuous and discrete-time population models.For example, [10,17,18] study global dynamics of maps with a carrying simplex by way of an index theorem.Baigent and Hou [3,4] utilise the carrying simplex in both continuous and discrete time models to construct Lyapunov functions on forward invariant sets, and Montes de Oca and Zeeman [22] use the carrying simplex concept iteratively to reduce a continuous-time competition model to an easily solved one-dimensional model.
That there exists such an invariant manifold under ecologically reasonable assumptions is intuitive.We impose that the origin is unstable, which means that small population densities grow; all the species cannot simultaneously go extinct.We also require that the per-capita growth rate of a given species decreases with increasing densities of all species (although we relax this to non-increasing for some densities later).This can be thought of as modelling competition for resources.In differential equation models for competition, these assumptions are typically sufficient for a carrying simplex to exist [11,14], but not in the discrete time case.We follow other authors [12,15,26,31] and add further conditions that render the map retrotone.Roughly speaking the retrotone property, which can be imposed through spectral properties of the derivative of the map [12,15,26], aside from modelling competitive interactions, puts restrictions on the maximal change in total population density over one generation, i.e. large radial changes under one application of the map are not permitted.The retrotone properties of the map render the carrying simplex unordered, so that it projects radially onto the probability simplex.Hence on the carrying simplex, given the frequency of the species, i.e. a point in the probability simplex, the radial coordinate, which is the total population density, is determined.
We argue that the radial representation, in which phase space is the cartesian product of the probability simplex and the positive real line, is a natural coordinate choice, and it is the main set of coordinates that we use here.In this description the carrying simplex is just the graph of a continuous functions, locally Lipschitz on the interior of the phase space, over the probability simplex.However, working with the radial representation presents technical difficulties at the boundary of phase space where derivatives can become unbounded.To resolve this issue we use the Kuratowski metric to establish Hausdorff convergence of the unordered manifolds generated by the graph transform approach.We generate one increasing and one decreasing sequence of Kuratowski convergent sequences of unordered manifolds and then we utilise the Harnack metric to show that the two limits are identical and identified as the carrying simplex.
The paper is organised as follows.In Section 1 we introduce our notation and give important definitions, such as for cone-orderings, unordered and weakly unordered sets, retrotone maps and weakly retrotone maps, and attractors of various classes of sets.
In particular, a definition of the carrying simplex is proposed (Definition 2.13).Since no standard definition has been so far settled on, we choose to base our proposal on definitions in [12] and [26], perhaps with more dynamical flavour added (property (iv)).We mention also some additional properties, (vi) up to (ix), that have appeared in some earlier papers.We will see later that all those properties are satisfied under our assumptions.Section 2 is concerned with proof of the existence of the carrying simplex.Subsection 2.1 deals with the dynamics of the map restricted to the boundary, and is crucial for establishing the existence of a bounded rectangle on which the map is retrotone/weakly retrotone and which contains the unique compact attractor of bounded sets.Conditions under which the map is retrotone/weakly retrotone on the bounded rectangle are established in Subsection 2.7 and Subsection 2.2 characterises the compact attractor of bounded sets of the map.In Subsection 2.3 we construct the carrying simplex/weak carrying simplex from sequences of unordered/weakly unordered manifolds using the graph transform and Kuratowski convergence radial coordinates, as well as give some of its properties.Attractor-repeller pairings are used to establish further properties of the carrying simplex/weak carrying simplex in Subsections 2.4, and Subsections 2.5 and 2.6 deal with asymptotic completeness and a Lipschitz representation of the carrying simplex.In Section 3 we discuss some examples that illustrate the main ideas of the paper.Finally in Section 4 we show how to use our results for retrotone [weakly retrotone] maps to derive conditions for the existence of a carrying simplex [weak carrying simplex] for competitive systems of ordinary differential equations.

