Carrying Simplices in Discrete Competitive Systems and

Age-structured Semelparous Populations

Odo Diekmann

Department of Mathematics, University of Utrecht

P.O. Box 80010, 3580 TA Utrecht

The Netherlands

Yi Wang1

Department of Mathematics, University of Science and Technology of China

Hefei, Anhui, 230026, P. R. China

and

Ping Yan

Department of Mathematics and Statistics, University of Helsinki

FIN-00014 Helsinki, Finland

1Supported by FANEDD and NSF of China, partially supported by the Academy of Finland.

Abstract. For discrete competitive dynamical systems, amenable general conditions are presented

to guarantee the existence of the carrying simplex and then these results are applied to age-structured

semelparous population models, as well as to an annual plant competition model.

1 Introduction

In population ecology, there are many mathematical models of competition in which an increase of

the population size or density of one species does have a negative effect on the per capita growth rate

of both the same and other species. The well-known construction of Smale [24] showed that math-

ematical models of competition between species could lead to differential equations with extremely

complicated dynamics. On the other hand, Hirsch [12] proved that there exists an (n−1)-dimensional

balanced attractor, called carrying simplex (see [13, 33]), attracting all nontrivial orbits provided

the system is totally competitive and the origin is a repeller. This then led to the insight that

n-dimensional competitive systems behave like (n−1)-dimensional general systems, for example the

Poincare-Bendixson theorem holds for 3-D competitive systems.

Recently, the well-known results of Hirsch have been generalized and the existence of the carrying

simplex for Kolmogorov competitive mappings has been verified by the second author in joint work

with Jiang [30]. More precisely, let T be a mapping from C to C, where C = {x ∈ Rn : xi ≥

0 for all i}, satisfying the following seven hypotheses:

(H1) T is a C2-diffeomorphism onto its image TC.

(H2) For each nonempty I ⊂ N := {1, 2, . . . , n}, the sets A = HI , H+I and H+

I have the property

that T (A) ⊂ A and T−1(A) ⊂ A, where HI = {x ∈ Rn : xj = 0 for j /∈ I}, H+I = C ∩ HI and

H+I = {x ∈ H+

I : xi > 0 for i ∈ I}.

(H3) For each nonempty subset I ⊂ N and x ∈ H+I , the I×I Jacobian matrix D(T |H+

I)(x)−1 =

(DT (x)−1)I = (DT−1(Tx))I À 0, where T |H+I

means the restriction of T on H+I .

(H4) If x ∈ C and y = Tx then [0, y] ⊂ T [0, x]. (The notation [0, y] is explained in Section 2.

We call the map T , which satisfies (H1)-(H4), a Kolmogorov competitive map.)

(H5) For each i ∈ N , T |H+{i}

has a unique fixed point ui > 0 with 0 < (d/dxi)(T |H+{i}

)(ui) < 1.

Hence ui attracts all orbits with nontrivial initial condition in H+{i}.

(H6) If x is a nontrivial p-periodic point of T and I ⊂ N is such that x ∈ H+I , then µI,p(x) < 1,

where µI,p(x) is the (necessarily real) eigenvalue of the mapping D(T |H+I

)p(x) with the smallest

modulus.

1

x3

x2

x1O

u3

u2

u1

Figure 1: Carrying simplex in 3-D discrete competitive system

(H7) For each nonempty subset I ⊂ N and x, y ∈ H+I , if 0 < Tix < Tiy for all i ∈ I, then

Tix

Tiy≥ xi

yifor all i ∈ I (where T = (T1, . . . , Tn)).

Hypotheses (H1)-(H6) were first introduced by H. L. Smith [25] and are motivated by applications

and in particular by techniques for dealing with the Poincare maps associated with time-periodic

differential equations. By introducing the additional mild hypothesis (H7), Wang and Jiang [30]

were able to prove

Theorem (Wang and Jiang [30]). Let T : C → C be a map satisfying the hypotheses (H1)–(H7).

Then there exists a compact invariant hypersurface Σ, called carrying simplex, such that

(a) For any x ∈ C \ {0}, there is some y ∈ Σ such that ‖T kx − T ky‖ → 0 as k → +∞;

(b) Σ is homeomorphic via radial projection to the (n − 1)-dim standard probability simplex

4 := {x ∈ C :∑n

i=1 xi = 1}.

Figure 1 shows the carrying simplex in the three dimensional case. The geometry and smoothness

of carrying simplices and the dynamics on carrying simplices have been widely investigated for

continuous-time cases (see [10, 16, 18, 19, 32, 33, 34]) and discrete-time cases (see [4, 14, 21, 25, 30]).

The theory of the carrying simplex has been applied successfully to many mathematical models

2

described by differential equations such as competitive Lotka-Volterra systems [33, 34], the growth

of phytoplankton in a chemostat [26, 27] and competitor-competitor-mutualist models [11, 15], etc.

However, it is somewhat of an embarrassment that the theory of the carrying simplex does not

seem to apply easily to discrete-time models. The main obstacle is that, although it can be easily

checked for the Poincare map associated with time-periodic differential equations (see [30]), the

hypothesis (H6) is actually very difficult, sometimes more or less hopeless, to check in discrete-time

models. This is indeed the situation, for example, when we investigate a class of nonlinear Leslie

models, describing the population dynamics of an age-structured semelparous species (see [5, 6, 7]).

