PERSISTENCE DEFINITIONS AND THEIR CONNECTIONS

Transcription

1 prceedings f the american mathematical sciety Vlume 109, Number 4, August 1990 PERSISTENCE DEFINITIONS AND THEIR CONNECTIONS H. I. FREEDMAN AND P. MOSON (Cmmunicated by Kenneth R. Meyer) Abstract. We give varius definitins f types f persistence f a dynamical system and establish a hierarchy amng them by prving implicatins and demnstrating cunterexamples. Under apprpriate cnditins, we shw that several f the definitins are equivalent. 1. Intrductin The main purpse f this paper is t cnsider varius frms f persistence (defined in the next sectin) in dynamical systems and t establish a hierarchy amng them. The wrk in this paper may be thught f as a cntinuatin f wrk dne in [2] (als see [3]). In [2], the cncepts f weak persistence, persistence and unifrm persistence were defined fr the first time fr dynamical systems in a lcally cmpact metric space with respect t sets with bundary and nnempty interir. Frm the definitins it was clear that unifrm persistence implies persistence, which in turn implies weak persistence. It was then shwn that, under certain circumstances, weak persistence implies unifrm persistence. In this paper, we define tw additinal cncepts f persistence, ne which is weaker than weak persistence, dented P^-weak persistence, and has been used in applicatins, and the ther which is a unifrm versin f weak persistence, dented weak unifrm persistence. Amng ther results, we prve that if the flw is dissipative r if the bundary is cmpact, then weak unifrm persistence is equivalent t unifrm persistence. Previus wrk in persistence thery has typically dealt with the abstract thery, with applicatins, r with bth. Criteria fr the equivalence f varius frms f persistence were cnsidered in [2, 3]. Criteria fr ne r mre frms Received by the editrs April 18, 1989 and, in revised frm, August 28, Mathematics Subject Classificatin (1985 Revisin). Primary 34C35; Secndary 58F25, 92A15. The first authr's research was partially supprted by the Natural Sciences and Engineering Research Cuncil f Canada, Grant N. NSERC A4823. The secnd authr's research was partially supprted by the Hungarian Natinal Fundatin fr Scientific Research and was carried ut while the authr was a WUSC research assciate at the University f Alberta American Mathematical Sciety /90 $1.00+ $.25 per page License r cpyright restrictins may apply t redistributin; see

2 1026 H. I. FREEDMAN AND P. MOSON f persistence t hld in general dynamical systems were given in [3, 4, 7, 10, 11, 13, 18, 19], in Ltka-Vlterra systems in [10, 11, 13, 14], in infinite dimensinal systems in [1, 9]. Persistence thery as applied t ppulatin survival r extinctin was discussed in [5, 6, 12, 14, 15, 18]. There are als many papers dealing with persistence thery exclusively fr discrete dynamical and semi-dynamical systems which are cited in the abve references. In a theretical sense persistence definitins can be interpreted as a cmplete instability r as bundedness-like qualitative cncepts (see [17, Chapter VI]). Our therem abut the equivalence f weak unifrm persistence t unifrm persistence is a mdificatin f a therem by V. A. Pliss [16] stating the equivalence f weak ultimate bundedness and ultimate bundedness fr the case f rdinary differential equatins with peridic cefficients. The paper is rganized as fllws. In 2 we frmulate the main definitins and ntatins. Sectin 3 is devted t the cnnectin between weak unifrm persistence and unifrm persistence. In 4 we present sme cunterexamples prving the independence f these persistence definitins. Finally in 5 we cnsider the special cases f autnmus and peridic Klmgrv systems in 2. Definitins and ntatins Here we briefly recall the definitins and ntatin intrduced in [2], and define sme new cncepts as well. Let e be a lcally cmpact metric space, and E = E (the clsure f E) C e. We cnsider the cntinuus flw F = (E, R,n), where n: E x R > E, n(n(x, t), s) n(x, t + s), Vx G E, Vs, t R. T avid trivialities assume that neither E (the interir f E) nr de (the bundary f E) is empty. We call the flw & dissipative if Vx G E the climit set A+(x) ^ 4> and Çl(3r) = \Jx(zEh+(x) has cmpact clsure. We shall say that (i) SF is weakly persistent (WP) if Vx G E, limsupi/(7r(x, t), de) > 0, t-t+oo (ii) SF is (strngly) persistent (P) if Vx G E, liminfí3f(7r(x, t), de) > 0, t»+ (iii) y is weakly unifrmly persistent (WUP) if 3e0 > 0 3 Vx G E, lim sup(d(n(x, t)), de) > e0. / >+ (iv) y is unifrmly persistent (UP) if 3e0 > 0 9 Vx G E, lim inf(ú?(7i(x, t)), de) > en License r cpyright restrictins may apply t redistributin; see

