Carrying Simplices in Discrete Competitive Systems and Age-structured Semelparous Populations

Transcription

1 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 Wang 1 Department of Mathematics, University of Science and Technology of China Hefei, Anhui, , P. R. China and Ping Yan Department of Mathematics and Statistics, University of Helsinki FIN Helsinki, Finland 1 Supported by FANEDD and NSF of China, partially supported by the Academy of Finland.

2 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 mathematical 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 Poincaré-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 R n : x i 0 for all i}, satisfying the following seven hypotheses: (H1) T is a C 2 -diffeomorphism onto its image T C. (H2) For each nonempty I N := {1, 2,..., n}, the sets A = H I, H + I and Ḣ+ I that T (A) A and T 1 (A) A, where H I = {x R n : x j = 0 for j / I}, H + I Ḣ + I = {x H+ I : x i > 0 for i I}. have the property = C H I and (H3) For each nonempty subset I N and x Ḣ+ I, the I I Jacobian matrix D(T H + )(x) 1 = I (DT (x) 1 ) I = (DT 1 (T x)) I 0, where T H + means the restriction of T on H + I. I (H4) If x C and y = T x 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 u i > 0 with 0 < (d/dx i )(T H + {i} )(u i ) < 1. Hence u i 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 Ḣ+ I, then µ I,p(x) < 1, where µ I,p (x) is the (necessarily real) eigenvalue of the mapping D(T H + ) p (x) with the smallest I modulus. 1

3 x 3 u 3 x 2 u 2 O u 1 x 1 Figure 1: Carrying simplex in 3-D discrete competitive system (H7) For each nonempty subset I N and x, y T i x T i y x i for all i I (where T = (T 1,..., T n )). y i Ḣ+ I, if 0 < T ix < T i y for all i I, then 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 Poincaré 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 k x T k y 0 as k + ; (b) Σ is homeomorphic via radial projection to the (n 1)-dim standard probability simplex := {x C : n i=1 x i = 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

4 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 Poincaré 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. Examples 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 = (h 0,, h k 1 ) and L(h) = h k 1 h h h k 2 0. (1.2) Here N(t) = (N 0 (t),, N k 1 (t)) and N i (t) is the density of the i-th age class in year t, i = 0, 1,, k 1. h i (0 i k 2) is the survival probability of the i-th class, while h k 1 is the per capita expected number of offspring of the (k 1)-th class. For each i = 0, k 1, h i has, for instance, the Beverton-Holt form h i (E) = σ i 1 + g i E, (1.3) 3

5 where σ i, g i are positive constants with k 1 i=0 g i = 1. c = (c 0,, c k 1 ) is a nonnegative constant vector with k 1 i=0 c i = 1. Biologically, c i is called the age-specific impact on the environmental condition, and g i 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 semelparous 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) N(k) such that T j (N(0)) = N j (k) = ( k 1 i=0 h j+i (E i ) ) N j (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 interesting 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 C 1 -diffeomorphism onto its image T C; (H7 ) For nonempty subsets I J N, x Ḣ+ I T i x T i y > x i for all i I. y i and y Ḣ+ J, if T ix < T i y for all i I then Without adopting the hypothesis (H6), we have as our main result 4

6 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 k x T k y 0 as k +. (b)σ is homeomorphic via radial projection to the (n 1)-dim standard probability simplex := {x C : n i=1 x i = 1}. Remark 1.1. The hypothesis example, according to (H2), we can rewrite the mapping T as: (H7 ) can be easily checked in various discrete-time models. For T (x 1,, x n ) = (x 1 G 1 (x), x 2 G 2 (x),, x n G n (x)), for x C, where G i (x) := T i (x) x i if x i 0 T i x i (x) if x i = 0, is a differentiable continuous function (if, for instance, T C 2 ) for i = 1,, n. Assume that G i (x) < 0 for all i, j N and x C (as in differential equations we might call this total x j competition ). Then a straightforward calculation yields that (H7 ) holds. For age-structured semelparous population models, although the condition G i x j (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 l x T l y 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