Semelparous species are those whose individuals reproduce only once and die afterwards. Ex-

amples include many plants, Pacific salmon, cicada’s and many other insects. For many species, in

particular many cicada species, the period in between being born and going to reproduce is strictly

fixed at, say, k years. The population then subdivides into subpopulations according to the year

of birth modulo k (or, equivalently, the year of reproduction modulo k ). Such a subpopulation is

called a year class. Year classes mate and reproduce k years later, so are reproductively isolated

from other classes. However, they may still interact by influencing each others living conditions,

e.g. by competition for food or space. So competitive interaction between individuals is modeled

via a feedback loop involving variable environmental conditions (cf. [8, 9]). Mathematically, the

discrete-time model can be expressed as (see [5] for Biennials and [6] for the general case)N(t + 1) = L (h(E(t))) N(t)

E(t) = c · N(t), t = 0, 1, 2 · · · ,

(1.1)

where h = (h0, · · · , hk−1) and

L(h) =

0 0 · · · 0 hk−1

h0 0 · · · 0 0

0 h1 · · · 0 0

· · ·

0 0 · · · hk−2 0

. (1.2)

Here N(t) = (N0(t), · · · , Nk−1(t)) and Ni(t) is the density of the i-th age class in year t, i =

0, 1, · · · , k − 1. hi(0 ≤ i ≤ k − 2) is the survival probability of the i-th class, while hk−1 is the per

capita expected number of offspring of the (k − 1)-th class. For each i = 0, · · · k − 1, hi has, for

instance, the Beverton-Holt form

hi(E) =σi

1 + giE, (1.3)

3

where σi, gi are positive constants with∑k−1

i=0 gi = 1. c = (c0, · · · , ck−1) is a nonnegative constant

vector with∑k−1

i=0 ci = 1. Biologically, ci is called the age-specific impact on the environmental

condition, and gi is called the sensitivity to the environment. Observe that the one-dimensional

environmental condition E has an influence on survival and reproduction, but is also influenced by

the population size and composition.

Working with Davydova and van Gils, the first author [6, 7] investigated the dynamics of semel-

parous populations and found various phenomena, such as competitive exclusion (also called single

year class (SYC) behaviour in [3, 31], or synchronization in [20]), coexistence, vertical bifurcation

and the possibility of an attracting heteroclinic boundary cycle. One of the main techniques of the

analysis in [6, 7] is to consider the “full-life-cycle map” T , which is defined as the kth-iterate of the

map featuring in (1.1), i.e., define the map T : C → C;N(0) 7→ N(k) such that

Tj(N(0)) = Nj(k) =

(k−1∏i=0

hj+i(Ei)

)Nj(0), (1.4)

j = 0, · · · , k−1. Extensive numerical simulation of the combined bifurcation diagram of SYC-points

and MYC-points (M for “Multiple”) for the case k = 3 confirmed that the dynamics of T is very

similar to the stable phase classification of the 3-D competitive Lotka-Volterra systems (see [33]).

So, it is reasonable to expect that the full-life-cycle map is competitive so that a carrying simplex

exists.

Exploiting the monotonicity of the one-dimensional environmental condition E under some in-

teresting inverse iteration and the properties of cyclic shift, one can eventually verify that the

full-life-cycle map T indeed satisfies the Hypotheses (H1)-(H5) and (H7) (see the details in Section

4). However, to check (H6) is actually more or less hopeless. Observe that (H6) is of key importance

to prove the existence of the carrying simplex (see [25, 30]).

The objective of this paper is to provide, by following a different approach, amenable general

conditions for Kolmogorov competitive maps that guarantee the existence of the carrying simplex .

More precisely, we modified the hypotheses (H1) and (H7) as follows:

(H1’) T is a C1-diffeomorphism onto its image TC;

(H7’) For nonempty subsets I ⊆ J ⊆ N , x ∈ H+I and y ∈ H+

J , if Tix < Tiy for all i ∈ I thenTix

Tiy>

xi

yifor all i ∈ I.

Without adopting the hypothesis (H6), we have as our main result

4

Theorem 1.1. Let T : C → C be a map satisfying the hypotheses (H1’),(H2)–(H5) and (H7’). Then

there exists a compact invariant hypersurface Σ, called carrying simplex, such that

(a) For any x ∈ C \ {0}, there is some y ∈ Σ such that ‖T kx − T ky‖ → 0 as k → +∞.

(b)Σ is homeomorphic via radial projection to the (n− 1)-dim standard probability simplex 4 :=

{x ∈ C :∑n

i=1 xi = 1}.

Remark 1.1. The hypothesis (H7’) can be easily checked in various discrete-time models. For

example, according to (H2), we can rewrite the mapping T as:

T (x1, · · · , xn) = (x1G1(x), x2G2(x), · · · , xnGn(x)), for x ∈ C,

where

Gi(x) :=

Ti(x)

xiif xi 6= 0

∂Ti

∂xi(x) if xi = 0,

is a differentiable continuous function (if, for instance, T ∈ C2) for i = 1, · · · , n. Assume that∂Gi

∂xj(x) < 0 for all i, j ∈ N and x ∈ C (as in differential equations we might call this “total

competition”). Then a straightforward calculation yields that (H7’) holds.

For age-structured semelparous population models, although the condition∂Gi

∂xj(x) < 0 does not

always hold (see the example presented in Remark 4.3), hypothesis (H7’) can still be obtained by

the monotonicity of the environmental condition E. Thus we obtain an affirmative answer for the

conjecture concerning the existence of a carrying simplex in [6, 7], that is,

Theorem 1.2. Let T : C → C be the full-life-cycle map of the age-structured semelparous population

model with Beverton-Holt type nonlinearity. Then there exists a carrying simplex Σ such that

(a) For any x ∈ C \ {0}, there is some y ∈ Σ such that ‖T lx − T ly‖ → 0 as l → +∞.

(b)Σ is homeomorphic via radial projection to the (k − 1)-dim standard probability simplex.

As a corollary we derive a far reaching generalization of the result in [7] concerning the existence

of a heteroclinic cycle at the boundary of the positive cone in the three-dimensional case.

Corollary 1.3. In the setting of Theorem 1.2, specialize to k = 3 and assume that the restriction

of T to an invariant coordinate plane has no other fixed point than those on the axes. Then T has

a heteroclinic cycle at the boundary of C connecting the three fixed points on the axes.

5

Proof. The intersection of Σ with a coordinate plane is an invariant line segment connecting the

two fixed points on the axes. Since there are no interior fixed points on this line segment, the

restriction of T to the segment must be a monotone one-dimensional map. Accordingly all orbits on

the segment have one fixed point as the α-limit set and the other as the ω-limit set. By the cyclic

symmetry, a fixed point on an axis is necessarily an ω-limit point in one plane and an α-limit point

in the other.

