MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL. 1. Introduction

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1 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL J. M. CUSHING, SHANDELLE M. HENSON, AND CHANTEL C. BLACKBURN Abstract. We show that a discrete time, two species competition model with Ricker (eponential) nonlinearities can ehibit multiple mied-tpe attractors. B this is meant dnamic scenarios in which there are simultaneousl present both coeistence attractors (in which both species are present) and eclusion attractors (in which one species is absent). Recent studies have investigated the inclusion of life ccle stages in competition models as casual mechanism for the eistence of these kinds of multiple attractors. In this paper we investigate the role of nonlinearities in competition models without life ccle stages.. Introduction In [8] the authors utilize a competition model to eplain an unusual coeistence result observed and studied b T. Park and his collaborators in a series of classic eperiments involving two species of insects (from the genus Tribolium) [4], [5], []. The eplanation offered in [8] is based on a single species model (called the LPA model) designed eplicitl to account for the dnamics of the species involved. The LPA model has an impressive track record, spanning several decades, of describing and predicting the dnamics of Tribolium populations, under a variet of circumstances in controlled laborator eperiments dnamics that range from equilibrium and periodic ccles to quasi-periodic and chaotic attractors [], []. This histor of success adds credence to the two species competition model used in [8] (called the competition LPA model) and significant weight to the eplanation given for the observed case of coeistence. The eplanation entails, however, some unusual aspects with regard to classic competition theor, including non-equilibrium dnamics, coeistence under increased intensit of inter-specific competition, and the occurrence of multiple mied-tpe attractors. B multiple mied-tpe attractors we mean a scenario that includes at least one coeistence attractor and at least one eclusion attractor. A coeistence attractor is one in which both species are present. An eclusion attractor is one in which at least one species is absent and at least one species is present. Park observed the coeistence case in an eperimental treatment that also included cases of competitive eclusion, that is to sa, he observed a case of what we have termed to be multiple mied-tpe attractors. Competition theor is primaril an equilibrium theor that is eemplified, for eample, b the classic Lotka-Volterra model and its limited number of asmptotic outcomes: a globall attracting coeistence equilibrium; a globall attracting eclusion equilibrium; or two attracting eclusion equilibria. (In this contet globall attracting means within the positive cone of state space.) These three equilibration alternatives are illustrated b the Leslie-Gower model [] (the discrete analog of the famous Lotka-Volterra differential Date: 7 Ma 007. Ke words and phrases. Competitive eclusion principle, coeistence ccles, multiple attractors. Grant Support: J. M. Cushing and Shandelle M. Henson were supported in part b NSF grant DMS AMS classification (MSC000): Primar 9D40, 9D5; Secondar 9A. Special Acknowledgment: WethankR.F.Costantino,J.Edmunds,S.L.Robertson,andS.LeVargeforhelpfuldiscussions.

2 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL equation model) t+ = b t + s t (.) +c t + c t t+ = b t + s t +c t + c t where t =0,,, and the b i > 0 are the inherent birth rates, s i (0 s i < ) the survival rates, and c ij > 0 the densit dependent effects on newborn recruitment [], [], [6]. Leslie et al. used this model to stud the Tribolium eperiments, but it is incapable of eplaining the observed case of multiple miedtpe attractors. On the other hand, the competition LPA model used in [8] ehibits a greater variet of competition scenarios, including ones with multiple mied-tpe attractors (also see [6], [7]). The competition LPA model, although applied specificall to species of Tribolium in [], is nonetheless a rather general model that, unlike the Leslie-Gower model (or a Lotka-Volterra tpe model in general), accounts for life ccle stages in the competing species. Therefore, the LPA model serves to illustrate that in general (when more biological details are included) competition theor is likel to be considerabl more complicated and varied than that represented b classic Lotka-Volterra tpes of models. The competition LPA model is, like the Leslie-Gower model (.), a discrete time (difference equation) model. It differs from the Leslie-Gower model, however, in two basic was: the state variables of the LPA model account for three life ccle stages for each species (which mathematicall introduces time delas and makes the model higher dimensional) and it utilizes stronger (overcompensator) nonlinearities. A natural question to ask is which of these two mechanisms most accounts for non-lotka-volterra dnamic scenarios and, in particular, for the occurrence of multiple mied-tpe attractors? With regard to the first mechanism, it is shown in [4] that a result of introducing onl a single life ccle stage (specificall, a juvenile stage) in just one species in a Leslie-Gower model (.) can indeed result in multiple mied-tpe attractors specificall the occurrence of eclusion equilibria in the presence of coeistence -ccles (provided inter-specific competition is sufficientl strong). A more robust occurrence of multiple attractors (equilibrium and ccles) of mied tpe occurs if both species are given a juvenile stage [5]. Our goal here is investigate the second mechanism, namel the role of the nonlinearit in the occurrence of multiple mied-tpe attractors. We do this b introducing a Ricker-tpe nonlinearit into the Leslie-Gower model (.) : t+ = b t ep ( c t c t )+s t (.) t+ = b t ep ( c t c t )+s t. In Section we show that this Ricker competition model cannot displa multiple equilibrium attractors of mied tpe, a feature it therefore has in common with the Leslie-Gower model (.) and classic Lotka- Volterra theor. We will show in Section, however, that the Ricker model (.) can ehibit scenarios with multiple mied-tpe attractors in which periodic ccles are present. We provide formal proofs of this possibilit (mathematical details appear in the Appendi) for the case of -ccle and equilibrium scenarios. An investigation for scenarios involving higher period ccles (or quasi-periodic or chaotic attractors) remains to be carried out, although we give in Section 4 a numerical eample involving higher period ccles and quasi-periodic attractors.