7 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 = I N, let H I = {x R n : x j = 0 for j / I}. For two vectors x, y H 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 subscript on, <, is dropped if I = N). Let C = {x R n : x 0} be the usual nonnegative cone. The interior of C is the open cone C = {x R n : x 0} and the boundary of C is C. We also let H + I = C H I and Ḣ+ I = {x H + I : x i > 0 for i I}. For any two points x y in R n we define the closed order interval [x, y] = {z R n : x z y} and open order interval [[x, y]] = {z R n : x z y}. A set in R n 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 X A, 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) = {T m x : T m x 0 and m Z + } (O (x) = {T m x : T m x 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 n x T C for 0 n N but T (N+1) x / T C, or T n x T C 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 : T n k x y(k ) for some sequence n k + in Z} and the α-limit set of x by α(x) = {y C : T n k x y(k ) for some sequence n k + in Z}. Note that if O + (x) is 6

8 compact in C, then the ω-limit set of x is nonempty and invariant, i.e., T ω(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 T x < T y, then x < y. Proposition 2.2. For each I N, T is strongly competitive in the interior Ḣ+ I of H + I, i.e., if x H + I, y Ḣ+ I and T x < I T y, then x I y. Furthermore, let u = (u 1, u 2,, u n ), where u i are the fixed points introduced in the statement of (H5). Here we are abusing notation and allowing u i 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 = T k [0, u] k=1 is a nonempty, order-convex global compact attractor of T in C. M := Γ is an unordered invariant compact set containing the fixed points u i, 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 T k C : T j y 0 as j } k=1 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 u i, 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) = {p 0, p 1,, p m 1 }, T i p j = p (i+j) mod m and p 0 = 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

9 Fix(T ) the set of the fixed points of T. A point z is in the lower boundary S of a set S R n provided there is a sequence {y n } in S converging to z with y n z (denoted by y n z), but no sequence {x n } in S converging to z with x n z (denoted by x n z). Then, for i {0, 1,, m 1}, we define, as in [29], M(p i ) = {x C : T nm x p i as n + }; M (p i ) = {x M(p i ) : T nm x p i for sufficiently large n N}; V (p i ) = (M (p i )), and M (O(p)) = V (O(p)) = m 1 i=0 m 1 i=0 M (p i ); V (p i ). 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 T x < T y, then x < y. Strong Competition. For each I N, T is strongly competitive in the interior Ḣ+ I of H + I, i.e., if x H + I, y Ḣ+ I and T x < I T y, then x I y. Proposition 2.6 (Wang and Jiang [29]). Let T : C C be a C 1 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 n x + as n +. Therefore, 8

10 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 1 x < T 1 y. T is strongly competitive in Ḣ+ I Γ if and only if T 1 is monotone in Ḣ+ I Γ, i.e., if x H+ I Γ, y Ḣ+ I Γ with x < I y, then T 1 x I T 1 y. 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 C 1 strongly monotone mappings. Proposition 2.7. Let x Ḣ+ I Γ be an m-periodic point of T. Then there exists a local backward invariant with respect to T m, totally ordered by I one-dimensional submanifold W Ḣ+ I Γ, W I x, homeomorphic to the interval [0, 1) such that precisely one of the following three alternatives applies: (i) T m w I w for all w W \ {x}; (ii) T m w I w for all w W \ {x}; (iii) there is a sequence x k W F ix(t m ) such that x k 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 , 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 n k + such that T n k x z. Without loss of generality assume that T n k y 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 is a decreasing sequence. Thus, y T n x T n y λ := (λ 1,, λ n ) (1,, 1), a contradiction to T n k x z T n k y as n k. Thus we have proved α(x) α(y) =. Suppose that α(x) C. Then one can find a w α(x) C and a sequence n k + such that T n k x w α(x) C S C = M C. Without loss of generality we assume T n i 9