This paper is organized as follows. In Section 2 we introduce some notations, give relevant

definitions and preliminaries which will be important in our proofs. Theorem 1.1 is proved in Section

3. Section 4 is devoted to the study of the existence of a carrying simplex for the age-structured

semelparous population. As another application of our result, in Section 5, we will show that some

annual plant competition model (see, e.g. [23]) also has a carrying simplex.

2 Notations and Preliminary Results

Given ∅ 6= I ⊂ N , let HI = {x ∈ Rn : xj = 0 for j /∈ I}. For two vectors x, y ∈ HI , we write

x ≤I y if xi ≤ yi for all i ∈ I, and x ¿I y if xi < yi for all i ∈ I. If x ≤I y but x 6= y we write

x <I y(the subscript on ≤, <,¿ is dropped if I = N). Let C = {x ∈ Rn : x ≥ 0} be the usual

nonnegative cone. The interior of C is the open cone C◦ = {x ∈ Rn : x À 0} and the boundary of

C is ∂C. We also let H+I = C ∩ HI and H+

I = {x ∈ H+I : xi > 0 for i ∈ I}. For any two points

x ≤ y in Rn we define the closed order interval [x, y] = {z ∈ Rn : x ≤ z ≤ y} and open order interval

[[x, y]] = {z ∈ Rn : x ¿ z ¿ y}. A set in Rn is order convex if it contains the order closed intervals

defined by each pair of its elements. If A is a subset of a topological space X, A denotes the closure

of A in X. The boundary of A relative to X is denoted by ∂XA, or ∂A if X = C. A subset A in C

is called unordered if A does not contain two points related by <.

Let T : C → C satisfy (H1)’ and (H2)-(H5). The forward (backward) orbit of x ∈ C in C is

defined by O+(x) = {Tmx : Tmx ≥ 0 and m ∈ Z+} (O−(x) = {Tmx : Tmx ≥ 0 and m ∈ Z−},

where Z+(Z−) denotes the set of nonnegative (nonpositive) integers. The orbit of x ∈ C in C is

defined by O(x) = O+(x) ∪ O−(x). Let x ∈ C. Then either there exists some N ∈ N such that

T−nx ∈ TC for 0 ≤ n ≤ N but T−(N+1)x /∈ TC, or T−nx ∈ TC for any n ∈ N. In the first

case, we say that such an x does not have a full backward orbit. The ω-limit set of x is defined

by ω(x) = {y ∈ C : Tnkx → y(k → ∞) for some sequence nk → +∞ in Z} and the α-limit set of x

by α(x) = {y ∈ C : T−nkx → y(k → ∞) for some sequence nk → +∞ in Z}. Note that if O+(x) is

6

compact in C, then the ω-limit set of x is nonempty and invariant, i.e., Tω(x) = ω(x). Furthermore,

the α-limit set of x is nonempty and invariant provided x has a full backward orbit and O−(x) is

compact in C.

From the Hypotheses (H1’) and (H2)-(H5), one can obtain some properties of the map T (see

[25, Propositions 2.1 and 3.1]).

Proposition 2.1. If x, y ∈ C and Tx < Ty, then x < y.

Proposition 2.2. For each ∅ 6= I ⊆ N , T is strongly competitive in the interior H+I of H+

I , i.e., if

x ∈ H+I , y ∈ H+

I and Tx <I Ty, then x ¿I y.

Furthermore, let u = (u1, u2, · · · , un), where ui are the fixed points introduced in the statement

of (H5). Here we are abusing notation and allowing ui to denote a point in R or the corresponding

point on the boundary of C as required by the context. Then one has the following three propositions,

the proofs of which can also be found in [25].

Proposition 2.3. The set

Γ =∞⋂

k=1

T k[0, u]

is a nonempty, order-convex global compact attractor of T in C. M := ∂Γ is an unordered invariant

compact set containing the fixed points ui, 1 ≤ i ≤ n. Moreover, M is homeomorphic via radial

projection to the (n − 1)-dim standard probability simplex.

Proposition 2.4. The domain of repulsion of the origin

B(0) := {y ∈∞⋂

k=1

T kC : T−jy → 0 as j → ∞}

is a nonempty order-convex invariant open set in C. B(0) ⊂ Γ and S := ∂B(0) is an unordered

invariant compact set containing the fixed points ui, 1 ≤ i ≤ n. Moreover, S is homeomorphic via

radial projection to the (n − 1)-dim standard probability simplex.

Proposition 2.5 (Non-ordering of Limit-sets). Any ω- or α- limit set of x ∈ C◦ cannot contain

two points related by <.

Before ending this section, we shall state several known results which will be important in the

proof of the main result. In order to do this, we first introduce some crucial definitions and notations.

Let p be an m-periodic point and O(p) = {p0, p1, · · · , pm−1}, T ipj = p(i+j) mod m and p0 = p.

Frequently, O(p) is called a cycle or an m-cycle. If m = 1, we call p a fixed point. We denote by

7

Fix(T ) the set of the fixed points of T . A point z is in the lower boundary ∂−S of a set S ⊂ Rn

provided there is a sequence {yn} in S converging to z with yn À z (denoted by yn ↓ z), but no

sequence {xn} in S converging to z with xn ¿ z (denoted by xn ↑ z). Then, for i ∈ {0, 1, · · · ,m−1},

we define, as in [29],

M(pi) = {x ∈ C : T−nmx → pi as n → +∞};

M−(pi) = {x ∈ M(pi) : T−nmx ¿ pi for sufficiently large n ∈ N};

V−(pi) = ∂−(M−(pi)),

and

M−(O(p)) =m−1⋃i=0

M−(pi);

V−(O(p)) =m−1⋃i=0

V−(pi).

Working with Jiang, the second author [29] extended to time-periodic Kolmogorov systems a well

known result of M. W. Hirsch (see Theorem1.1 in [12]) for autonomous systems under the following

hypotheses:

Dissipation. There is a compact invariant set Γ, called the fundamental attractor, which uniformly

attracts each compact set of initial values.