3 persistence definitins 1027 The fllwing bvius relatins are valid fr these definitins: WUP,WPs A nnempty invariant set fr a flw which is the maximal invariant set in sme neighburhd f itself is called an islated invariant set. Let d&~ be the restrictin f 9 t de (assumed invariant under 9). The flw d9~ is islated if there exists a finite cvering Jif = {Mt} f Q(d&~) by pairwise disjint cmpact islated invariant sets Mx,..., Mk fr d9~ such that each M i is als islated fr W. Let M, N be islated invariant sets fr 9". We shall say that M is chained t N (M» TV) if there exists x M u N such that the a and ct>-limit sets f x belng crrespndingly t M and N. A chain f islated invariant sets Mx > 7l/2 Mk is a cyc/e if Mx = A/fc. Finally 39 is called acyclic if, fr sme islated cvering ^# f 1(39), n subset f J? = {At } frms a cycle. UP 3. Criteria fr the equivalence f WUP and UP In the previus sectin, we have intrduced a new cncept, namely weak unifrm persistence. As mentined there, UP => WUP. Here we state tw therems shwing when WUP = UP. Therem 1. Let SF be a cntinuus, dissipative flw n E with metric d. Let da? be the restrictin f 3~ t de (assumed invariant under SF). Then WUP^ UP. Prf. Suppse nt. Then there exist sequences 0 < ek R B lim/t_>+ ek = 0, and such that (1) liminfd(n(x.,t),de)<e,. {x^.}^,, {ek}kx>=x, xk E, One can assume ek < e0 V/c, where e0 is frm the definitin f W UP. Frm WUP it fllws that there exists a sequence {t^}^,, xk > 0 such that (2) d(n(xk,tk),de)>eq. As Çl(&~), the clsure f the unin f all «/-limit sets, is cmpact and E is lcally cmpact, we can find an pen set G such that ÏÏÇW) c G and G is cmpact. Assume that rk is chsen large enugh in the sense that if t > ik, then n(xk, t) G G. Frm (1) it can be seen that there exists a sequence {tk)kxlx, xk < tk and that (3) d(n(xk,tk),de)<ek. License r cpyright restrictins may apply t redistributin; see