Notation and definitions
We will need the following notation and definitions.
and for nonempty compact A, B ⊂ R d we denote by d H (A, B) their Hausdorff distance, . ., d} will denote convex cones.C + is often referred to as the first orthant and C ++ is its interior.
We denote N = {0, 1, 2, 3, . . .}.For a subset I ⊂ {1, . . ., d}, let For x, y ∈ C + , we write x ≤ y if x i ≤ y i for all i = 1, . . ., d, and x ≪ y if x i < y i for all i = 1, . . ., d.If x ≤ y but x ̸ = y we write x < y.The reverse relations are denoted by ≥, >, ≫.Two points x, y ∈ C + are said to be order-related if either x ≤ y or y ≤ x.
For x, y ∈ C + such that x ≤ y we define the order interval as We An important concept needed to describe the carrying simplex is that of unordered sets.We also use a weaker notion of weakly unordered sets (introduced (but not in name) in [15,Rem.
The set H(a) is weakly unordered.To show this, suppose to the contrary that there are a nonempty I ⊂ {1, . . ., d} and x, y ∈ H(a) ∩ (C I ) + with x ≪ I y, which means that x i < y i for all i ∈ I and x j = y j = 0 for all j ∈ {1, . . ., d} \ I.But then x i < a for all 1 ≤ i ≤ d, so x cannot belong to H(a).On the other hand, H(a) is not unordered: for example, ae 1 < ae and both belong to H(a).
Another important concept in the theory of carrying simplices is that of a retrotone map.Retrotonicity is the property that ensures that ordered points are ordered along backward orbits.
x, y ∈ B with F (x) < F (y), one has that x i < y i provided y i > 0.
A map F : C + → C + is weakly retrotone in B ⊂ C + provided for all I ⊂ {1, . . ., d} and any x, y ∈ B, if F (x) < F (y) and F i (x) < F i (y) for all i ∈ I then x < y and x i < y i for all i ∈ I.
From now on, we assume that F = (F 1 , . . ., F d ) : C + → C + is a continuous map.We will consider the dynamical system (F n ) ∞ n=0 on C + , where F 0 = Id C+ and for convenience we will write F n = (F n 1 , . . ., F n d ).For x ∈ C + we denote its forward orbit, O + (x), as By a backward orbit of x ∈ C + we understand a set {. . ., x −n−1 , x −n , . . ., x −2 , x −1 , x 0 } such that x 0 = x and x −n = F (x −n−1 ) for all n ∈ N (so as to allow for noninvertible maps).A total orbit of x ∈ C + is the union of a backward orbit of x and the forward orbit, O + (x).
Following the terminology as in [28] we say that B ⊂ C + attracts the set A ⊂ C + if for each ϵ > 0 there is n 0 ∈ N such that dist(F n (x), B) < ϵ for all n ≥ n 0 and all x ∈ A.
For a set A ⊂ C + define its ω-limit set as For B = C + we say simply compact attractor of bounded sets, which is the same as the compact attractor of neighbourhoods of compact sets.Proposition 1.6.[28, Thm.2.20 on p. 37] Let B ⊂ C + be forward invariant.The compact attractor of bounded sets is characterised as the set of all x ∈ B having bounded total orbits.
It is equivalent to the existence of a neighbourhood U of B in C + such that for each x ∈ U \ B there is n 0 ∈ N with the property that F n (x) / ∈ U for all n ≥ n 0 (see [28,Rem. 5.15 on p. 136]).
We say that F : Definition For the carrying simplex, the unorderedness in (i) appears in [11] (but not explicitly in [32] or [33]) in the case of competitive systems of ODEs, and in [27], [31], [26] in the discrete time case.For the weak carrying simplex, the weak unorderedness in (i) appears in [14] in the case of competitive systems of ODEs, and in [15] in the discrete time case.The fact that Σ is contained in the compact attractor of bounded sets Γ is seldom explicitly mentioned (as in [27]), but it follows from dissipativity assumed in other papers.
Property (ii) is usually mentioned explicitly (but in [26] the homeomorphism is defined in another way).
Invariance in (iii) is always mentioned.
To our knowledge, the only place where property (iv) has been explicitly stated is [15, p. 291].Indeed, in many papers it can be inferred from the property that the carrying simplex is obtained therein as the upper boundary of the repulsion basin of {0}, see, e.g., [27,31].
(vi) Σ is the boundary (relative to C + ) of Γ and Γ is order convex.In particular, (viii) Σ is characterised as the set of all x ∈ C + having total orbits that are bounded and bounded away from 0. (ix) The inverse (Π↾ Σ ) −1 of the orthogonal projection of Σ along e is Lipschitz con-tinuous.
As stated earlier, in the existing papers (vi) is one of the main ingredients in the proof of the existence of the carrying simplex (cf., for example, [26,Thm. 6.1]).
The characterisations given in (vii) and (viii) appear in [26].In the present paper they follow from abstract dynamical systems theory.
(ix) occurred first in [11].Since then it has seldom appeared.It has been frequently mentioned that the carrying simplex is unique.It follows from the conjunction of (i) and (ii), or from (iv) (using forward invariance of all faces), however in our approach it is simpler to use the additional property (viii).