Competition. T is competitive in C, i.e., if x, y ∈ C and Tx < Ty, then x < y.

Strong Competition. For each ∅ 6= I ⊆ N , T is strongly competitive in the interior H+I of H+

I , i.e.,

if x ∈ H+I , y ∈ H+

I and Tx <I Ty, then x ¿I y.

Proposition 2.6 (Wang and Jiang [29]). Let T : C → C be a C1 injective map satisfying hypotheses

of Dissipation, Competition and Strong Competition. Given x ∈ C◦ ∩ Γ, let L denote the ω- or α-

limit set of x. Assume that L is nonempty, not a cycle and L ⊂ C◦. Then precisely one of the

following two alternatives holds:

(a): L ⊂ M , where M is the boundary of Γ relative to C;

(b): there exists some positive m-periodic point q such that L ⊂ V−(O(q)).

It deserves to be noted that the fundamental attractor Γ of T is actually the fundamental repellor

of T−1. In terms of T−1, Γ is characterized as the set of points with bounded orbits, while x ∈ C \Γ

if and only if either x does not have a full backward orbit, or ‖T−nx‖ → +∞ as n → +∞. Therefore,

8

given any x ∈ Γ, α(x) is nonempty and invariant. Moreover, it is easy to see that T is competitive in

Γ if and only if T−1 is monotone in Γ, i.e., if x, y ∈ Γ with x < y, then T−1x < T−1y. T is strongly

competitive in H+I ∩ Γ if and only if T−1 is monotone in H+

I ∩ Γ, i.e., if x ∈ H+I ∩ Γ, y ∈ H+

I ∩ Γ

with x <I y, then T−1x ¿I T−1y.

Based on the discussion above, we have the following proposition, which is originally due to

Mierczynski [17, Theorem 2.1] who gave a preliminary classification of equilibria for C1 strongly

monotone mappings.

Proposition 2.7. Let x ∈ H+I ∩ Γ be an m-periodic point of T . Then there exists a local backward

invariant with respect to Tm, totally ordered by “¿I” one-dimensional submanifold W ⊂ H+I ∩ Γ,

W ¿I x, homeomorphic to the interval [0, 1) such that precisely one of the following three alternatives

applies:

(i) T−mw ¿I w for all w ∈ W \ {x};

(ii) T−mw ÀI w for all w ∈ W \ {x};

(iii) there is a sequence xk ∈ W ∩ Fix(Tm) such that xk → x as k → ∞.

3 The proof of the main result

In this section, we always assume that the hypotheses (H1’), (H2)-(H5) and (H7’) hold. Note that,

by Propositions 2.1-2.3, the hypotheses (H1’),(H2)-(H5) imply Dissipation, Competition, and Strong

competition. Therefore, the assumption of Proposition 2.6 always holds in this section.

We break the proof of the main result into several Lemmas.

Lemma 3.1. Suppose that S ∩ ∂C = M ∩ ∂C. Then given any 0 ¿ x ¿ y satisfying x ∈ S and

y ∈ M , we have α(x) ⊂ C◦ and α(x) ∩ α(y) = ∅.

Proof. Given any 0 ¿ x ¿ y satisfying x ∈ S and y ∈ M . Suppose that z ∈ α(x) ∩ α(y), then

there is a sequence nk → +∞ such that T−nkx → z. Without loss of generality assume that

T−nky → z′ ∈ α(y), then z ≤ z′. By Proposition 2.5, α(y) is unordered, which implies that z = z′.

On the other hand, it follows from (H7’) that, for each i ∈ N ,T−n

i x

T−ni y

is a decreasing sequence. Thus,

T−nx

T−ny→ λ := (λ1, · · · , λn) ¿ (1, · · · , 1), a contradiction to T−nkx → z ← T−nky as nk → ∞.

Thus we have proved α(x) ∩ α(y) = ∅.

Suppose that α(x) ∩ ∂C 6= ∅. Then one can find a w ∈ α(x) ∩ ∂C and a sequence nk → +∞

such that T−nkx → w ∈ α(x) ∩ ∂C ⊂ S ∩ ∂C = M ∩ ∂C. Without loss of generality we assume

9

that T−nky → w′ ∈ α(y) ⊂ M . Then w ≤ w′. Note that M is unordered, hence w = w′, which

contradicts (H7’) according to the same argument as in the paragraph above.

Lemma 3.2. All nontrivial periodic points of T belongs to S ∩ M .

Proof. Obviously all nontrivial periodic points of T belong to Γ. Let x be a nontrivial m-periodic

point of T , x ∈ H+I . Then, by Proposition 2.7, there is a local backward invariant with respect to

Tm, totally ordered by “¿I” one-dimensional submanifold W ⊂ H+I , W ¿I x, homeomorphic to

the interval [0, 1) such that either (i) T−mw ¿I w for all w ∈ W \ {x}, or (ii) T−mw ÀI w for all

w ∈ W \ {x}, or else (iii) there is a sequence xk ↑ x, xk ∈ W ∩ Fix(Tm). Case (iii) is impossible,

since then, by (H7’),(xk)i

xi>

T−mi xk

T−mi x

=(xk)i

xifor every i ∈ I, a contradiction. In case (ii), one can

obtain that α(w) = O(x) for all w ∈ W , consequently,T−ml

i w

xi→ 1(l → +∞) for every i ∈ I. On the

other hand, fix any i ∈ I, by (H7’), 1 >wi

xiÀ T−ml

i w

T−mli x

=T−ml

i w

xifor all l ∈ N and {T−ml

i w

xi}l∈N is

a decreasing sequence. HenceT−ml

i w

xi→ λ < 1 as l → +∞, a contradiction. So only case (i) holds,

and we can find an m-periodic point q ∈ H+J , J ⊆ I, with q ¿I w such that T−mlw → q(l → +∞)

for all w ∈ W . We claim that J = ∅. If not, then one can choose a j ∈ J such that 0 < qj < wj , and

hence again by (H7’), we haveqj

T−mlj w

=T−ml

j q

T−mlj w

→ λ < 1(l → +∞), contradicting T−mlw → q.