3 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL. Equilibria We can assume without loss in generalit (b scaling the units of and ) thatc ii =in the Ricker competition model (.). Therefore, we will consider, after relabeling c as c and c as c, the competition model t+ = b t ep ( t c t )+s t (.) t+ = b t ep ( c t t )+s t. The eclusion equilibria E, (ln n, 0), E, (0, ln n ) R of the Ricker competition model (.) are biologicall feasible (i.e., lie on the positive aes) if and onl if the inherent net reproductive numbers n i, b i / ( s i ) satisf n i >. Besides the trivial equilibrium E 0, (0, 0) and these two eclusion equilibria, there eists onl one other equilibrium: µ ln n c ln n E,, ln n c ln n. (.) c c c c The equilibrium E is a coeistence equilibrium if it lies in the positive cone R+, {(, ) :>0, >0}. Let S, (s,s ) R :0 s,s < ª denote the unit square in R. Lemma.. Assume (s,s ) S. Let ( t, t ) denote the solution of the Ricker competition model (.) with an initial condition ( 0, 0 ) ling in the closure R + of R+. If n < then lim t + t =0. If n < then lim t + t =0. Proof. If n < then all solutions of the linear equation u t+ = b u t + s u t satisf lim t + u t =0. From the inequalit 0 t+ b t + s t and u 0 = 0, an induction shows 0 t u t for all t =0,,,.A similar argument proves the assertion when n <. We assume throughout the rest of the paper that both inherent net reproductive numbers satisf n i >. In this case, all solutions of (.) are bounded and at least one species does not go etinct, as the following dissipativit and persistence theorem shows. The proof appears in the Appendi. Theorem.. Assume (s,s ) S and both n i > in (.). There eist positive constants α, β > 0 such that all solutions with ( 0, 0 ) R +/ {(0, 0)} satisf α lim inf ( t + t ) lim sup t + The equilibrium w =lnn, n = b/ ( s), of t + w t+ = bw t ep ( w t )+sw t ( t + t ) β. is (locall asmptoticall) stable if <n<n cr, ep (/ ( s)). A period doubling bifurcation occurs as n increases through n cr. It follows that a necessar condition for the stabilit of an eclusion equilibrium E i (i =or ) of the competition equations (.) is that the inherent net reproductive numbers n i satisf <n i <n cr i, ep (/ ( s i )). (.) The linearization principle provides sufficient conditions for stabilit according to the magnitude of the eigenvalues of the Jacobian J (, ) associated with (.) evaluated at an equilibrium point E i =( e, e ): Ã! ( s ) e c ( s ) e J ( e, e )=. (.4) c ( s ) e ( s ) e

4 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 4 The Jacobians of the equilibria E i, i =or, are triangular matrices whose eigenvalues appear along the diagonal. The equilibrium E i, i =or, is hperbolic if both eigenvalues ( s i )( ln n i )+s i, b j n c j i + s j, j 6= i have absolute value unequal to and, b the linearization principle [9], is (locall asmptoticall) stable if both have absolute value less than. Thus, a necessar condition that E i be hperbolic and stable is that c j > ln n j / ln n i, j 6= i. (.5) Sufficient for E i to be hperbolic and stable is that, in addition, the inequalities (.) hold. Theorem.. Assume (s,s ) S, that one of the inequalities (.5) holds, and that E is a coeistence equilibrium. Then E is unstable. Proof. If one of the inequalities (.5) holds and if E is a coeistence equilibrium, then the formula (.) for E implies c c < 0. A calculation shows +detj ( e, e ) trj ( e, e )=( c c ) e e ( s )( s ) < 0. The Jur criteria for instabilit impl that at least one eigenvalue of J ( e, e ) has magnitude greater than. It follows from Theorem. that if at least one eclusion equilibrium is (hperbolic and) stable, then either E is not a coeistence equilibrium or, if it is, it is unstable. Consequentl, with regard to equilibria, a mied-tpe multiple attractor scenario is impossible for the competition model (.). Thus, the Ricker competition model (.) and the classic Lotka-Volterra competition model have in common the impossibilit of multiple mied-tpe equilibrium attractors. In the net section we show, on the other hand, that it is possible for the Ricker model (.) to have multiple mied-tpe non-equilibrium attractors.. Multiple mied-tpe Attractors We want to investigate the possible occurrence of mied-tpe non-equilibrium attractors in the Ricker model (.) under smmetricall high inter-specific competition (as has been observed in more complicated models that include juvenile life ccle stages [6], [8], [], [4], [5]). To carr out this investigation b means of a single parameter problem, we introduce the notation r, c /c, c, c and re-write the competition model (.) as t+ = n ( s ) t ep ( t c t )+s t t+ = n ( s ) t ep ( rc t t )+s t (.) n i >, 0 s i <, and r, c > 0. Our goal is, for fied birth rates b i, survivorships s i and competition ratio r, to investigate the eistence and stabilit of non-equilibrium coeistence attractors as functions of the inter-specific competition intensit coefficient c. In this paper we restrict attention to coeistence -ccles. The source of these coeistence Both eigenvalues of a matri A have absolute value less than if and onl if, < det A< and ( + det A) < tra <+deta. At least one eigenvalue has absolute value greater if and onl if one of the inequalities is reversed.