Existence of a carrying simplex
We give two sets of assumptions which guarantee the existence of the carrying simplex (Theorem 2.2).
In the case of the first set, A1 up to A4, we start by assuming the existence of some bounded rectangle Λ such that F restricted to Λ, F ↾ Λ , is [weakly] retrotone (assumption A3).We work this way because not all maps with a carrying simplex are retrotone on all of C + (an example is given in Subsection 3.2).We prove then that F ↾ Λ , satisfies Definition 1.7 with instances of C + replaced by Λ.As [weak] retrotonicity is often difficult to prove directly, in Subsection 2.7 we give sufficient conditions, formulated in terms of the spectral radius of some matrix, for weak retrotonicity or retrotonicity to be satisfied.Those conditions are fulfilled for many discrete time competition models, as explained in Section 3.
The second set of assumptions, with A4 replaced by A4, covers the case when F is the time-one map of a competitive system of ODEs, and we utilise it in Subsection 4 to recover well-known conditions for the existence of a carrying simplex in a system of competitive ODEs.Then retrotonicity on the whole of C + is a consequence of the Müller-Kamke theory [19,24].On the other hand, A4 may be difficult to check, so it is replaced by A4.Now, the role of Λ can be played by any sufficiently large rectangle.
The main part of the present section, Subsection 2.3, contains a proof of the existence of a set Σ satisfying (i), (ii) and (iii) in Definition 1.7.Also, the additional property (vi) is proved there.
In the second step (subsection 2.4) we show that all points in C + eventually enter and stay in Λ, so that, with the help of the dynamical systems theory, Σ actually attracts any bounded A ⊂ C + with 0 / ∈ Ā (property (iv)).The map is not required to be retrotone outside Λ.As a by-product we obtain the satisfaction of the additional properties (vii) and (viii).Subsection 2.5 contains a proof of property (v) (so, only at that point can Σ be legitimately called the carrying simplex).A proof of the additional property (ix) is given in Subsection 2.6.
We make the following assumptions: for all i, and A4-b f i (x) > f i (y) for those i for which x i < y i ; In A3-a by a local homeomorphism we mean that for each x ∈ Λ there exist a relative, in Λ, neighbourhood U of x and a relative in From A1 it follows that for any ∅ ̸ = I ⊂ {1, . . ., d} there holds Remark 2. In A2, we assume that each axis has a fixed point of F at e i , but by rescaling we may deal with fixed point at q i e i for any set of q i > 0, i = 1, . . ., d.
Proof.By A3-a, the map F ↾ Λ is a local homeomorphism.Since Λ is compact, F ↾ Λ is a proper map.Further, Λ being connected, its image F (Λ) is connected, and, as it does not contain critical points (i.e., points where F is not locally invertible), we can apply [5,Lem. 2.3.4] to conclude that the cardinality of (F ↾ Λ ) −1 (y) is constant for all y ∈ F (Λ).As it follows from the Kolmogorov property and A1 that card F −1 (0) = 1, F ↾ Λ is injective, so, being continuous from a compact space, is a homeomorphism onto its image.
Sometimes instead of A3-A4 we make the following stronger assumptions: Under A3 we will occasionally need a modified form of A4, namely A4 for any x, y ∈ Λ, if F (x) < F (y) then A4-a f i (x) ≥ f i (y) for all i, and A4-b for those i for which F i (x) < F i (y) there holds either Similarly, under A3 ′ we will occasionally need a modified form of A4 ′ , namely As will be seen later, the assumptions A4 and A3 are not, in general, independent of each other.Our motivation is that we wish to strike a balance between assumptions that are reasonably general and, on the other hand, easy to check.
In particular, since A4 and A3-b imply A4, one may well ask why we have not chosen to assume the latter only.The reason is that in the case when F is given by a closedform formula and is not necessarily injective on the whole of C + (as, for instance, in the planar Ricker model, see subsection 3.2), the checking of whether A4 is satisfied could be a difficult task, whereas A4 is a simple consequence of the negativity of the relevant derivatives.On the other hand, when F is the time-one map in the semiflow generated by a competitive system of ODEs, both A4 and A3-b are fairly direct consequences of the Müller-Kamke theorem (see subsection 4), whereas we see no reason why A4 should be satisfied.
The remainder of this section is devoted to the proof of the following existence theorem for the weak carrying simplex or carrying simplex: Theorem 2.2.Under the assumptions A1-A3, and A4 or A4 where κ can be arbitrary (so that Λ could be all of C + ), there exists a weak carrying simplex Σ.If we assume additionally A3 ′ or A4 ′ , Σ is a carrying simplex.Moreover, Σ satisfies the additional properties (vi)-(ix).

Restriction of the dynamical system
The map G i is the dynamical rule F restricted to the forward invariant i-th axis, and In the remainder of the present subsection, the terms from the theory of dynamical systems, such as attractor, ω-limit set, etc. will refer to each one-dimensional dynamical system (G n i ) ∞ n=0 , i = 1, . . ., d.The following results are straightforward to prove, cf., e.g., [26,Lem. 6.6].

Existence of the compact attractor Γ of bounded sets
Throughout the present subsection we assume A1-A3.Later on, our assumptions will be successively strengthened.
Proof.We prove the lemma by induction on n.For n = 0 we have [0, a] = G 0 i ([0, a]).Assume that the inclusion holds for some n ∈ N.
In case (a), suppose to the contrary that there is x ∈ [0, ae] such that F n+1 (x) / ∈ [0, G n+1 (a)], which means that there are j ∈ {1, . . ., d} such that F n+1 j (x) > G n+1 j (a).Fix such a j.We have thus with F n (x), G n j (a)e j ∈ Λ, so, by weak retrotonicity (A3-b), F n j (x) > G n j (a), which contradicts our inductive assumption.The last equality is a consequence of Lemma 2.4.
From now on until the end of the present subsection we assume additionally A4 or A4.
Proof.The first sentences in (a) and (b) are direct consequences of Lemma 2.6.The second sentence in (a) follows since, by Lemma 2.
It should be remarked that in [27,Prop. 3.5] an analogue of Γ was defined as this same set Until the end of the present subsection, in case of A4 we assume furthermore that any positive number can serve as κ.
Lemma 2.9.For a bounded A ⊂ C + there is n 0 such that F n (A) ⊂ Λ for all n ≥ n 0 .In particular, Λ attracts bounded sets A ⊂ C + .
Proof.Let A ⊂ C + be a bounded set and choose a > 1 such that A ⊂ [0, ae].Since, by Proposition 2.5, for each i ∈ {1, . . ., d} in the dynamical system (G n i ) the set [0, 1] attracts [0, a], there exists n 0 ∈ N such that for all n ≥ n 0 and all i ∈ {1, . . ., d} the set Collecting the various results of this subsection together we obtain the following characterisation of Γ: Theorem 2.10.
(a) Γ is the compact attractor of bounded sets in C + .(b) Γ is characterised as the set of those x ∈ C + for which there exists a bounded total orbit.
A consequence of Theorem 2.10 is Lemma 2.11.For x ∈ C + the following are equivalent.
(2) There is a total orbit of x, contained in Λ.
(3) There is a bounded backward orbit of x.
Γ is the largest compact invariant set in C + (see [28, Thm.2.19 on p. 37]).Further, since Γ ⊂ Λ and, by Lemma 2.1, −∞ is a (two-sided) dynamical system on the compact metric space Γ (and Γ is the largest subset of C + with that property).