Therefore, J = ∅, that is, q = 0. So T−lw → 0 as l → +∞ for all w ∈ W , which implies that x ∈ S.

Similarly, we can obtain that x ∈ M . Indeed, if x /∈ M then one can find a z ∈ Γ∩ H+I , x ¿I z.

Then every point in [x, z] has a unique full backward orbit. By appealing to Proposition 2.7 and

(H7’) again, we obtain that there is a local backward invariant with respect to Tm, totally ordered

by “¿I” one-dimensional submanifold W ⊂ H+I , x ¿I W , homeomorphic to the interval [0, 1) such

that T−mw ÀI w for all w ∈ W \ {x}. Thus there exists an m-periodic point r ∈ Γ∩ H+I such that

T−mlw → r(l → +∞) for all w ∈ W . Note I 6= ∅, this can not happen by the same reason in the

end of the previous paragraph. Thus we have proved that x ∈ M .

Theorem 3.1. We have the following conclusions:

(a): Σ := S = M ;

(b): For any x ∈ C \ {0}, there is some y ∈ Σ such that ‖T kx − T ky‖ → 0 as k → +∞.

Proof. The proof goes by induction on n. If n = 1, then Γ is an interval [0, ui] where ui is some fixed

point of T in some axis, which is induced from the hypothesis (H5). Then the statement is trivial.

10

From now on we assume n > 1. The induction hypothesis is that Theorem 3.1 holds for systems

in Rm if m < n.

Given any ∅ 6= I ( N , the hypotheses of (H1’), (H2)-(H5) and (H7’) are inherited by the

restriction of the system to H+I . Therefore, by the induction hypothesis we conclude that S ∩H+

I =

M ∩ H+I and every trajectory in H+

I \ {0} is attracted to S ∩ H+I = M ∩ H+

I . Therefore, one can

see that S ∩ ∂C = M ∩ ∂C.

Now we shall prove S = M . It suffices to prove that S ∩C◦ = M ∩C◦. Note that every ray from

the origin through a point of C◦ meets S and M in points x and y, then it suffices to prove x = y.

Suppose that for some ray x 6= y. Then 0 ¿ x ¿ y. It follows from Lemma 3.1 that α(x) ⊂ C◦ and

α(x) ∩ α(y) = ∅.

If α(x) is a cycle, then, by Lemma 3.2, we have α(x) ⊂ M . Then one can find z ∈ α(x) ⊂ M ,

w ∈ α(y) ⊂ M such that z ≤ w, but α(x)∩α(y) = ∅, so z < w. Thus M contains two related points

z and w, contradicting Proposition 2.3.

If α(x) is not a cycle, then, by Proposition 2.6, we have α(x) ⊂ V−(O(q)) for some positive

m-periodic point q, or α(x) ⊂ M . Suppose that α(x) ⊂ V−(O(q)) holds. Then M−(O(q)) is

nonempty. So one can find some z ∈ C such that T−mkz → q(k → +∞) with T−mkz ¿ q for

sufficiently large k ∈ N. Fix an k0 ∈ N sufficiently large and let z′ = T−mk0z, then 0 ¿ z′ ¿ q

and T−mkz′ → q(k → +∞). However, by (H7’),T−mk

i z′

qi=

T−mki z′

T−mki q

→ λi < 1 for each i ∈ N , a

contradiction. Hence α(x) ⊂ M . Then again one can find z ∈ α(x) ⊂ M , w ∈ α(y) ⊂ M such that

z ≤ w, but α(x)∩α(y) = ∅, so z < w. Thus M contains two related points z and w, a contradiction.

Therefore we have completed the proof that S = M .

The statement (b) directly comes from [30, Corollary 1.3].

Note that Theorem 1.1 on the existence of the carrying simplex is a corollary of Theorem 3.1

and Propositions 2.3 and 2.4.

4 Age-Structured semelparous populations

Consider a semelparous species with a life cycle of exactly k years, so consisting of k reproductively

isolated year classes. Once a year class goes extinct, it remains extinct. We then say that the year

class is “missing”. The periodical insects [3] are those for which all but one year classes are missing.

The prime example are the Magicicadas (The k = 17 species had its most recent emergence in the

North-East United States in 2004).

11

We consider the discrete-time model described by (1.1) and the associated full-life-cycle map

(1.4). We want to check that the hypotheses (H1’), (H2)-(H5) and (H7’) from Section 1 hold for the

discrete-time system (1.4).

Firstly, it is easy to see that all coordinate axes and faces are invariant under the full-life-

cycle map T , which implies that (H2) holds. Secondly, (H5) is actually concerned with the one-

dimensional dynamics. Since we consider Beverton-Holt type nonlinearities, one can easily see

that the corresponding one-dimensional map shows convergence of all nontrivial orbits towards the

positive steady state (which is hyperbolic) for values of parameters for which the positive steady

state exists (or, equivalently, the basic reproduction ratio exceeds one).

Proposition 4.1. Assume (H1’), (H2) and (H3). Then (H4) holds.

Proof. The arguments are similar to those for proving Proposition 2.1 in [14]. For completeness,

we provide the details. If one of the points x, y is 0 then by (H2) the other is 0, too, and there is

nothing to prove. So assume that both x and y are nonzero. Again by (H2) there is a nonempty

I ⊂ N such that both x and y are in H+I . Take any z ∈ [0, y], 0 6= z 6= y, and consider the segment

Σ ⊂ H+I joining y and z, Σ = {sy + (1 − s)z : 0 ≤ s ≤ 1}. Since TC is a relatively open subset

of C, it follows that (Σ \ {z}) ∩ TC is a relatively open subset of Σ containing y. Put σ ∈ [0, 1)

to be the infimum of those τ ∈ (0, 1) such that {sy + (1 − s)z : τ ≤ s ≤ 1} ⊂ TC, and define

Σ′ := {sy + (1 − s)z : σ < s < 1}. As Σ′ ⊂ ([0, y] \ {y}) ∩ H+I , it follows from (H3) that T−1Σ′ is

a linearly ordered by ¿I subset of [[0, x]] and that T sends bijectively T−1Σ′ onto Σ′. As T−1Σ′ is

contained in the compact set [0, x]I , there is ζ := inf T−1Σ′. We have

Tζ = T ( lims→σ+

T−1(sy + (1 − s)z)) = lims→σ+

T (T−1(sy + (1 − s)z)) = σy + (1 − σ)z.