4 1028 H. I. FREEDMAN AND P. MOSON Since 9 and is cntinuus there exists a sequence {6k} such that 6k [xk, tk] (4) d(7t(xkjk),de) = e0, and, fr 6k < t <tk, (5) d(n(xk,t),de)<e0. Let xk = n(xk, 6k). Frm the chice f xk and 6k it is clear that xk G, which is a cmpact set, s we can assume that {xk}kx>=x is a cnvergent sequence (therwise we wuld chse a cnvergent subsequence). Let lim/t^+ xk = x G G. It fllws frm (4) that d(x, de) = e0, s x' E. Let tk = tk 6k. Frm the grup prperty f n and (3), (5), it fllws that (6) d(n(xk,tk),de)<ek(<e0) and, if 0 < t < tk, (7) d(n(xk,t),de)<e0. Nw cnsider the rbit with initial pint x. It fllws frm WUP that there exists x > 0 such that (8) d(n(x, x), de) > e0. Since E is invariant under 9 we can find 0 < m < e0 such that (9) d(n(x,t),de)>m, 0<t<x. It fllws frm the cntinuity is sufficiently large then (10) d(n(xk,x'),de)>eq and f n, Hmk_^+xk x, and (8), (9) that if k (11) d(n(xk,t),de)>m(>ek), 0 < t < x. Inequalities (10), (11) cntradict (6), (7). In fact if x <tk, then (10) cntradicts (7), if x > tk then (11) cntradicts (6). These cntradictins prve the therem! G Therem 2. In Therem 1, if the dissipativity hypthesis is replaced by a hypthesis that de is cmpact, then again WUP => UP. Prf. The existence f a cnvergent sequence {x.} c H {x\x E, d(x, de) < e0} fllws frm the cmpactness f de. Using H instead f G, the prf fllws as in Therem 1. D Crllary 3. If there exists e0 > 0 3 fr allx E, 3t(x)> 0 9 d(n(x, x(x)),de) > e0 then 9 is weakly unifrmly persistent with cnstant e0. Prf. Suppse nt. Then there is an x E 3 limsupl_^+ d(n(x, t), de) = e, < e0. But then, fr all sufficiently large t > T, d(n(x, t), de) < e0. Let License r cpyright restrictins may apply t redistributin; see

5 PERSISTENCE DEFINITIONS 1029 x = x(x, T). Frm the grup prperty f n and the previus inequality it fllws that d(n(x, t),3e)<s0 W>0, which cntradicts the assumptin f the statement. Remark 4. Withut additinal assumptins, WUP is nt equivalent t UP. T shw this, let E = R+, and 9 the flw generated by the simplest Ltka- Vlterra system x = x(a - by), y - y(-c + dx), a, b, c, d > 0. All interir rbits are clsed arund the fixed pint (f, ), and, fr any e0, 0 < e0 < min(f, ), 9 is WUP, but nt UP. 4. Equivalences and nnequivalences f persistence definitins As already nted the fllwing implicatins hld: UP => WUP => WP and UP => P => WP. In the previus sectin we have shwn that if the flw is dissipative, then WUP => UP, i.e., UP and WUP are equivalent. In this case ne has WUP = UP => P => WP. Here we shw that, even with dissipativity, P =*> UP and WP *> P. Hwever, if certain additinal assumptins (acyclicity, islatedness f d?~) are made, then we nte (see [2]) that WP => UP s that all definitins are equivalent. Example 5 ( WP #> P). A cunterexample demnstrating the trajectries f a dissipative, weakly persistent, but nt persistent, system can be seen in Figure 1. The detailed descriptin f the system is described belw (see Figure 1). Let E = R+, AF be the flw generated by the system (12) x(. = xtft{xx, x2), i =1,2. Figure 1. License r cpyright restrictins may apply t redistributin; see