Construction of the carrying simplex
In the present subsection we always assume A1-A3, and A4 or A4.At some places we assume A3 ′ or A4 ′ .For convenience we write F instead of F ↾ Λ .Denote by T the radial projection of C + \ {0} onto the unit probability simplex ∆, T (x) := x/∥x∥ 1 .
Following [2], let U [resp.U] stand for the set of bounded and weakly unordered [resp.unordered] hypersurfaces contained in Λ that are homeomorphic to the standard probability simplex ∆ via radial projection.In particular, a hypersurface S ∈ U is at a positive distance from the origin, as the radial projection is not defined at 0.
It follows that for any S ∈ U the inverse of the restriction (T ↾ S ) −1 can be written as where R : ∆ → (0, ∞) is a continuous function (called the radial representation of S).In other words, For S ∈ U, its complement C + \ S is the union of two disjoint sets, a bounded one, S − , and an unbounded one, S + .One has S − = [0, 1)S, S + = (1, ∞)S.We will write ∂S for S ∩ ∂C + .∂S = bd C+ S.
The following consequence of the weak unorderedness of an element of U will be used several times, so we formulate it as a separate lemma.Lemma 2.12.Let S ∈ U.There are no two points x < y on S such that x i < y i for all i ∈ I(y).
Proof.Suppose to the contrary that there are such x and y.Then x, y ∈ (C I(y) ) + with x ≪ I(y) y, which contradicts the fact that S is weakly unordered.
Since F is continuous on C + and S ⊂ Λ is compact, F (S) is compact, and, by Lemma 2.7(a), F (S) is a subset of Λ ′ .Define F on Λ ′ as F = F/∥F ∥ 1 , in other words, F = T • F .Then F is a continuous map on Λ ′ , and consequently the continuous map F ↾ S : S → ∆ is proper.By Lemma 2.1, F is invertible, and T ↾ F (S) is locally invertible at each element of F (S), since otherwise F (S) would not be weakly unordered, which has been excluded in the previous paragraph.Hence F ↾ S is locally invertible at each x ∈ S, and we can apply [5,Cor. 2.3.6] to conclude that F ↾ S is a homeomorphism onto ∆.Therefore, F (S) ∈ U.
Let S ∈ U.If there are x, y ∈ S such that F (x) < F (y) then, by weak retrotonicity (A3-b), x < y, which contradicts the unorderedness of S.
Assume that A3 ′ or A4 ′ holds, and let S ∈ U. Suppose to the contrary that F (S) is not unordered, that is, there are x, y ∈ S such that F (x) < F (y).
In the case of A3 ′ it follows from retrotonicity (A3 ′ -b) that x i < y i for all i ∈ I(y) = I(F (y)), which is in contradiction to Lemma 2.12.
We consider now the case of A4 ′ .We already know that F (S) is weakly unordered, so, as a consequence of Lemma 2.12, I(F (y)) is the disjoint union of two nonempty sets, J := { i ∈ I(F (y)) : 0 < F i (x) = F i (y) } and K := { i ∈ I(F (y)) : F i (x) < F i (y) }.By weak retrotonicity (A3-b), x i ≤ y i for all i ∈ I(y) = I(F (y)), with x i < y i for all i ∈ K. Applying again Lemma 2.12, this time to S, we obtain that x j = y j for at least one j ∈ J. Fix such a j.But A4 ′ gives us f j (x) > f j (y), hence F j (x) = x j f j (x) > y j f j (y) = F j (y), which contradicts the fact that j ∈ J.
We now introduce a partial order relation for the hypersurfaces in U.For S, S ′ ∈ U, let R, R ′ denote their respective radial representations.We write We assume the convention that 0 Î S for any S ∈ U.
For S ≼ S ′ we denote and for S Î S ′ we denote In other words, ⟨S, Notice that S ≼ S ′ if and only if S ⊂ ⟨0, S ′ ⟩.
The following lemma shows that the volume ⟨S, S ′ ⟩ between two hypersurfaces in U is the union of order intervals and hence is order convex.
Lemma 2.14.For S, S ′ ∈ U with S ≼ S ′ , and is order convex.
Proof.The '⊂' inclusion is straightforward.Assume that z ∈ [x, y] with x ∈ S and y ∈ S ′ .There are ξ ∈ S and η ∈ S ′ such that z = αξ = βη.We claim that α ≥ 1 and β ≤ 1.Indeed, suppose that α < 1.Then x ≤ z ≪ I ξ, where I := I(z) = I(ξ), with both x and ξ in S. If I(x) = I then x ≪ I ξ, which contradicts the weak unorderedness of S. If not, we can find x ∈ S ∩ (C ++ ) I so close to x that x ≪ I ξ, which again contradicts the weak unorderedness of S. The case β > 1 is excluded in much the same way.We have thus z ∈ [R(T (z)), R ′ (T (z))]T (z), with ξ = R(T (z))T (z) and η = R ′ (T (z))T (z).Finally, as a consequence of the equality, ⟨S, S ′ ⟩ is order convex.
The property described in the result below has appeared in the literature, see, e.g., [16,Prop. 2.1].As the assumptions of F made in the present paper are different (e.g.weak retrotonicity), we have decided to give its reasonably complete proof.
Lemma 2.15.Assume A1-A3.Then for any x, y ∈ Λ, if F (x) ≤ F (y), then Proof.Suppose that F (x) ≤ F (y).If F (x) = F (y), then as, by Lemma 2.1, F is a homeomorphism of Λ onto its image, x = y.This leaves the case F (x) < F (y), when the statement (i) follows from A3-b.
Regarding (ii), the conclusion is obvious if y = 0. Assume y > 0 with support I = I(y).Then, by Lemma 2. Lemma 2.17.F preserves the ≼ and Î relations on U.
Proof.Let S, S ′ ∈ U, S ≼ S ′ , which means that S ⊂ ⟨0, S ′ ⟩.Then, by Lemma 2.16, Assume that S Î S ′ .By the previous paragraph, F (S) ≼ F (S ′ ), and by the fact that F ↾ Λ is a homeomorphism onto its image, F (S) and F (S ′ ) are disjoint, consequently F (S) Î F (S ′ ).
Proof.(Recall that we are not assuming that F is C 1 ).As U in the definition of uniform repeller we take ⟨0, S 0 ⟩.It follows from the choice of S 0 that there is δ > 0 such that f i (x) ≥ 1 + δ, i ∈ {1, . . ., d}, consequently ∥F (x)∥ 1 ≥ (1 + δ)∥x∥ 1 , for all x ∈ U .So, if x ∈ U \ {0} we have that there exists n 0 ∈ N with the property that F n0 (x) / ∈ U (n 0 is not larger than the least nonnegative integer n such that (1 + δ) n ∥x∥ 1 ≥ ε, where ε > 0 is as in the definition of S 0 ).Since F ↾ Λ is a homeomorphism onto its image contained in Λ, there holds F n (x) / ∈ U for all n ≥ n 0 .
The sequence S n := F n (S 0 ) ∈ U by Lemma 2.13.Denote by R n : ∆ → (0, ∞) the radial representation of S n .By our choice of S 0 , we have S 0 Î S 1 .Lemma 2.17 implies S n Î S n+1 for all n = 1, 2, . . . .As a consequence, for each u ∈ ∆ the sequence (R n (u)) ∞ n=0 is strictly increasing.Let R * stand for its pointwise limit.We define We recall the definition of Kuratowski limit.
Recall that F denotes here F ↾ Λ , so the invariance of S means that, first, if x ∈ S then F (x) ∈ S, and, second, if x ∈ Λ is such that F (x) ∈ S then x ∈ S.
Proof.Part (1) is a consequence of the fact that for the compact Λ, the Kuratowski convergence and the convergence in the Hausdorff metric are the same, see [1,Prop. 4.4.15].
As a consequence, d H (S n+1 , S) → 0, as n → ∞.But S n+1 = F (S n ), hence d H (F (S n ), S) → 0 as n → ∞.On the other hand, F (S n ) converge to F ( S) in the Hausdorff metric, so S = F ( S).
S is invariant under F .
Proof.We claim that Indeed, the above follows directly from A4-b.In case of A4, by weak retrotonicity in Λ (A3-b), for each i ∈ I there holds x i < y i , and we can apply A4-b.Therefore, Now with the aid of the Harnack metric we may establish Theorem 2.26.S * = S = q S = S * .
Proof.We start by proving that S = q S. Suppose not.This means, in view of Lemmas 2.20(1) and 2.22 (1), that there is v ∈ ∆ such that the half-line starting at 0 and passing through v intersects S at x and intersects q S at y, where x < y.Put I := I(x).We have x, y ∈ (C I ) ++ and x ≪ I y.
By Lemmas 2.21(2) and 2.23(2), S and q S are invariant.From weak retrotonicity (A3-b) it follows that F −n (x) ≪ I F −n (y) for all n ∈ N. We write and similarly for α(y).
Proof.Since ∆ is compact, we use the fact that uniform convergence (to a function that is necessarily continuous) is equivalent to continuous convergence: We set Σ := S * .
Observe that, by Proposition 2.27(c)-(d), Σ satisfies (i), (ii) and (iii) in the definition of the carrying simplex [weak carrying simplex].Notice also that by taking ε sufficiently small in the definition of S 0 we see that Σ attracts all points in Λ \ {0}.
As it is straightforward that Σ is the relative boundary of ⟨0, Σ⟩ in C + , the satisfaction of the additional property (vi) follows from Theorem 2.28 together with Lemma 2.14.