5 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 5 -ccles will be a competitive eclusion -ccle, that is to sa, a -ccle on a coordinate ais that undergoes a loss of stabilit. In the absence of species t the dnamics of species t are governed b the Ricker model equation t+ = b t ep ( t )+s t. (.) A period doubling bifurcation occurs at the critical value b cr, ( s )ep(/( s )) of b at which point the equilibrium =lnn equals cr, / ( s ). This bifurcation results in a (locall asmptoticall) stable -ccle 0 0 (.) 0 < < 0 for b greater than but near b cr, ( s )ep(/( s )), (.4) which we write as b ' b cr.thetwopoints0, of the -ccle (.) satisf the equations 0 = n ( s ) ep ( )+s = n ( s ) 0 ep ( 0)+s 0. This -ccle is stable (b the linearization principle) because the product of the derivative of the map (.) evaluated at 0 and at is less than one in absolute value, i.e. holds under (.4). The -ccle (.) ields an eclusion -ccle λ, Q j=0 n ( s ) j ep j + s < (.5) (0, 0) (0, ) (0, 0) (0, ) (.6) of the Ricker competition model (.). This -ccle is stable on the (invariant) -ais under the assumption (.4). Our first goal is to stud the stabilit of this eclusion -ccle in the, -plane and determine how it depends on the competition intensit c. Specificall, we will show a planar loss of stabilit occurs at a critical value c of c, the result of which is a (transcritical) bifurcation of non-eclusion -ccles. B the linearization principle, the eclusion -ccle (.6) is (locall asmptoticall) stable if the spectral radius of the matri J(0,0)J(0, ) is less than one. A calculation shows this matri is triangular and its eigenvalues are λ = Q j=0 n ( s )ep cj + s > 0, λ = λ. Under the assumption (.4), 0 <λ < (see (.5)). As a function of c, thefirst eigenvalue λ = λ (c) is decreasing and satisfies λ (0) = (n ( s )+s ) >, lim λ (c) =s s <. c + It follows that there eists a unique c > 0 such that λ (c )=.

6 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 6 Theorem.. Assume (s,s ) S and that b >b cr is such that (.) has a stable -ccle. Let c denote the unique positive root of the equation Q j=0 n ( s )ep cj + s =. (.7) The eclusion -ccle (.6) of the competition model (.) is (locall asmptoticall) stable for c>c and unstable for c<c. The loss of stabilit of the eclusion -ccle (.6) described in Theorem. suggests the occurrence of a bifurcation of planar -ccles from the eclusion -ccle (.6). -ccles of the map defined b (.) correspond to fied points of the composite map. The point (, ) =(0,0), corresponding to the eclusion -ccle (.6), is a fied point of the composite for all values of c. On the other hand, a positive fied point (, ) R+ of the composite corresponds to a coeistence -ccle. Positive fied points of the composite satisf the equations (obtained from the composite equations after and are cancelled) f(,, c) =0, g(,, c) =0 (.8) where f(,, c), + b e c + s b ep b e c + s c b e rc + s + s g(,, c), + b e rc + s b ep rc b e c + s b e rc + s + s. Note that b the wa that c is defined, the point (, ) =(0,0) still satisfies these equations when c = c, i. e., f(0,0,c )=0and g(0,0,c )=0. The Implicit Function Theorem implies the eistence of a solution branch (,, c) =(, (), c()) of equations (.8) that passes through this point, i.e., a branch such that (0,(0),c(0)) = (0,0,c ), provided the Jacobian of f and g with respect to and is non-singular when evaluated at (, ) =(0,0). Itisdifficult in general to relate this non-singularit condition in a simple wa to the parameters b i and s i in the competition model (.). In the Appendi it is shown, however, that the non-singularit condition does hold for b ' b cr. The analsis utilizes the lowest order terms in Lapunov- Schmidt epansions of the bifurcating eclusion -ccle (.6), which in turn are then used to estimate the bifurcation value c of the bifurcating -ccles generated b the solution branch (,, c) =(, (),c()). In that analsis, attention is restricted to b ling on the interval I, {b : s <b <b cr }, b cr, ( s )ep(/ ( s )). For b I the Ricker equation t+ = n ( s ) t ep ( t c t )+s t has a stable equilibrium. Theorem.. Assume (s,s ) S and b I. Ifb ' b cr, then a branch of coeistence -ccles bifurcates from the eclusion -ccle (.6) at c = c. B Theorem., the eclusion -ccle (.6) loses stabilit as c decreases through c. B the echange of stabilit principle ([0], p. 6) the bifurcating coeistence -ccles guaranteed b Theorem. are (locall asmptoticall) stable if the eist for c / c (and unstable if the eist for c ' c ). Accordingl, our net goal is to determine the conditions under which the bifurcating coeistence -ccles occur for c / c.that is to sa, we want to determine when c 0 (0) < 0 for the solution branch (,, c) =(, (),c()) of equations