Clearly σ = 0, since otherwise, by (H2) both ζ and Tζ would be in ∂H+I , and that contradicts our

choice of σ. Therefore σ = 0 and Tζ = z.

Now it remains to show (H1’), (H3) and (H7’) for (1.4). In order to verify (H1’) and (H3), we

need the following Propositions:

Proposition 4.2. T is injective, T is competitive in C and strongly competitive in H+I , ∅ 6= I ⊂

{0, · · · , k − 1}.

Proof. Let T : Rk → Rk, (T x)i = xihi(c · x) for i = 0, · · · , k − 1. We claim that T is injective, T is

competitive in C and strongly competitive in H+I , ∅ 6= I ⊂ {0, · · · , k − 1}.

12

In fact, let v = T x, then xi = vi/hi(E) for i = 0, · · · , k − 1, where E is equal to c · x. Hence

E = c · x =k−1∑i=0

civi

hi(E),

which implies that

1 =k−1∑i=0

civi

Ehi(E). (4.1)

Since hi is of Beverton-Holt type, Ehi(E) is strictly increasing as a function of E. So for given

v there is at most one solution E for (4.1). Once E is determined, so is x via xi = vi/hi(E) for

i = 0, · · · , k − 1. Hence T is injective. Moreover, as a function of v, the rhs of (4.1) is strictly

increasing. Therefore E is a strictly increasing function of v. The formula xi = vi/hi(E) implies

that the same holds for all component of x, which implies that T is competitive in C, and even

strongly competitive in H+I . Thus we have proved the claim.

It is not difficult to see that the full-life-cycle map T = (ST )k, where

S =

0 · · · 0 1

1 · · · 0 0

· · ·

0 · · · 1 0

is the cyclic forward shift on Rk. Therefore, T is injective, competitive in C and strongly competitive

in H+I , ∅ 6= I ⊂ {0, · · · , k − 1}.

Remark 4.1. Based on Proposition 4.2 and the expression (1.4), it is easy to check that (H1’) and

(H3) hold.

Proposition 4.3. (H7’) holds for the full-life-cycle map T defined by (1.4).

Proof. Denote by P = ST with T introduced in Proposition 4.2 above, then the full-life-cycle map

T = P k. Obviously, P is also injective and competitive in C. Based on the argument following the

formula (4.1) in the proof of Proposition 4.2, we let Ej(x) denote the unique solution of

1 =k−1∑i=0

ci(P j+1x)i

Ehi(E), (4.2)

for j = 0, · · · , k − 1.

Let I = {k1, · · · , ki} ⊆ {0, · · · , k − 1} be an index set, define σjI := {(k1 + j) mod k, · · · , (ki +

j) mod k} for j ∈ N. It is easy to see that σkI = I. Now, given any nonempty index subset

13

I ⊆ J ⊆ N , let x ∈ H+I and y ∈ H+

J with Tix < Tiy for all i ∈ I. Then one has P kx < P ky and

P ki x < P k

i y for all i ∈ I. Moreover, by the competitiveness of P and the property of the shift S , one

has P jx < P jy and P ji x < P j

i y for all i ∈ σjI and j = 1, · · · , k, which implies that P j+1x < P j+1y

and P j+1x ¿σj+1I P j+1y for all j = 0, · · · , k − 1. It follows from (4.2) and the strict monotonicity

of E with v, where v is as in (4.1), that Ej(x) < Ej(y) for all j = 0, · · · , k − 1. Hence,∏k−1j=0 hj+i(Ej(x))∏k−1j=0 hj+i(Ej(y))

> 1,

for all i ∈ J . Note thatTix

Tiy=

(P kx)i

(P ky)i=

∏k−1j=0 hj+i(Ej(x))∏k−1j=0 hj+i(Ej(y))

· xi

yi,

for all i = 0, · · · , k−1. Then we haveTix

Tiy>

xi

yifor all i ∈ I. Thus we have completed the proof.

Remark 4.2. When hi(E) =σi

1 + giEone might try to derive an explicit expression for (1.4) but

the expression will be extremely complicated and not yield any insights.

Remark 4.3. Although (H7’) holds for the age-structured semelparous model, the condition∂Gi

∂xj<

0 in Remark 1.1 does not always hold. For example, consider k = 2 and the iteration as

N0(t + 1) =R0N1(t)1 + N0(t)

, N1(t + 1) = N0(t),

where R0 is the positive constant representing the basic reproduction ratio. Then an easy calculation

shows that

N0(t + 2) =R0N0(t)

1 +R0N1(t)1 + N0(t)

= G0(N0(t), N1(t))N0(t)

with G0(N0, N1) = R0

1+R0N11+N0

, so ∂G0∂N0

> 0 if N1 > 0.

Note that Theorem 1.2 on the existence of the carrying simplex for the age-structured semelparous

model is, now that we have verified all hypotheses, a corollary of Theorem 1.1.

5 Application to an annual plant competition model

The annual plant competition model we consider here was first derived by Atkinson [2] and Allen et

al. [1] (see also a discrete-time model formulated by Jones and Perry in the book [22]). In particular,

14

the three-species annual plant model can be simplified and expressed asx1(n + 1) =

2(1 − b)x1(n)1 + x1(n) + α1x2(n) + β1x3(n)

+ bx1(n),

x2(n + 1) =2(1 − b)x2(n)

1 + β2x1(n) + x2(n) + α2x3(n)+ bx2(n),

x3(n + 1) =2(1 − b)x3(n)

1 + α3x1(n) + β3x2(n) + x3(n)+ bx3(n),

(5.1)

where xi(n)(i = 1, 2, 3) is the density of species i at time n, b and αi, βj are the seedbank and

competition coefficients, respectively, and

0 < b < 1, αi > 0, βi > 0, αi 6= 1, βi 6= 1 (5.2)

for all i = 1, 2, 3.