Back to Dynamics on C +
Assume A1-A3, and A4 or A4 where in case of A4 we assume furthermore that any positive number or ∞ can serve as κ.From now on, F is considered defined on the whole of C + again.
Recall that, by Theorem 2.10(b), Γ is the compact attractor of bounded sets in C + , which is the same as the compact attractor of neighbourhoods of compact sets in C + .In the context of Conley's attractor-repeller pairs [6] we may decompose Λ into an attractor Σ, a repeller {0} and a set of connecting orbits.According to [28, Theorem 5.17 on p. 137], the compact attractor Γ of neighbourhoods of compact sets in C + is the union of pairwise disjoint sets, with the following properties: We claim that E = (0, 1)Σ.Indeed, because F↾ Γ is a homeomorphism of Γ onto itself and Σ is invariant, E ∩ Σ = ∅.As 0 / ∈ E ⊂ Γ, and Γ \ {0} is the disjoint union of (0, 1)Σ and Σ, there holds E = (0, 1)Σ and H = Σ.
So we have the following classification of points in x ∈ C + .
We have thus obtained property (iv) in the definition of the carrying simplex [weak carrying simplex], as well as the additional properties (vii) and (viii).
It follows from general results on attractors, see, e.g., [28, Thms.2.39 and 2.40], that Σ, being a compact attractor of neighbourhoods of compact sets, is stable, meaning that for each neighbourhood U of Σ in C + there exists a neighbourhood V of Σ in C + such that F n (V ) ⊂ U for all n ∈ N. Indeed, by our construction, { ⟪S n , S n ⟫ : n ∈ N } is a base of relatively open forward invariant neighbourhoods of Σ, from which the stability of Σ follows in a straightforward way.