7 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 7 (.8). We can utilize the Lapunov-Schmidt epansions used in the Appendi to established Theorem. to calculate an epansion of c 0 (0) for b near b cr, the lowest order terms of which determine the sign of c0 (0). Details appear in the Appendi. To describe the results of this analsis, we need some further notation. We partition the unit square into the union S = S S, S S = where S is the set of points (s,s ) S that satisf either 0 s s < or satisf 0 s <s < and 6(s s ) 8 6s s + ( s ) s > 0 and where S is the set of points (s,s ) S that satisf 0 s <s < or 6(s s ) 8 6s s + ( s ) s 0. See Figure. Two critical numbers b ±,lingintheintervali and satisfing b <b+,aredefined b (4.0) and (4.) in the Appendi. Also defined in the Appendi, b formula (4.), is a critical value r of the ratio r. Insert FIGURE near here Theorem.. Assume b I. For b ' b cr, and c / c the bifurcating coeistence -ccles of the competition model (.) (guaranteed b Theorem.) are stable in either of the following cases. () (s,s ) S and either (a) b <b <b + (b) b <b and r<r (c) b >b + and r<r ; () (s,s ) S and r<r. The subinterval b <b <b + in Theorem.4 is centered on the value µ s ( s )+s ( s ) ( s )ep, s ( s )( s ) which supplies a rough estimate for those b for which the theorem applies. According to (.5), the eclusion equilibrium E is stable if rc > ln n / ln n. (.9) For b b cr it follows that n cr ep (/ ( s )) and from Lemma 4. in the Appendi that c ( s )lnn. Thus, if r>r, 4( s ) ln n,thenforc / c and b ' b cr the inequalit (.9) holds and E is stable. In order for both the coeistence -ccles and the eclusion equilibrium to be stable in the cases (b,c) and () of Theorem., it is required that r <r<r. Necessar for this requirement is r <r. This inequalit is characterized in Lemma 4.6 of the Appendi. The results, together with Theorem., lead to the following theorem. Theorem.4. Assume b I. For b ' b cr,andc/ c the eclusion equilibrium E and the bifurcating coeistence -ccles of the competition model (.) are both stable if (s,s ) S and one of the following cases holds:

8 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 8 () b <b <b + () b / b and r <r<r () b ' b + and r <r<r. This theorem provides conditions on the parameters in the competition model (.) under which there are multiple mied-tpe attractors (specificall, a -ccle and an equilibrium). It follows from Lemma 4.6 of the Appendi that in the cases not covered in Theorem.4 (namel when (s,s ) S or when (s,s ) S and b is near the endpoints s and b cr of the interval I) either the -ccle is unstable or the equilibrium E is unstable. 4. Discussion The Ricker competition model (.) can possess multiple mied-tpe attractors. Theorem.4 provides some conditions under which there eist both a stable eclusion equilibrium and a stable coeistence -ccle. That theorem deals with values of b greater than (but near) the critical period doubling bifurcation value b cr,valuesofb less than the critical value b cr, and with the inter-specific competition coefficient c near a specified critical value c. The theorem also requires that the survivorships (s,s ) lie in the region S of Figure. This latter assumption means that the survivorship s of species is larger than the survivorship s of species. Therefore, Theorem.4 requires that there be an asmmetr between the two species in the sense that one species has a high reproductive rate and low survivorship in contrast to the other species, which has a low reproductive rate and a high survivorship. Figure illustrates the eistence of multiple mied-tpe attractors under these conditions. Insert FIGURE near here Theorem.4 implies the local bifurcation of stable coeistence -ccle onl for c sufficientl large, namel, near the critical point c. An interesting question concerns the global etent of this bifurcating branch of -ccles. What is the spectrum of c values for which these coeistence -ccles occur? Numerous numerical eplorations have shown that the bifurcation sequence displaed in Figure is tpical. As c decreases, and the coeistence -ccles bifurcate from the eclusion -ccle on the -ais at c = c, there eists a second critical value of c at which the coeistence -ccles lose stabilit because of an invariant loop (Sacker/Naimark or discrete Hopf) bifurcation. The resulting coeistence (double) invariant loops persist until c reaches a third critical value at which the loops disappear in a global heteroclinic bifurcation. See Figures and 4. Insert FIGURES & 4 near here In this paper we have shown that the Ricker competition model (.) cannot displa a multiple miedtpe attractor scenario with onl equilibria. On the other hand, Theorem.4 shows that multiple miedtpe attractor scenarios are possible with non-equilibrium attractors, specificall, with stable competitive eclusion equilibria and stable coeistence -ccles. Multiple mied-tpe attractors scenarios are also possible for model (.) that involve other combinations of higher period ccles, quasi-periodic (as in Figure 4) and even chaotic attractors. Figure 5 shows one eample. The analsis of such multiple attractor cases remains an open problem. Insert FIGURE 5 near here