Roeger and Allen [23] studied the local dynamics and Hopf bifurcations of the discrete-time model

(5.1) under the additional assumption that the three species compete in the rock-scissors-paper type,

i.e., 0 < αi < 1 < βi, i = 1, 2, 3. Their analysis and numerical simulations strongly suggest that a

two-dimensional carrying simplex exists for (5.1). In this section, we will show that this is indeed

even true with no restriction of rock-scissors-paper manner. Therefore, all the interesting dynamics

(for example, center manifold of the positive equilibrium, periodic solutions and heteroclinic cycles,

etc.) are on the two-dimensional carrying simplex. Moreover, we can deduce that the dynamics is

trivial if there is no positive equilibrium (see the following Corollary 5.2). We do believe that the

existence of the carrying simplex will shed light on the investigation of the global behaviour and the

complete classification of (5.1). We will leave this as a future research topic.

Now we focus on the proof of the existence of the carrying simplex. Define the map T : C → C

such that

T

x1

x2

x3

=

2(1−b)x1

1+x1+α1x2+β1x3+ bx1

2(1−b)x21+β2x1+x2+α2x3

+ bx2

2(1−b)x31+α3x1+β3x2+x3

+ bx3

, (5.3)

where (x1, x2, x3) ∈ C := R3+ = {(x1, x2, x3) : xi ≥ 0, for i = 1, 2, 3}.

We need to check that the hypotheses (H1’), (H2)-(H5) and (H7’) in Section 1 do hold for T .

Obviously, (H2) holds. As for (H5), it is easy to see that E1(1, 0, 0), E2(0, 1, 0), E3(0, 0, 1) are the

equilibria on each axis, respectively. Moreover, they attract all the nontrivial orbits in each axis

respectively. Now we need the following lemmas:

Lemma 5.1. T satisfies (H1’) and (H3).

15

Proof. We denote by DT (x) the derivative of T at x ∈ C. It follows from (5.3) that

DT (x) = DT

x1

x2

x3

=

m11 m12 m13

m21 m22 m23

m31 m32 m33

, (5.4)

where m11 = 2(1−b)(1+α1x2+β1x3)(1+x1+α1x2+β1x3)2

+ b, m12 = −2(1−b)α1x1(1+x1+α1x2+β1x3)2

, m13 = −2(1−b)β1x1(1+x1+α1x2+β1x3)2

, m21 =−2(1−b)β2x2

(1+β2x1+x2+α2x3)2, m22 = 2(1−b)(1+β2x1+α2x3)

(1+β2x1+x2+α2x3)2+b, m23 = −2(1−b)α2x2

(1+β2x1+x2+α2x3)2, m31 = −2(1−b)α3x3

(1+α3x1+β3x2+x3)2,

m32 = −2(1−b)β3x3(1+α3x1+β3x2+x3)2

, m33 = 2(1−b)(1+α3x1+β3x2)(1+α3x1+β3x2+x3)2

+ b.

After a lengthy calculation we obtain that det DT (x) > 0, hence DT (x) is invertible for every

x ∈ C. Furthermore, it is also not difficult to show that (DT (x)−1)I À 0 for every x ∈ H+I and the

nonempty subset I ⊂ {1, 2, 3}. Thus we have proved (H3).

Since DT (x) is invertible for every x ∈ C, T is a local homeomorphism. In order to prove that

T is a global homeomorphism, by [28, Theorem 4.1], we need to show that T is competitive, i.e., if

x, y ∈ C and Tx < Ty then x < y. To end this, let x, y ∈ C and Tx < Ty. Define the segment

Ip = {λTx + (1 − λ)Ty : 0 ≤ λ ≤ 1}.

Since T is a local homeomorphism, it is not difficult to show that there exists a continuous path

γ : [0, 1] → C with γ(0) = x, γ(1) = y such that T (γ([0, 1])) = Ip and T is one-to-one from γ onto Ip.

For any z ∈ Ip, there exists a unique tz ∈ [0, 1] such that T (γ(tz)) = z. Furthermore, there exists

a neighborhood Uz ⊂ C of z and a neighborhood Vγ(tz) ⊂ C of γ(tz) such that T : Vγ(tz) → Uz

is a homeomorphism. Note that (H3) holds. Then, by [25, Proposition 3.1(iv)], T is a competitive

map from Vγ(tz) to Uz, that is, if w, v ∈ Vγ(tz), Tw, Tv ∈ Uz and Tw < Tv then w < v. Clearly,

{Uz ∩ Ip : z ∈ Ip} forms an open cover of Ip. The compactness of Ip guarantees the existence

of a finite subcover {Uj ∩ Ip : 1 ≤ j ≤ l} of Ip. Similarly as above, one can choose a finitely

many neighborhoods Vγ(tj) with tj ∈ [0, 1] such that T : Vγ(tj) → Uj is a homeomorphism for each

j = 1, · · · , l. Since Ip is totally ordered, one can choose zj ∈ Uj ∩ Ip (j = 1, 2, ..., l) such that

Tx = T (γ(0)) ≤ z1 = T (γ(t1)) < z2 = T (γ(t2)) < · · · < zl = T (γ(tl)) ≤ T (γ(1)) = Ty.

Note that T is a homeomorphism, and hence competitive, from Vγ(tj) onto Uj (j = 1, 2, ..., l). Then

we obtain

x = γ(0) ≤ γ(t1) < · · · < γ(tl) ≤ γ(1) = y.

Consequently, we have proved that T is competitive, which implies that T is a C1-diffeomorphism

onto its image TC, i.e., (H1’) holds.

16

From the discussion above and Proposition 4.1, we obtain that (H1’), (H2), (H3) (H4) and

(H5) hold. It is obvious from Remark 1.1 and the expression for the map T that (H7’) holds. By

Theorem 1.1, we have the following corollary on the existence of the carrying simplex in the annual

plant competition model.