Asymptotic completeness
Assume A1-A3, and A4 or A4 where in case of A4 we assume furthermore that any positive number can serve as κ.
Proof.As the case x ∈ Σ is obvious, in view of Lemma 2.9 and Theorem 2.29, by replacing x with some of its iterates, we can assume that either F n (x) ∈ (0, 1)Σ or In the case A n is compact and nonempty.Pick y ∈ A. By construction, F n (x) < F n (y) for all n ∈ N. Suppose to the contrary that ∥F Since the ≤ relation is preserved in the limit, we have u ≤ v, consequently u < v.By Theorem 2.29, both u and v belong to Σ, which contradicts the unorderedness of Σ (Proposition 2.27(c)).
In the case The sets A n are compact and nonempty.We claim that A n+1 ⊂ A n .Indeed, let η ∈ A n+1 , which means that F n+1 (η) ∈ Σ and A n is compact and nonempty.Pick y ∈ A. By construction, F n (y) < F n (x) for all n ∈ N. The rest of the proof goes as in the previous paragraph.
Without the additional assumptions stated in Proposition 2.31 we have the weaker result.
Proposition 2.32.For each x ∈ Γ \ {0} there exists y ∈ Σ such that lim , where I := I(x).As in the proof of Proposition 2.31 we obtain the existence of y ∈ Σ ∩ (C I ) + such that F n (x) < F n (y) for all n ∈ N. Take a subsequence As a consequence, 0 = u i < v i is impossible, that is, either with the help of Lemma 2.24 that ), a contradiction.Since u = v for any convergent subsequence, the statement of the proposition holds.
Therefore, (v) is satisfied.Hence, since now on, Σ can be legitimately called the [weak] carrying simplex.

Lipschitz Property
We formulate the simple geometrical result: Lemma 2.33.Let S ∈ U. Then (a) Π↾ S is injective, consequently, a homeomorphism onto its image; Proof.If Π↾ S were not injective, there would be x, y ∈ S with y = x + αe for some α > 0, which contradicts Lemma 2.12.
In view of the linearity of Π and the fact that no two points in S are in the ≪ relation, we will prove (b) if we show that for any nonzero u ∈ R d \ (C ++ ∪ (−C ++ )) there holds Let u ∈ R d be arbitrary nonzero.Again by linearity, we can restrict ourselves to the and, by the Pythagorean theorem, As Σ ∈ U, the additional property (ix) follows.

2.7.
A sufficient condition for retrotonicity of the map F when C 1 In the case of discrete-time models checking whether A3 or A3 ′ is satisfied may not be an easy task.In the present subsection we give a simple criterion when F is C 1 .Recall that the spectral radius ρ(P ) of a square matrix P is the modulus of the eigenvalue with maximum modulus.A nonsingular M -matrix is a square matrix P = ρ 0 I − Z where Z is nonnegative and ρ 0 exceeds the spectral radius of Z and that a P -matrix is a square matrix P with positive principal minors.It is a standard result that every nonsingular M -matrix is a P -matrix.Moreover, every eigenvalue of a nonsingular M -matrix has positive real part and every real eigenvalue of a P -matrix is positive [13].
In [8] Gale and Nikaidô proved an important result on the invertibility of maps whose derivatives are P -matrices on rectangular subsets of R d : If Ω ⊂ R d is a rectangle and F : Ω → R d is a continuously differentiable map such that DF (x) is a P -matrix for all x ∈ Ω then F is injective in Ω.
Our standing assumption in the present subsection is that F is a Kolmogorov map satisfying A1 ′ , A2, C and GN, where has spectral radius ρ(Z(x)) < 1 for all x ∈ [0, e] \ {0}.
Sometimes instead of C a stronger assumption is made: We pause a little to reflect on the C 1 property.The standard definition of a C 1 map on C + is that it can be extended to a C 1 map on an open subset of R d containing C + .
In fact, since C + is sufficiently regular, it follows from Whitney's extension theorem (see, e.g., [25]) that the definitions are just the same as in the case of functions defined on open sets, with derivatives replaced by one-sided derivatives where necessary.As usual, DF (x) denotes the Jacobian matrix of F at the point x.If F is invertible with differentiable inverse F −1 then by DF −1 we mean the derivative of F −1 ; this matrix is to be distinguished from (DF ) −1 , the matrix inverse of DF .has spectral radius less than 1.
Remark 5. Regarding GN, observe that for all x ∈ C + there holds It follows from GN, by the continuity of the spectral radius of a matrix, that for any sufficiently small κ > 0 there holds ρ(x) < 1 for all x ∈ [0, (1 + κ)e] \ {0}.
Proof.By Lemma 2.34, DF (x) is invertible at each x ∈ Λ, so the inverse function theorem implies that F ↾ Λ is a local C 1 diffeomorphism.Applying Lemma 2.1 gives us that F ↾ Λ is indeed a C 1 diffeomorphism.For an alternative approach, see [8,Thm. 4].
We proceed to the proof of part (b).We will prove weak retrotonicity by induction on the dimension d.For d = 1 it follows from GN that F is increasing on the segment [0, 1 + κ], so the required property holds.Now, let d > 1 and assume that weak retrotonicity holds for any system satisfying A1 ′ , C and GN, of dimension < d.Suppose to the contrary that there exist x, y ∈ Λ and (up to possible relabelling) 1 ≤ m ≤ d such that F (x) ≤ F (y) but x i > y i for i = 1, 2, . . ., m x i ≤ y i for i = m + 1, . . ., d.
First suppose m = d, i.e.F (x) ≤ F (y) but x ≫ y.As noted in the proof of Lemma 2.34, under the stated conditions DF is a P -matrix on Λ.Thus by [8,Thm. 3], for x, y ∈ Λ, the inequalities F (x) ≤ F (y) and x ≥ y only have the solution x = y, and so the case m = d is not possible.This leaves the case 1 ≤ m < d.We identify the subspace { (ξ 1 , . . ., ξ m , 0, . . ., Observe that H satisfies all the conditions imposed on F , in particular DH has positive diagonal entries and non-positive off-diagonal entries.For i = 1, 2, . . ., m one has We collect what we have just proved as Proposition 2.36.