9 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 9 References [] R.F.Costantino,R.A.Desharnais,J.M.Cushing,B.Dennis,S.M.Henson,andA.A.King,Theflour beetle Tribolium as an effective tool of discover, Advances in Ecological Research, Volume 7, 005, 0-4 [] J. M. Cushing, R. F. Costantino, B. Dennis, R. A. Desharnais and S. M. Henson, Chaos in Ecolog: Eperimental Nonlinear Dnamics, Academic Press, New York, 00 [] J. M. Cushing, S. LeVarge, N. Chitnis, S. M. Henson, Some discrete competition models and the competitive eclusion principle, Journal of Difference Equations and Applications 0, No. -5 (004), 9-5 [4] J. M. Cushing and S. LeVarge, Some discrete competition models and the principle of competitive eclusion, Difference Equations and Discrete Dnamical Sstems, Proceedings of the Ninth International Conference, World Scientific, 005 [5] J. M. Cushing, S. M. Henson and Lih-Ing Roeger, A competition model for species with juvenile-adult life ccle stages, Journal of Biological Dnamics, No. (007), 0- [6] J. Edmunds, A Stud of a Stage-structured Model of Two Competing Species, Ph.D. dissertation, The Universit of Arizona, 00 [7] J. Edmunds, Multiple Attractors in a Discrete Model of Two Competing Species, preprint [8] J. Edmunds, J. M. Cushing, R. F. Costantino, S. M. Henson, B. Dennis and R. A. Desharnais, Park s Tribolium competition eperiments: a non-equilibrium species coeistence hpothesis, Journal of Animal Ecolog 7 (00), 70-7 [9] S. N. Eladi, An Introduction to Difference Equations, rd Edition, Springer-Verlag, New York, 005 [0] H. Kielhofer, Bifurcation Theor: An Introduction with Applications to PDEs, Series in Applied Mathematical Sciences, Vol. 56, Springer, Berlin, 004, p. 6 [] M. Kulenović and O. Merino, Competitive-Eclusion versus Competitive-Coeistence for Sstems in the Plane, Discrete and Continuous Dnamical Sstems Ser. B 6 (006), 4-56 [] P. H. Leslie and J. C. Gower, The properties of a stochastic model for two competing species, Biometrika 45 (958), 6-0. [] P. H. Leslie, T. Park and D. M. Mertz, The effect of varing the initial numbers on the outcome of competition between two Tribolium species, JournalofAnimalEcolog7 (968), 9 [4] T. Park, Eperimental studies of interspecies competition. III, Relation of initial species proportion to the competitive outcome in populations of Tribolium, Phsiological Zoolog 0 (957), -40 [5] T. Park, P. H. Leslie and D. B. Mertz, Genetic strains and competition in populations of Tribolium, Phsiological Zoolog 7 (964), 97 6 [6] H. L. Smith, Planar competitive and cooperative difference equations, JournalofDifference Equations and Applications (998), 5-57 APPENDIX The proof of Theorem. utilizes the following lemma. Lemma 4.. Consider the difference equation z t+ = bz t ep ( kz t )+sz t with z 0,b,k > 0, s 0 and b + s >. There eist positive constants α, β > 0 (independent of z 0 ) such that the solution satisfies α lim inf t + z t lim sup t + z t β. Proof. The maimum of bz ep ( kz) for z 0 is bk e. The inequalities 0 <z t+ bk e + s t and an induction show 0 <z t u t, t =0,,, (4.) where u t is the solution of the linear difference equation u t+ = bk e + su t with u 0 = z 0. Since s< it follows that lim t + u t = bk e / ( s) > 0. For an number β>bk e / ( s), ansolutionu t satisfies u t <βfor all large t. B(4.)itfollowsthatthereeistsat = t (z 0 ) such that z t β for t t. (4.)