11 that T n k y 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 Ḣ+ I. Then, by Proposition 2.7, there is a local backward invariant with respect to T m, totally ordered by I one-dimensional submanifold W Ḣ+ I, W I x, homeomorphic to the interval [0, 1) such that either (i) T m w I w for all w W \ {x}, or (ii) T m w I w for all w W \ {x}, or else (iii) there is a sequence x k x, x k W F ix(t m ). Case (iii) is impossible, > T m i x k x = (xk ) i for every i I, a contradiction. In case (ii), one can since then, by (H7 ), (xk ) i x i T m i obtain that α(w) = O(x) for all w W, consequently, T ml i other hand, fix any i I, by (H7 ), 1 > w i x i a decreasing sequence. Hence T ml i w x i x i T ml i w T ml i x i x = T w 1(l + ) for every i I. On the ml i w x i for all l N and { T ml i x i w } l N is λ < 1 as l +, a contradiction. So only case (i) holds, and we can find an m-periodic point q Ḣ+ J, J I, with q I w such that T ml w q(l + ) for all w W. We claim that J =. If not, then one can choose a j J such that 0 < q j < w j, and q j hence again by (H7 ), we have T ml j w = T ml j q T ml j w λ < 1(l + ), contradicting T ml w q. Therefore, J =, that is, q = 0. So T l w 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 Ḣ+ 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 T m, totally ordered by I one-dimensional submanifold W Ḣ+ I, x I W, homeomorphic to the interval [0, 1) such that T m w I w for all w W \ {x}. Thus there exists an m-periodic point r Γ Ḣ+ I such that T ml w r(l + ) for all w W. Note I, 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 k x T k y 0 as k +. Proof. The proof goes by induction on n. If n = 1, then Γ is an interval [0, u i ] where u i is some fixed point of T in some axis, which is induced from the hypothesis (H5). Then the statement is trivial. 10

12 From now on we assume n > 1. The induction hypothesis is that Theorem 3.1 holds for systems in R m if m < n. Given any = 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 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 mk z q(k + ) with T mk z q for sufficiently large k N. Fix an k 0 N sufficiently large and let z = T mk 0 z, then 0 z q and T mk z q(k + ). However, by (H7 ), T mk i z = T mk i z q i T mk i q λ i < 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 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

13 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-lifecycle map T, which implies that (H2) holds. Secondly, (H5) is actually concerned with the onedimensional 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 Ḣ+ I. Take any z [0, y], 0 z y, and consider the segment Σ H + I joining y and z, Σ = {sy + (1 s)z : 0 s 1}. Since T C is a relatively open subset of C, it follows that (Σ \ {z}) T C 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} T C, and define Σ := {sy + (1 s)z : σ < s < 1}. As Σ ([0, y] \ {y}) Ḣ+ I, it follows from (H3) that T 1 Σ is a linearly ordered by I subset of [[0, x]] and that T sends bijectively T 1 Σ onto Σ. As T 1 Σ is contained in the compact set [0, x] I, there is ζ := inf T 1 Σ. We have T ζ = T ( lim s σ+ T 1 (sy + (1 s)z)) = lim s σ+ T (T 1 (sy + (1 s)z)) = σy + (1 σ)z. Clearly σ = 0, since otherwise, by (H2) both ζ and T ζ would be in H + I, and that contradicts our choice of σ. Therefore σ = 0 and T ζ = 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 {0,, k 1}. Ḣ+ I, = I Proof. Let ˆT : R k R k, ( ˆT x) i = x i h i (c x) for i = 0,, k 1. We claim that ˆT is injective, ˆT is competitive in C and strongly competitive in Ḣ+ I, I {0,, k 1}. 12