6 1030 H. I. FREEDMAN AND P. MOSON T btain the required f (xx, x2), we first cnsider the system (13) xi = xlgi(xx,x2), which has the fllwing prperties: (i) There are fur equilibria in R+, 0(0,0), Px(a, 0), P2(0, b), L(lx, l2), a, b, lx, l2> 0. (ii) O, Px, P2 are saddle pints cnnected by séparatrices as fllws: Tx dentes the rbit alng the x, axis frm O t Px, T2 dentes the 2 rbit alng the x2 axis frm P2 t O, Y dentes the rbit lying in R+ frm Px t P2. We assume that the curve T can be parametrized by the relatin x2 = <p-(xx ), 0 < x, < a. We then smthly extend cp n R+ s that cp(xx) <0 fr xx> a. We dente the pen regin bunded by T{, T2 and Y as D, i.e., D = {(xx, x2)\0 < x, < a, 0 < x2 < cp(xx)}. (iii) L is an unstable fcus lying in D. (iv) All slutins with initial values lying in D\{L} have L as their a-limit sets and r, u Y2 u Y as their cu-limit sets. Nte that such a system can be cnstructed (see [8, pp ]). We nw define { (x2-cp(xx))2gi(xx,x2), (xx,x2) D /;(*!, X2) - < 2 ^D2,-= t -(x2 - <p(xx)), (x,, x2) G R+\D. Then system (12) has the prperties that x2 = <p(xx) is a curve f equilibria and slutins initiating in R+\D apprach Y. Hence (12) is dissipative (in 2 fact trajectries initiating in R+\D apprach Y). Trajectries initiating in D have the same prperties as described fr system (13), and hence system (12) weakly persists, but des nt persist. Example 6 (P *> UP). If, in the previus example, ne replaces the unstable fcus L by a center and all rbits in the interir f T, L)Y2uY are clsed, then a persistent but nt unifrmly persistent dissipative system is btained. Remark 1. If dissipativity des nt hld, then clearly all fur definitins are independent. Hwever, if 3E is cmpact, then, under the same assumptins, i.e., the flw 3 F is islated and acyclic (except fr dissipativity) as in [2], WP P <=> UP hlds. The prf f this is analagus t the prf in [2]. 5. KOLMOGOROV SYSTEMS In this sectin we cnsider autnmus Klmgrv systems f the frm (14) x( = x^x,,..., xn), i = 1,..., n, as well as nnautnmus, peridic Klmgrv systems (15) x, = x(.g((i, x.,..., x) = x,.g,.(/ + T,xx,...,xn), i= I,..., n, T > 0. License r cpyright restrictins may apply t redistributin; see

7 PERSISTENCE DEFINITIONS 1031 Fr the flw described by (14), E = R" and 3E is invariant under this flw. Als, the definitins WP, P, WUP, UP may be written in crdinate frm as fllws fr slutins with initial values in R" : (i) (14) is WP if limsup^min1<;<n{x/(i)}>0; (ii) (14) is P if liminf;_x (f)>0, V/= 1,...,«; (iii) (14) is WUP if 3e0 > 0 3 limsup^^minix^/)} > e0 ; (iv) (14) is UP if 3e0 > 0 3 liminf,^ x.(t) > e0 V/ = 1,...,«. In additin t the abve, ne can define fr system (14) a weaker frm f weak persistence, WWP, as fllws: (v) (14) is WWP if limsup/_>00jci(r) > 0 Vi = 1,..., n. We nte that in several papers in the literature (see e.g. [5]), what was defined as WP was actually WWP. Clearly WP => W WP. The next example shws that the reverse is nt true, even if (14) is dissipative. Example 8. Cnsider the nntransitive cmpetitin mdel described in [15, 18], x, = x, ( 1 - x, - ax2 - ßx3) (16) x2 = x2(l - ßxx - x2 - ax3) x3 = x3(l - ax, - ßx2 -x3), where a + ß > 2, a < 1. There are five equilibria, O(0,0,0), Px ( 1, 0, 0), P2(0, 1,0), P3(0,0, 1), L(l,l, I), where l = (l+a + ß)~x. P, / = 1, 2, 3 are hyperblic saddle pints such that P2 and Px are cnnected by a separatrix rbit T21 in the x, - x2 plane and directed frm P2 t Px. Y2X can be parametrized by a relatin f the frm x2 = cp2x(xx), x3 = 0, 0 < x, < 1. Here we extend cp2x in a smth manner in R+ s that cp2x(xx) < 0 fr x, > 1. Similarly there exists T13 defining X[ = ç»13(x3), x2 = 0, and T32 defining x3 = cpl2(x2), x, = 0. As was shwn in [15], every rbit with psitive initial cnditins (except fr tw singular rbits which tend t L and L itself) has T = T21 ur]3ur32 as its a>-limit set. Hence this is an example f a system which exhibits WWP, but nt WP. We nw define a new system by multiplying the right hand sides f (16) by [x3 + (x2 - <p2x(xx))2][x22 + (x, - <pxi(x}))2][x2 + (x3 -?>32(x2))2]. Fr this new system all pints lying n Y are equilibria. All ther rbits tgether with their rientatins are the same as fr rbits f (16). We nte that fr this example, the system remains WWP, but nt WP, and all the cnditins f the main therem f [2], i.e., dissipativity, islatedness, acyclicity except WP (replaced by WWP) are satisfied. The pints f Y were fixed t btain the acyclic cnditin. S, frm WWP under these cnditins, it des nt fllw that UP P WUP «WP. Finally, we cnsider system (15). This system may be transfrmed int an autnmus system (but nt f Klmgrv type) by intrducing the crdinate License r cpyright restrictins may apply t redistributin; see