10 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 0 For an α satisfing 0 <α<βthe smooth function f(z) =bz ep ( kz) +sz is positive on the interval α z β. Since f(0) = 0, the minimum of f(z) on the interval α z β occurs at z = α if α is sufficientl small: min α z β f(z) =f(α).for all sufficientl small α>0 it follows from f 0 (0) = b + s> and from the mean value theorem that f(α) >α.pick a number m between b + s and (their mean, for eample). B the continuit of f 0 there eists an α>0 so small the f 0 (z) >m> for z on the interval 0 <z<α.for z on this interval we have, b the mean value theorem, that f(z) =f 0 (ξ)z for some ξ satisfing 0 ξ z. Thus, f(z) >mzfor z on the interval 0 ξ z. Thus, z t+ >mz t or z t >m t z 0 for as long as z t <α. It follows that from an point in the interval 0 <z 0 <α(for an α>0 sufficientl small) the solution z t will eceed α in a finite number of steps. At this point, we know that for t t the solution satisfies z t β and that if for some t t it happens that z t <αthen there eists a t >t such that z t α. B induction it follows that for all subsequent t t we have z t α. This follows from the fact that α z β = f(z) min α z β f(z) =f(α) >α. In summar, we have shown that for an z 0 > 0 there eists a time t > 0 such that α<z t <βfor all t t and the lemma follows immediatel. Proof of Theorem.. If 0 =0or 0 =0the result follows from Lemma 4.. Suppose ( 0, 0 ) R+/ {(0, 0)}. Note that the maimum of the function b ep( ) for 0 is be. The inequalities 0 < t+ b e + s t and 0 < t+ b e + s t, together with a straightforward induction, show that 0 < t u t and 0 < t v t where u t and v t satisf the linear difference equations u t+ = b e + s u t and v t+ = b e + s v t with initial conditions u 0 = 0 and v 0 = 0. Because s i <, we have that lim t + u t = b e ( s ) and lim t + v t = b e ( s ).Asaresult 0 < lim sup t b e ( s ), t + 0 < lim sup t b e ( s ). t + Define β, b e ( s ) + b e ( s ). From the inequalities b t ep ( k ( t + t )) + s t b t ep ( t c t )+s t = t+ b t ep ( k ( t + t )) + s t b t ep ( c t t )+s t = t+, where k, ma {,c,c },s, min {s,s }, and b, min {b,b }, we obtain (b addition) the inequalit b ( t + t )ep( k( t + t )) + s ( t + t ) t+ + t+. An induction shows 0 <w t t+ + t+ (4.) where w t satisfies the difference equation w t+ = bw t ep ( kw t )+sw t (4.4) with w 0 = > 0. Note that because both n i > we have that b + s>. Lemma 4. implies the eistence of a constant α>0such that α lim inf t + w t which, together with (4.), implies α lim inf t + ( t + t ). ProofofTheorem..Define z = 0 and w = c c and re-write the composite, fied point equations (.8) as p(, z, w) =0, q(, z, w) =0 (4.5)

11 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL where p(, z, w), f(, 0 + z,c + w) and q(, z, w), g(, 0 + z,c + w).b the Implicit Function Theorem there eists a (unique, analtic) solution pair z = z() and w = w() of (4.5), for on an open interval containing =0,thatsatisfies z(0) = w(0) = 0 provided δ, p z q w p w q z (0,0,0) 6=0. A straightforward calculation shows g w (0,0,0) =0and hence δ, p w q z (0,0,0). (4.6) The solution (, ) =(, ()) is a fied point of the composite equations (.8) for c = c() that corresponds to (i.e., is the first component of) a -ccle point of the competition equation (.). When =0, and hence c = c and = 0, this branch of -ccles intersects the eclusion ccle (.6). For ' 0 the fied point (, ()) R+ corresponds to a coeistence -ccle of (.). The proof of Theorem. will be complete when we show that δ 6= 0for b >b cr sufficientl close to b cr, i.e., for b ' b cr. This investigation makes use of approimations obtained from a parameterization of the bifurcating -ccles. The first step is to obtain approimations of the eclusion -ccle on the -ais. Lemma 4.. Assume (s,s ) S and b ' b cr. The bifurcating stable -ccles (.) of the Ricker equation t+ = b t ep ( t )+s t, 0 > 0 have, for ε 0, the representations b = b cr + 6s s + ε + O ε 6( s ) 0 = + ε + s +6s ε + O ε s 6( s ) = ε + +s s 6( s ) ε + O ε. Proof. The point 0 6= lnn on a -ccle (.) of the Ricker equation is a fied point of the composite map. The fied point equation reduces, after the cancellation of, to + b e + s b ep b e + s + s =0. To center this equation on the equilibrium ln n,letz = ln n and re-write the equation as h (z,b )=0 where h (z,b ), + s +( s ) e z s + b ep s (z +lnn ) (z +lnn )( s ) e z. Since h(0,b )=0for all b and since we are interested in fied points z 6= 0(i.e., 6= lnn ), we define k(z,b )=h(z,b )/z and re-write the equation for z and b as k(z, b )=0. We let ε = z and calculate the lower order coefficients in the epansion b = b cr + β ε + β ε + O ε of the solution of this equation. From the epansion k ε, b cr + β ε + β ε ( s ) = β ε b cr (s ) 6 ³s 6s +6β e s ε + O ε