14 In fact, let v = ˆT x, then x i = v i /h i (E) for i = 0,, k 1, where E is equal to c x. Hence which implies that E = c x = 1 = k 1 i=0 k 1 i=0 c i v i h i (E), c i v i Eh i (E). (4.1) Since h i is of Beverton-Holt type, Eh i (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 x i = v i /h i (E) for i = 0,, k 1. Hence ˆT 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 x i = v i /h i (E) implies that the same holds for all component of x, which implies that ˆT is competitive in C, and even strongly competitive in Ḣ+ I. Thus we have proved the claim. It is not difficult to see that the full-life-cycle map T = (S ˆT ) k, where S = is the cyclic forward shift on R k. Therefore, T is injective, competitive in C and strongly competitive in Ḣ+ I, 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 = S ˆT with ˆT 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 E j (x) denote the unique solution of for j = 0,, k 1. 1 = k 1 i=0 c i (P j+1 x) i, (4.2) Eh i (E) Let I = {k 1,, k i } {0,, k 1} be an index set, define σ j I := {(k 1 + j) mod k,, (k i + j) mod k} for j N. It is easy to see that σ k I = I. Now, given any nonempty index subset 13

15 I J N, let x Ḣ+ I and y Ḣ+ J with T ix < T i y for all i I. Then one has P k x < P k y and Pi kx < P i k y for all i I. Moreover, by the competitiveness of P and the property of the shift S, one has P j x < P j y and P j i x < P j i y for all i σj I and j = 1,, k, which implies that P j+1 x < P j+1 y and P j+1 x σ j+1 I P j+1 y 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 E j (x) < E j (y) for all j = 0,, k 1. Hence, for all i J. Note that T i x T i y = (P k x) i (P k y) i = k 1 j=0 h j+i(e j (x)) k 1 j=0 h j+i(e j (y)) > 1, k 1 j=0 h j+i(e j (x)) k 1 j=0 h j+i(e j (y)) xi, y i for all i = 0,, k 1. Then we have T ix T i y > x i y i for all i I. Thus we have completed the proof. σ i Remark 4.2. When h i (E) = one might try to derive an explicit expression for (1.4) but 1 + g i E 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 G i x j < 0 in Remark 1.1 does not always hold. For example, consider k = 2 and the iteration as N 0 (t + 1) = R 0N 1 (t) 1 + N 0 (t), N 1(t + 1) = N 0 (t), where R 0 is the positive constant representing the basic reproduction ratio. Then an easy calculation shows that N 0 (t + 2) = R 0 N 0 (t) 1 + R 0N 1 (t) 1 + N 0 (t) with G 0 (N 0, N 1 ) = R 0, so G 1+ R 0 N 0 1 N 0 > 0 if N 1 > 0. 1+N 0 = G 0 (N 0 (t), N 1 (t))n 0 (t) 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 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

16 the three-species annual plant model can be simplified and expressed as 2(1 b)x 1 (n) x 1 (n + 1) = 1 + x 1 (n) + α 1 x 2 (n) + β 1 x 3 (n) + bx 1(n), 2(1 b)x 2 (n) x 2 (n + 1) = 1 + β 2 x 1 (n) + x 2 (n) + α 2 x 3 (n) + bx 2(n), 2(1 b)x 3 (n) x 3 (n + 1) = 1 + α 3 x 1 (n) + β 3 x 2 (n) + x 3 (n) + bx 3(n), (5.1) where x i (n)(i = 1, 2, 3) is the density of species i at time n, b and α i, β j competition coefficients, respectively, and are the seedbank and for all i = 1, 2, 3. 0 < b < 1, α i > 0, β i > 0, α i 1, β i 1 (5.2) 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 < α i < 1 < β i, 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 x 1 x 2 x 3 = 2(1 b)x 1 1+x 1 +α 1 x 2 +β 1 x 3 + bx 1 2(1 b)x 2 1+β 2 x 1 +x 2 +α 2 x 3 + bx 2 2(1 b)x 3 1+α 3x 1+β 3x 2+x 3 + bx 3 where (x 1, x 2, x 3 ) C := R 3 + = {(x 1, x 2, x 3 ) : x i 0, for i = 1, 2, 3}., (5.3) 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 E 1 (1, 0, 0), E 2 (0, 1, 0), E 3 (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