Examples
The purpose of the present section is to give simple examples relating to well-known models from theoretical ecology to illustrate the applicability of our results.The examples given in this section cover the situation when F is given by some closed-form formula.Direct checking whether A3 is satisfied appeared to be a hopeless task, so we use GN instead.On the other hand, it is easy to check C or C ′ .In the case of Beverton-Holt (Subsection 3.1) or Atkinson-Allen (Subsection 3.3) F is a (weakly) retrotone homeomorphism on the whole of C + , whereas for the Ricker case (Subsection 3.2) the 'box' Λ on which F is a (weakly) retrotone homeomorphism cannot be too large.
Since the boundary of Ω in C + is the graph of a decreasing function we can take Λ = [0, 1 + κ] 2 , where κ > 0 is so small that In view of Proposition 2.35(b) we have the following.

Arbitrary d: Atkinson-Allen map
The Atkinson-Allen map is given by where b ∈ (0, 1) and A > 0. A more slightly general map, which includes (8), called the generalized competitive Atkinson-Allen map, was studied in [9] where the map was shown to have a carrying simplex for a large range of positive parameter values.
In [15], these results are extended to cover cases where some parameters are zero.
If the range of b is extended to include b = 0 the map (8) reduces to a Leslie-Gower map, and this is known to have a carrying simplex for all A ≫ 0 [9] and all A > 0 with a ii > 0 when i = 1, . . ., d [15].If the range of b is extended to include b = 1, F is the identity map and there is no carrying simplex.
For (8) we have f is continuously differentiable on C + and f i (x) ≥ b > 0 so that assumption A1 ′ . is satisfied.It is also easy to check that f i (e i ) = 1 so assumption A4 is satisfied.As

An application to competitive systems of ODEs
This section is devoted to the case when F is a time-one map of the semiflow generated by an autonomous competitive system of ODEs.Even when the vector field is given by some formula, it is seldom possible to find a formula for F .We have at our disposal, however, the Müller-Kamke theory [19,24], which allows us to show that F is a (weakly) retrotone homeomorphism onto its image.Further, A4 is a consequence of the competitivity property of the ODE system.Consider an autonomous system of ordinary differential equations of the form dx i dt = x i g i (x), i ∈ {1, . . ., d}, x = (x 1 , . . ., where g = (g 1 , . . ., g d ) : C + → R d .Such systems are called Kolmogorov systems of ODEs.
The first assumption is H1 g is of class C 1 .
The local flow (continuous-time dynamical system) Φ on a subset D(Φ) of R × C + satisfies the following properties: (EF0) D(Φ) is an open subset of R × C + containing { (0, x) : x ∈ C + }, and Φ is C 1 ; (EF1) Φ(0; x) = x for all x ∈ C + ; (EF2) Φ(t 2 ; Φ(t 1 ; x)) = Φ(t 1 + t 2 ; x), which is to be interpreted so that if for some t 1 , t 2 ∈ R and x ∈ C + one of the sides as well as Φ(t 1 , x) exist then the other side exists, too, and the equality holds.
By (EF1)-(EF2), we have the formula Φ(−t, Φ(t; x)) = x, (10) provided that Φ(t; x) exists.In particular, it follows that for each t > 0 the map Φ(t; •) is a C 1 diffeomorphism onto its image.We will give now a representation of its inverse in terms of a solution of system (9) with P replaced with −P .For x ∈ C + and fixed t ∈ R such that Φ(t; x) exists we put ϕ(s) := Φ(s; x), s ∈ (τ min (x), τ max (x)).Let χ(θ) := ϕ(t − θ).There holds, where ˙= We formulate the result of the above calculation as the following.
The next assumptions are: H2 for each 1 ≤ i ≤ d there holds g i (e i ) = 0; H3 Dg(x) ≤ 0 with its diagonal entries negative, for all x ∈ C + .
Sometimes instead of H3 we make the following stronger assumption: H3 ′ Dg(x) ≪ 0, for all x ∈ C + .
We deduce from H2 and the negativity of the diagonal terms in H3 that the vector field restricted to an i-axis takes value 0 at 0 and e i , has positive direction between 0 and e i and negative direction to the right of e i .In view of that, from the nonpositivity property of the off-diagonal entries it follows that if x ∈ d i=1 [0, 1 + κ] then Φ(t; x) ∈ d i=1 [0, 1 + κ] for t ≥ 0 as long as Φ(t; x) exists, for any κ > 1.In particular, from the standard ODEs extension theorem we obtain that Φ(t; x) exists for any t ≥ 0 and any x ∈ C + .
For x ∈ C ++ and i ∈ {1, . . ., d} we write By the continuous dependence of solutions of ODEs on initial values and the continuity of g i , the formula (12) for f i extends to the whole of C + .Since F is continuous on C + , we have that F = diag[Id] f with f given by ( 12) on ∂C + , too.Therefore A1 is fulfilled.
The fulfillment of A2 follows directly from H2.
For each s ∈ [0, 1] there holds g(Φ(s, x)) − g(Φ(s, y)) = It follows from H3 that the matrix on the right-hand side of (13) has nonpositive entries, and negative diagonal entries.Under H3 ′ , that matrix has all entries negative.
Assume now that x, y ∈ C + are such that F (x) < F (y). Applying Lemma 4.2 and comparing the signs of the entries/coordinates on the right-hand side of (13) gives the desired inequalities for g i (Φ(s, x)) − g i (Φ(s, y)), s ∈ [0, 1).
We can thus apply Theorem 2.2 to obtain the existence of the carrying simplex [weak carrying simplex] for F : face of C I , where k = card I, and let (C I ) ++ := { x ∈ (C I ) + : x i > 0 for all i ∈ I } denote the relative interior of (C I ) + .∂(C I ) + := (C I ) + \ (C I ) ++ is the relative boundary of (C I ) + .A 1-dimensional face is referred to as an i-th axis, where I = {i}.Instead of (C {i} ) + , etc., we write (C i ) + , etc.For x ∈ C + let I(x) stand for the (unique) subset of {1, . . ., d} such that x ∈ (C I(x) ) ++ (often I(x) is called the support of x).Ā denotes the closure of A. Let D be a closed subset of C + .A ⊂ D is said to be relatively open in D if there is an open subset U of R d such that A = U ∩ D. Int D A (called the relative interior of A in D) stands for the largest subset of A that is relatively open in D, and bd D A = Ā \ Int D A. For x ∈ D, we say U ⊂ D is relative neighbourhood of x in D if there exists a neighbourhood V of x in R d such that U = V ∩ D. Neighbourhoods/relative neighbourhoods are tacitly assumed to be open/relatively open.
say B ⊂ C + is order convex if for any x, y ∈ B with x ≤ y one has [x, y] ⊂ B. Let ∅ ̸ = I ⊂ {1, . . ., d}.For x, y ∈ (C I ) + , we write x ≤ I y if x i ≤ y i for all i ∈ I, and x ≪ I y if x i < y i for all i ∈ I.If x ≤ I y but x ̸ = y we write x < I y.The reverse relations are denoted by ≥ I , > I , ≫ I .∆ := {x ∈ C + : d i=1 x i = 1} denotes the standard probability (d − 1)-simplex.For I ⊂ {1, . . ., d}, ∆ I := ∆ ∩ (C I ) + .For I ⊂ {1, . . ., d}, by π I we understand the orthogonal projection of C + onto (C I ) + .Π denotes the orthogonal projection along e onto V := e ⊥ , Πu = u − (u • ê)ê = u − (u • e)e/d.Here e i is the vector with a one at the i-th position and zeros elsewhere and e = d i=1 e i , ê = e/ √ d.