Corollary 5.1. Let T : C → C be the map (5.3) of the annual plant competition model. Then there

exists a carrying simplex Σ such that the conclusions in Theorem 1.1 hold.

Moreover, by the results of [4] we have the following global dynamics of the annual plant com-

petition model.

Corollary 5.2. Let T be in Corollary 5.1. Assume also that there is no positive equilibrium of

(5.3), then every orbit of (5.3) converges to some equilibrium on the boundary of C.

Remark 5.1. It is easy to see that (5.3) has no positive equilibrium if and only if the following

linear system of equations x1 + α1x2 + β1x3 = 1,

β2x1 + x2 + α2x3 = 1,

α3x1 + β3x2 + x3 = 1.

has no positive solution.

References

[1] L. J. S. Allen, E. J. Allen and D. N. Atkinson, Integrodidderence equations applied to plant

dispersal, competition, and control, In: S. Ruan, G. Wolkowicz and J. Wu, eds, Differential

Equations with Applications to Biology Fields Institute Communications, American Mathe-

matical Society, Providence, RI, 21 (1998), 15-30.

[2] D. N. Atkinson Mathematical models for plant competition and dispersal, Texas Tech Univer-

sity, Lubbock, TX79409, 1997.

[3] M.G. Bulmer, Periodical insects, The Amer. Natural., 111 (1977), 1099-1117.

[4] J. Campos, R. Ortega and A. Tineo, Homeomorphisms of the disk with trivial dynamics and

extinction of competitive systems, J. Differential Equations, 138 (1997), 157-170.

[5] N.V. Davydova, O. Diekmann and S.A. van Gils, Year class coexistence or competitive exclusion

for strict biennials?, J. Math. Biol., 46 (2003), 95-131.

17

[6] N.V. Davydova, O. Diekmann and S.A. van Gils, On Circulant Populations. I. The algebra of

Semelparity, J. Lin. Algebra and Applications, 398 (2005), 185-243.

[7] O. Diekmann, N.V. Davydova and S.A. van Gils, On a boom and bust year class cycle, J.

Difference Equations and Applications, 11 (2005), 327-335.

[8] O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J.A.J. Metz and H.R. Thieme, On the

formulation and analysis of general deterministic structured population models. II. Nonlinear

theory, J. Math. Biol., 43 (2001), 157-189.

[9] O. Diekmann, M. Gyllenberg and J.A.J. Metz, Steady state analysis of structured population

models, TPB 63 (2003), 309-338.

[10] P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems

with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234.

[11] M. Gyllenberg and Y. Wang, Dynamics of the periodic type-K competitive Kolmogorov systems,

J. Differential Equations, 205 (2004), 50-76.

[12] M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III.

Competing species, Nonlinearity, 1 (1988), 51-71.

[13] M. Hirsch and H. Smith, Monotone Dynamical Systems, to appear Handbook of Differential

Equations, Ordinary Differential Equations (second volume), Elsevier, to appear.

[14] J. Jiang, J. Mierczynski and Y. Wang, Smoothness of the carrying simplex for discrete-time

competitive dynamical systems: a characterization of neat embedding, preprint, 2006.

[15] X. Liang, J. Jiang, The dynamical behaviour of type-K competitive Kolmogorov systems and

its application to three-dimensional type-K competitive Lotka-Volterra systems, Nolinearity,

16 (2003), 785-801.

[16] J. Mierczynski, The C1 property of carrying simplices for a competitive systems of ODEs, J.

Differential Equations, 111 (1994), 385-409.

[17] J. Mierczynski, p-arcs in strongly monotone discrete-time dynamical systems. Differential Inte-

gral Equations, 7 (1994), 1473–1494.

[18] J. Mierczynski, On smoothness of carrying simplicies, Proc. Amer. Math. Soc., 127 (1999),

543-551.

18

[19] J. Mierczynski, Smoothness of carrying simplices for three-dimensional competitive systems: a

counterexample, Dynam. Contin. Discrete Impuls. Systems, 6 (1999), 147–154.

[20] E. MjØlhus, A. Wikan, T. Solberg, On synchronization in semelparous populations, J. Math.

Biol., 50 (2005), 1-21.

[21] R. Ortega and A. Tineo, An exclusion principle for periodic competitive systems in three di-

mension, Nonlinear Analysis, T.M.A., 31 (1998), 883-893.

[22] A. Pakes and R. Maller, Mathematical ecology of plant species competition: a class of determin-

istic models for binary mixtures of plant genotypes, Cambridge University Press, Cambridge,

1990.

[23] L.-I. W. Roeger and L. J. S. Allen, Discrete May-Leonard competition model I, J. Diff. Equ.

Appl., 10 (2004), 77-98.

[24] S. Smale, On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7.

[25] H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive

maps, J. Differential Equations, 64 (1986), 165-194.

[26] H. L. Smith and P. Waltman, “The Theory of the Chemostat”, Cambridge Univ. Press, 1995.

[27] H. L. Smith and B, Li, Competition for essential resources: A brief review, Fields Institute

Communications, 36 (2003), 213-227.

[28] Y. Wang and J. Jiang, The general properties of discrete-time competitive dynamics systems,

J. Differential Equations, 176 (2001), 470-493.

[29] Y. Wang and J. Jiang, The long-run behavior of periodic competitive Kolmogorov systems,

Nonlinear Analysis: Real World Applications, 3 (2002), 471-485.

[30] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simples for the discrete-time

competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632.

[31] A. Wikan and E. MjØlhus, Overcompensatory recruitment and generation delay in discrete

age-structured population models, J. Math. Biol., 35 (1996), 195-239.

[32] D. Xiao and W. Li, Limit cycles for competitive three dimensional Lotka-Volterra system, J.

Differential Equations, 164 (2000), 1-15.

19

[33] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dy-

namics and Stability of Systems, 8 (1993), 189-217.

[34] E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra

systems, Trans. Amer. Math. Soc., 355 (2003), 713-734.

20