12 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL we conclude that β =0and β = b cr 6s s + /6( s ), which ields the epansion for b in the Lemma. Then from n, ln (b / ( s )) we obtain ln n = + s +6s ε + O ε 4 s 6( s ) and see that the fied point = z +lnn has the epansion given in the Lemma. We can calculate the epansion of the second point = b 0 e 0 + s 0 on the -ccle from the epansions for b and 0.The result is that given in the statement of the Lemma. Lemma 4.. Assume (s,s ) S and b ' b cr. The critical value c (at which the eclusion -ccle (.6) loses stabilit) has, for ε 0, the representation c = ( s )lnn + µ s ( s ) ln n s ε + O ε. Proof. The critical value c is the unique root of the equation (.7). B means of the epansions from Lemma 4. and this equation, we seek the coefficients in the epansion c = c 0 + c ε + c ε + O ε of the root. Substitution of these ε-epansions into the left hand side of (.7) and epanding in ε results, to first order, in µ µ s + b ep c 0 4b µ c ep c 0 ε + O ε = s s s and consequentl c 0 =( s )(lnn ) / and c =0. An epansion of the left hand side to second order then results in s c + s + c 0 s ( s ) c 0 ε + O ε = s and hence c + s + c 0 s ( s ) c 0 =0which, when solved for c, leads to the formula in the Lemma. Using the ε epansions provided b Lemmas 4. and 4., we can obtain ε epansions, and hence lower order approimations, of δ for b ' b cr for ε 0. To do this, we need to calculate (with the aid of a smbolic computer program) the partial derivatives p z (0,0,0), p (0,0,0), q z (0,0,0), q (0,0,0) and p w (0,0,0). These have complicated formulas onl one of which we displa here: p w (0,0,0) = 0b s e 0 c 0b h ³ ³ ep s + b e 0 s ³s + b e 0 0c. ³ + b +s + b e 0 e c i 0 Note that p w (0,0,0) < 0. Fromtheε-epansions in Lemmas 4. and 4. we find Notice that p z (0,0,0) = ( s )( s )lnn (s + s s ln n s ln n ) ε + O ε p (0,0,0) = s ³ r ( s ) (ln n ) 4 + O(ε) q z (0,0,0) = s 6s s + ε + O ε q (0,0,0) = r ( s ) ln n 4 ( ( s )) ε + O ε. q z (0,0,0) < 0 for b ' b cr (4.7)

13 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL (b ' b cr isthesameasε 0). This is because 6s s +> 0 for 0 s. Therefore, b (4.6) δ 6= 0 for b ' b cr and the proof of Theorem. is complete. Proof of Theorem.. Our goal is to determine when c 0 (0) = w 0 (0) < 0. Fromtheequationsp(, z(),w()) = 0, q(, z(),w()) = 0 we obtain b implicit differentiation that w 0 (0) = ρδ where ρ, p q z p z q (0,0,0). (4.8) It follows that w 0 (0) < 0 if δ and ρ have opposite signs. In the proof of Theorem. above we showed that p w (0,0,0) < 0 and consequentl b (4.6) the sign of δ isthesameasq z (0,0,0). It follows that for b ' b cr, δ is negative (see (4.7)) and we conclude that w 0 (0) < 0 if ρ>0. Forb ' b cr the formula (4.8) for ρ and the epansions calculated in the proof of Theorem. ield ρ = Ωε + O ε where with Ω 0, ( s )( s ) 6s s + / and Ω = Ω 0 + Ω r (4.9) Ω, ( s )( s ) ln n m(ln n ) 4 m(ξ), 4 s + +6( s )(( s ) s +( s ) s ) ξ s ( s ) ( s ) ξ. It follows that w 0 (0) < 0 for b ' b cr if Ω > 0 (and w 0 (0) < 0 if Ω < 0). Lemma 4.4. Assume (s,s ) S and b ' b cr.letc denotetheuniquepositiverootoftheequation(.7). If Ω > 0 then the bifurcating branch of coeistence -ccles occurs for c / c and the -ccles are (locall asmptoticall) stable. If Ω < 0 then the bifurcating branch of coeistence -ccles in Ω occurs for c ' c and the -ccles are unstable. The proof of Theorem. will be complete when we determine the parameter values s,s,b, and r for which Ω > 0. Since Ω 0 > 0 for (s,s ) S, the sign of Ω in (4.9) depends on that of Ω, which in turn is the sign of the factor m(ln n ).Thetermm(ζ) is a quadratic polnomial in ξ. Restricting our attention to b I, we need onl investigate the sign of m(ξ) for ξ on the interval 0 <ξ</ ( s ) which we denote b I.Note that m 00 (ξ) = 6s ( s )( s ) < 0 and at the endpoints of I we find that m(0) = 4 s + < 0 and m((/ ( s )) = 4 6s s + < 0.The maimum of the quadratic m(ξ) occurs at ξ ma, s ( s )+s ( s ) > 0. s ( s )( s ) It follows that m(ξ) > 0, and hence Ω > 0 if and onl if ξ ma I, m (ξ ma ) > 0 and ξ.lies between the two roots of m(ξ) r 6(( s ) s +( s ) s ) ± 6 ³6(s s ) 8(6s ξ ± s +)( s ) s =, (4.0) 6s ( s )( s ) which in this case lie in I. Calculations show ξ ma I, m (ξ ma ) > 0 if and onl if (s,s ) S. We conclude that Ω > 0 if (s,s ) S and ξ lies between ξ ±. If, on the other hand, (s,s ) S and ξ does not lie between ξ ± or if (s,s ) / S (i.e., if (s,s ) S ), then m(ln n ) < 0 and hence Ω < 0. In these