Example 1 . 3 .
2.1(f)]): Definition 1.1.A set B ⊂ C + is said to be unordered if no two distinct points of B are ordered by the < relation.A set B ⊂ C + is said to be weakly unordered if for any ∅ ̸ = I ⊂ {1, . . ., d} no two distinct points of B ∩ (C I ) + are ordered by the ≪ I relation.Example 1.2.The standard probability simplex ∆ ⊂ C + is unordered.Let d > 1.For a > 0 consider the set

Definition 2 .
19. S is the Kuratowski limit of the sequence (S n ) ∞ n=0 if the following conditions are satisfied: (K1) For each x ∈ S there is a sequence (x n ) ∞ n=0 such that x n ∈ S n and lim n→∞ x n = x.(K2) For any sequence (x nk ) ∞ k=1 with x nk ∈ S nk and n k −→ k→∞ ∞, if lim k→∞ x nk = x then x ∈ S. See [1, Def.4.4.13].Recall that d H stands for the Hausdorff metric on the family 2 Λ of nonempty compact subsets of Λ. (2 Λ , d H ) is a compact metric space (see [1, Thm.4.4.15]).Lemma 2.20.(1) S * ⊂ S. (2) S * is weakly unordered.Proof.(1) is a consequence of the definitions of S * and S. Suppose to the contrary that x, y ∈ S * ∩ (C I ) + are such that x ≪ I y.By construction, there are u, v ∈ ∆ I such that x = lim n→∞ R n (u)u and y = lim n→∞ R n (v)v.Since the relation ≪ I is relatively open in
To prove the latter, observe that for any subsequence such that lim k→∞ R nk (u nk )u nk = x there holds, by (K2) in Definition 2.19 and Theorem 2.26, x ∈ S * .As, by the continuity of T , lim k→∞T (R nk (u nk )u nk ) = T (x) and T (R nk (u nk )u nk ) = u nk → u as k → ∞,we have T (x) = u, hence x = R * (u)u.This proves (a), the part of part (b) being similar.Since S * equals, by Theorem 2.26, the Kuratowski limit S, it is compact, and as the radial projection T ↾ S * : S * → ∆ is a continuous bijection, S * is homeomorphic to ∆.By Lemma 2.20(2), S * is weakly unordered, consequently S * ∈ U.The last sentence in part (c) is a consequence of the equality S * = S, Lemma 2.21(2) and Lemma 2.13.Part (d) is, again in view of S * = S, a consequence of Lemma 2.21(2).

Lemma 3 . 1 . 7 )
For the Ricker map (5) assumptions C and GN are satisfied if r, s > 0, a, b ≥ 0 and r + s < 1 + rs(1 − ab) < 2 (If r, s, a, b > 0 and (7) holds then C ′ and GN are satisfied, so that there is a carrying simplex consisting of a curve that connects the two axial fixed points (r, 0) and (0, s).

1 )
when b ∈ (0, 1), which confirms assumption GN.Hence the Atkinson-Allen map has a carrying simplex for all values of b ∈ (0, 1) and A > 0. Figure2shows an example of the carrying simplex for the Atkinson-Allen map (8) when d = 3 and A = of orbits, all of which are attracted to the carrying simplex.
B such that for any compact A ⊂ B there is a relative neighbourhood U of A in B such that Γ attracts U .Such a set Γ is unique (see [28, Thm.2.19 on p. 37]).