8 1032 H. I. FREEDMAN AND P. MOSON 8 = t. Then we have the transfrmed system (17) xi = xifi(9,xl,...,xn), i=l,...,n 6=1, which defines a flw n the cylindrical phase space E = Sx x R", where pints (0, x,,..., xn) and (6 + T, xx,..., xn) cincide. Here 3E = Sx x3rn+. The persistence definitins are the same as fr system (14) (there is n cnditin fr 6). We nte that in this case dissipativity fllws whenever there exists a p > 0 such that, fr all slutins, limsup^^ x(i) < p (see [16]). Acknwledgment We thank the referee fr sme useful remarks especially cncerning 4. References 1. T. Burtn and V. Hutsn, Repellers in systems with infinite delay, J. Math. Anal. Appl. 137 (1989), G. Butler, H. I. Freedman and P. Waltman, Unifrmly persistent systems, Prc. Amer. Math. Sc. 96(1986), _, Persistence in dynamical systems, J. Differential Equatins 63 (1986), A. Fnda, Unifrmly persistent semidynamical systems, Prc. Amer. Math. Sc. 104 (1988), H. I. Freedman and P. Waltman, Persistence in mdels f three interacting predatr-prey ppulatins, Math. Bisci. 68 (1984), _, Persistence in a mdel f three cmpetitive ppulatins, Math. Bisci. 73 (1985), T. C. Gard, Unifrm persistence in multispecies ppulatin mdels, Math. Bisci. 85 (1987), J. Guckenheimer and Ph. Hlmes, Nnlinear scillatins, dynamical systems, and bifurcatin f vectr fields, Applied Math. Sei. 42 (1983). 9. J. K. Hale and P. Waltman, Persistence in infinite-dimensinal systems, SIAM J. Math. Anal. 20(1989), J. Hfbauer and K. Sigmund, Permanence fr replicatr equatins, Lecture Ntes in Ecnmics and Math. Systems, vl. 287, Springer (1987), U.V. Hutsn, A therem n average Liapunv functins, Mnatsh. Math. 98 (1984), V. Hutsn and R. Law, Permanent cexistence in general mdels f three interacting species, J. Math. Bil. 21 (1985), V. Hutsn and K. Schmitt, Permanence in dynamical systems, preprint. 14. G. Kirlinger, Permanence in Ltka-Vlterra equatins: Linked predatr-prey systems, Math. Bisci. 82(1986), R. M. May and W. J. Lenard, Nnlinear aspects f cmpetitin between three species, SIAM J. Appl. Math. 29 (1975), V. A. Pliss, Nnlcal prblems f the thery f scillatins, Academic Press, New Yrk, H. Ruche, P. Habets and M. Laly, Stability thery by Liapunv's direct methd, Appl. Math. Sei. 22 (1977). License r cpyright restrictins may apply t redistributin; see

9 PERSISTENCE DEFINITIONS P. Schuster, K. Sigmund and R. Wlff, On c-limits fr cmpetitin between three species, SIAM J. Appl. Math. 37 (1979), J. Hfbauer and J. W-H. S, Unifrm persistence and repe/lersfr maps, Prc. Amer. Math. Sc. 107(1989), Department f Mathematics, University f Alberta, Edmntn, Alberta T6G 2G1 Canada Department f Mathematics, Technical University f Budapest, Budapest, Hungary, H-1521 License r cpyright restrictins may apply t redistributin; see