메뉴 건너뛰기




Volumn 34, Issue 5, 2003, Pages 1007-1038

Robust permanence for ecological differential equations, minimax, and discretizations

Author keywords

Average Liapunov functions; Conley index; Discretizations of Kolmogorov type; Evolutionary games; Invariant measures; Liapunov exponent; Minimax; Population dynamics; Robust permanence

Indexed keywords


EID: 0141820163     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141001392815     Document Type: Article
Times cited : (65)

References (69)
  • 1
    • 84968476142 scopus 로고
    • Computation of topological degree using interval arithmetic and applications
    • O. ABERTH. Computation of topological degree using interval arithmetic and applications, Math. Comp., 62 (1994), pp. 171-178.
    • (1994) Math. Comp. , vol.62 , pp. 171-178
    • Aberth, O.1
  • 3
    • 6144229366 scopus 로고    scopus 로고
    • From attractor to chaotic saddle: A tale of transverse instability
    • P. ASHWIN, J. BUESCU, AND I. N. STEWART, From attractor to chaotic saddle: A tale of transverse instability, Nonlinearity, 9 (1996), pp. 703-737.
    • (1996) Nonlinearity , vol.9 , pp. 703-737
    • Ashwin, P.1    Buescu, J.2    Stewart, I.N.3
  • 5
    • 0242673200 scopus 로고
    • Zubov's stability criterion
    • R. W. BASS, Zubov's stability criterion, Bol. Soc. Mat. Mexicana, 4 (1959), pp. 26-29.
    • (1959) Bol. Soc. Mat. Mexicana , vol.4 , pp. 26-29
    • Bass, R.W.1
  • 6
    • 0001408436 scopus 로고    scopus 로고
    • Asymptotic pseudotrajectories and chain recurrent flows, with applications
    • M. BENAIM AND M. W. HIRSCH, Asymptotic pseudotrajectories and chain recurrent flows, with applications, J. Dynam. Differential Equations, 8 (1996), pp. 141-176.
    • (1996) J. Dynam. Differential Equations , vol.8 , pp. 141-176
    • Benaim, M.1    Hirsch, M.W.2
  • 7
    • 0007108344 scopus 로고
    • Center manifolds of dynamical systems under discretization
    • W. J. BEYX AND J. LORENZ, Center manifolds of dynamical systems under discretization, Numer. Funct. Anal. Optim., 9 (1987), pp. 381-414.
    • (1987) Numer. Funct. Anal. Optim. , vol.9 , pp. 381-414
    • Beyx, W.J.1    Lorenz, J.2
  • 9
    • 0242673199 scopus 로고
    • High order approximation of implicitly defined maps
    • A. BRESSAN, High order approximation of implicitly defined maps, Ann. Mat. Pura Appl., 137 (1984), pp. 163-173.
    • (1984) Ann. Mat. Pura Appl. , vol.137 , pp. 163-173
    • Bressan, A.1
  • 10
    • 84972517829 scopus 로고
    • A further generalization of the Schauder fixed point theorem
    • F. E. BROWDER, A further generalization of the Schauder fixed point theorem, Duke Math. J., 32 (1965), pp. 575-578.
    • (1965) Duke Math. J. , vol.32 , pp. 575-578
    • Browder, F.E.1
  • 13
    • 0242673201 scopus 로고
    • Saturated invariant sets and the boundary behaviour of differential systems
    • A. CAPIETTO AND B. M. GARAY, Saturated invariant sets and the boundary behaviour of differential systems, J. Math. Anal. Appl., 176 (1993), pp. 166-181.
    • (1993) J. Math. Anal. Appl. , vol.176 , pp. 166-181
    • Capietto, A.1    Garay, B.M.2
  • 14
    • 0000371241 scopus 로고    scopus 로고
    • On the approximation of complicated dynamical behaviour
    • M. DELLNITZ AND O. JUNGE, On the approximation of complicated dynamical behaviour, SIAM J. Numer. Anal., 36 (1999), pp. 491-515.
    • (1999) SIAM J. Numer. Anal. , vol.36 , pp. 491-515
    • Dellnitz, M.1    Junge, O.2
  • 15
    • 48749138514 scopus 로고
    • Persistence in models of three interacting predator-prey populations
    • H. I. FREEDMAN AND P. WALTMAN, Persistence in models of three interacting predator-prey populations, Math. Biosci., 68 (1984), pp. 213-231.
    • (1984) Math. Biosci. , vol.68 , pp. 213-231
    • Freedman, H.I.1    Waltman, P.2
  • 16
    • 38249023378 scopus 로고
    • Uniform persistence and chain recurrence
    • B. M. GARAY, Uniform persistence and chain recurrence, J. Math. Anal. Appl., 139 (1989), pp. 372-381.
    • (1989) J. Math. Anal. Appl. , vol.139 , pp. 372-381
    • Garay, B.M.1
  • 17
    • 0032368419 scopus 로고    scopus 로고
    • The discretized flow on domains of attraction: A structural stability result
    • B. M. GARAY, The discretized flow on domains of attraction: A structural stability result, IMA J. Numer. Anal., 18 (1998), pp. 77-90.
    • (1998) IMA J. Numer. Anal. , vol.18 , pp. 77-90
    • Garay, B.M.1
  • 18
    • 0242421400 scopus 로고    scopus 로고
    • Some remarks on Wazewski's retract principle