17 Proof. We denote by DT (x) the derivative of T at x C. It follows from (5.3) that x 1 m 11 m 12 m 13 DT (x) = DT x 2 = m 21 m 22 m 23, (5.4) m 31 m 32 m 33 where m 11 = 2(1 b)(1+α1x2+β1x3) (1+x 1+α 1x 2+β 1x 3) + b, m 2 12 = 2(1 b)β 2 x 2 (1+β 2 x 1 +x 2 +α 2 x 3 ), m 2 22 = 2(1 b)(1+β 2x 1 +α 2 x 3 ) (1+β 2 x 1 +x 2 +α 2 x 3 ) +b, m 2 23 = 2(1 b)β m 32 = 3 x 3 (1+α 3 x 1 +β 3 x 2 +x 3 ), m 2 33 = 2(1 b)(1+α 3x 1 +β 3 x 2 ) (1+α 3 x 1 +β 3 x 2 +x 3 ) + b. 2 x 3 2(1 b)α 1x 1 (1+x 1+α 1x 2+β 1x 3) 2, m 13 = 2(1 b)α 2 x 2 (1+β 2 x 1 +x 2 +α 2 x 3 ) 2, m 31 = 2(1 b)β 1x 1 (1+x 1+α 1x 2+β 1x 3) 2, m 21 = 2(1 b)α 3 x 3 (1+α 3 x 1 +β 3 x 2 +x 3 ) 2, 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 0 for every x Ḣ+ I nonempty subset I {1, 2, 3}. Thus we have proved (H3). and the 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 T x < T y then x < y. To end this, let x, y C and T x < T y. Define the segment I p = {λt x + (1 λ)t y : 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])) = I p and T is one-to-one from γ onto I p. For any z I p, there exists a unique t z [0, 1] such that T (γ(t z )) = z. Furthermore, there exists a neighborhood U z C of z and a neighborhood V γ(tz) C of γ(t z ) such that T : V γ(tz) U z is a homeomorphism. Note that (H3) holds. Then, by [25, Proposition 3.1(iv)], T is a competitive map from V γ(tz ) to U z, that is, if w, v V γ(tz ), T w, T v U z and T w < T v then w < v. Clearly, {U z I p : z I p } forms an open cover of I p. The compactness of I p guarantees the existence of a finite subcover {U j I p : 1 j l} of I p. Similarly as above, one can choose a finitely many neighborhoods V γ(tj ) with t j [0, 1] such that T : V γ(tj ) U j is a homeomorphism for each j = 1,, l. Since I p is totally ordered, one can choose z j U j I p (j = 1, 2,..., l) such that T x = T (γ(0)) z 1 = T (γ(t 1 )) < z 2 = T (γ(t 2 )) < < z l = T (γ(t l )) T (γ(1)) = T y. Note that T is a homeomorphism, and hence competitive, from V γ(tj ) onto U j (j = 1, 2,..., l). Then we obtain x = γ(0) γ(t 1 ) < < γ(t l ) γ(1) = y. Consequently, we have proved that T is competitive, which implies that T is a C 1 -diffeomorphism onto its image T C, i.e., (H1 ) holds. 16

18 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 competition 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 x 1 + α 1 x 2 + β 1 x 3 = 1, β 2 x 1 + x 2 + α 2 x 3 = 1, α 3 x 1 + β 3 x 2 + x 3 = 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 Mathematical Society, Providence, RI, 21 (1998), [2] D. N. Atkinson Mathematical models for plant competition and dispersal, Texas Tech University, Lubbock, TX79409, [3] M.G. Bulmer, Periodical insects, The Amer. Natural., 111 (1977), [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), [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),

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