14 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 4 cases Ω = Ω 0 + rω > 0 if and onl if r<r where r, Ω 0 /Ω = 6 6s s + ( s ) ln n m(ln n ). (4.) The roots (4.0) correspond to b ± =( s )ep ξ ±. (4.) We summarize these results in the following lemma. Lemma 4.5. Assume (s,s ) S and b I. Also assume b ' b cr,andc/ c. Then Ω > 0 in either of the following cases. () (s,s ) S and either (a) b <b <b + (b) b <b and r<r or (c) b >b + and r<r ; () (s,s ) S and r<r. Theorem. follows immediatel from Lemmas 4.4 and 4.5. In order to satisf r<r (to obtain the stabilit of the bifurcating coeistence -ccles in cases (b,c) and () of Theorem.) and also r>r (to obtain the stabilit of E ), it is necessar that r <r.this inequalit is equivalent to 0 <m(ξ)+4 6s s + where ξ =lnn I.(Recallm(ln n ) < 0 in cases (b) and () of Theorem.). This inequalit does not hold at the endpoints of the interval I. This is because m(0) + 4 6s s + = ( s ) s < 0 and m(/ ( s )) + 4 6s s + =0.Incase () the maimum of m (ξ) occurs at the right endpoint ξ =/ ( s ) and as a result the inequalit r <r does not hold. In cases (b,c) then inequalit holds at and near the roots ξ ± of m(ξ). Lemma 4.6. In case () of Lemma 4.5 r r. In cases (b,c) of Lemma 4.5, r <r if and onl if b / b or b ' b +.

15 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 5 FIGURE CAPTIONS FIGURE : The unit square S for the survivorship parameters s and s in the competition model (.) is partitioned into to sub-regions S and S corresponding to the two case in Theorems.. FIGURE : Each plot shows a solution of the Ricker competition model (.) with b =8,b =0, s =0.65, s =0, r =.and c =.9. In plot (a) the initial conditions ( 0, 0 )=(0.,.5) lead to competitive eclusion. In (b) the initial conditions ( 0, 0 )=(0.9,.5) lead to a competitive coeistence -ccle. See Figure (a). FIGURE : A sequence of phase plane plots shows the bifurcation of stable coeistence -ccles from the eclusion -ccles on the -ais in the Ricker competition model (.) as the competition coefficient c decreases through the critical value c.5. Model parameters are b =8,b =0, s =0.65, s =0, and r =.. Plot (a) shows a sequence of stable -ccles (open circles with connecting lines) that eventuall destabilize and give rise to stable, double invariant loops as shown in plot (b). In plot (c) the double invariant loops eventuall collide, under further decreases in c, and undergo a global, heteroclinic bifurcation involving the (saddle) coeistence equilibrium, the eclusion (saddle) equilibrium, the eclusion (saddle) -ccle located and their stable and unstable manifolds. For the parameter values in these plots, the eclusion equilibrium E :(, ) (.86, 0) is also stable and hence these plots contain multiple mied-tpe attractors. FIGURE 4: Each graph shows a solution of the Ricker competition model (.) with b =8,b =0, s =0.65, s =0, r =.and c =.8. In plot (a) the initial conditions ( 0, 0 )=(0.,.5) lead to competitive eclusion. In plot (b) the initial conditions ( 0, 0 )=(0.0,.5) lead to a competitive coeistence quasi-periodic oscillation (see Figure (b,c)). FIGURE 5: A sequence of phase plane plots shows the bifurcation of stable coeistence 4-ccles from the eclusion 4-ccles on the -ais in the Ricker competition model (.) as c decreases from the critical value c Model parameters are b =8,b =4, s =0.8, s =0, r =0.8 and c =.9. Plot(a)shows a sequence of 4-ccles the undergoes a period-halving bifurcation to -ccles which ultimatel destabilize and give rise to stable, double invariant loops. As c decreases further, plot (b) shows the double invariant loops, which occasionall period lock, eventuall giving rise to chaotic attractors. The chaotic attractors suddenl disappears when an interior crisis occurs at a critical value of c. For the parameter values in these plots, the eclusion equilibrium E :(, ) (.69, 0) is also stable and hence these plots contain multiple mied-tpe attractors.

16 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL s 0.6 S S s Figure (a) (b) t t Figure a Figure b c =.5 c =. c =. c = c =.9 (a) (b) c =.85 c =.85 c = Figure a c = (c) Figure b c =.79 c = Figure c

17 MULTIPLE MIXED-TYPE ATTRACTORS IN A COMPETITION MODEL 7 c =.8 c = t t Figure 4a Figure 4b 5 4 c = 4.6 c = 4. c = 4. c =.5 (a) c = c = (b) c =.65 c =.6 c =.5 c =.44 c =.4 c = Figure 5a Figure 5b Department of Mathematics, Interdisciplinar Program in Applied Mathematics, Universit of Arizona, Tucson, AZ 857 USA address: cushing@math.arizona.edu URL: Department of Mathematics, Andrews Universit, Berrien Springs, MI 4904 USA address: henson@andrews.edu URL: Department of Mathematics, Universit of Arizona, Tucson, AZ 857 USA address: cblackburn@math.arizona.edu