    • B. M. GARAY, Some remarks on Wazewski's retract principle, Univ. Iagel. Acta Math., 36 (1998), pp. 97-105.
    • (1998) Univ. Iagel. Acta Math. , vol.36 , pp. 97-105
    • Garay, B.M.1
  • 19
    • 48849083289 scopus 로고    scopus 로고
    • A functional equation characterizing monomial functions used in permanence theory for ecological differential equations
    • submitted
    • B. M. GARAY, A functional equation characterizing monomial functions used in permanence theory for ecological differential equations, Univ. Iagel. Acta. Math., submitted.
    • Univ. Iagel. Acta. Math.
    • Garay, B.M.1
  • 22
    • 0002578327 scopus 로고    scopus 로고
    • Discretization near compact invariant sets
    • B. M. GARAY AND P. E. KLOEDEN, Discretization near compact invariant sets, Random Comput. Dynam., 5 (1997), pp. 93-123.
    • (1997) Random Comput. Dynam. , vol.5 , pp. 93-123
    • Garay, B.M.1    Kloeden, P.E.2
  • 23
    • 0035402158 scopus 로고    scopus 로고
    • Numerical flow-box theorems under structural assumptions
    • B. M. GARAY AND P. L. SIMON, Numerical flow-box theorems under structural assumptions, IMA J. Numer. Anal., 21 (2001), pp. 733-749.
    • (2001) IMA J. Numer. Anal. , vol.21 , pp. 733-749
    • Garay, B.M.1    Simon, P.L.2
  • 24
    • 0026931058 scopus 로고
    • Time averages for heteroclinic attractors
    • A. GAUNERSDORFER, Time averages for heteroclinic attractors, SIAM J. Appl. Math., 52 (1992), pp. 1476-1489.
    • (1992) SIAM J. Appl. Math. , vol.52 , pp. 1476-1489
    • Gaunersdorfer, A.1
  • 25
    • 0034654717 scopus 로고    scopus 로고
    • Convergence rates of perturbed attracting sets under vanishing perturbation
    • L. GRÜNE, Convergence rates of perturbed attracting sets under vanishing perturbation, J. Math. Anal. Appl., 244 (2000), pp. 369-392.
    • (2000) J. Math. Anal. Appl. , vol.244 , pp. 369-392
    • Grüne, L.1
  • 26
    • 0002008380 scopus 로고    scopus 로고
    • Chain transitivity, attractivity and strong repellors for semidynamical systems
    • M. W. HIRSCH, H. L. SMITH, AND X.-Q. ZHAO, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), pp. 107-131.
    • (2001) J. Dynam. Differential Equations , vol.13 , pp. 107-131
    • Hirsch, M.W.1    Smith, H.L.2    Zhao, X.-Q.3
  • 27
    • 0000373191 scopus 로고
    • A general cooperation theorem for hypercycles
    • J. HOFBAUER, A general cooperation theorem for hypercycles, Monatsh. Math., 91 (1981), pp. 233-240.
    • (1981) Monatsh. Math. , vol.91 , pp. 233-240
    • Hofbauer, J.1
  • 28
    • 0001862504 scopus 로고
    • A unified approach to persistence
    • J. HOFBAUER, A unified approach to persistence, Acta Appl. Math., 14 (1989), pp. 11-22.
    • (1989) Acta Appl. Math. , vol.14 , pp. 11-22
    • Hofbauer, J.1
  • 29
    • 0001391737 scopus 로고
    • An index theorem for dissipative semiflows
    • J. HOFBAUER, An index theorem for dissipative semiflows, Rocky Mountain J. Math., 20 (1990), pp. 1017-1031.
    • (1990) Rocky Mountain J. Math. , vol.20 , pp. 1017-1031
    • Hofbauer, J.1
  • 30
    • 0242421401 scopus 로고
    • Permanence on a Halfspace and the Ergodic Theorem
    • manuscript, Orlando, FL
    • J. HOFBAUER, Permanence on a Halfspace and the Ergodic Theorem, manuscript, presented at SIAM meeting, Orlando, FL, 1990.
    • (1990) SIAM Meeting
    • Hofbauer, J.1
  • 31
    • 0242589333 scopus 로고
    • Heteroclinic cycles in ecological differential equations
    • J. HOFBAUER, Heteroclinic cycles in ecological differential equations, Tatra Mt. Math. Publ., 4 (1994), pp. 105-116.
    • (1994) Tatra Mt. Math. Publ. , vol.4 , pp. 105-116
    • Hofbauer, J.1
  • 33
    • 0023516861 scopus 로고
    • Coexistence for systems governed by difference equations of Lotka-Volterra type
    • J. HOFBAUER, V. HUTSON, AND W. JANSEN, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol., 25 (1987), pp. 553-570.
    • (1987) J. Math. Biol. , vol.25 , pp. 553-570
    • Hofbauer, J.1    Hutson, V.2    Jansen, W.3
  • 34
    • 0034348033 scopus 로고    scopus 로고
    • Sophisticated imitation in cyclic games
    • J. HOFBAUER AND K. SCHLAG, Sophisticated imitation in cyclic games, J. Evol. Economics, 10 (2000), pp. 523-543.
    • (2000) J. Evol. Economics , vol.10 , pp. 523-543
    • Hofbauer, J.1    Schlag, K.2
  • 36
    • 84966222841 scopus 로고
    • Uniform persistence and repellors for maps
    • J. HOFBAUER AND J. W.-H. So, Uniform persistence and repellors for maps, Proc. Amer. Math. Soc., 107 (1989), pp. 1137-1142.
    • (1989) Proc. Amer. Math. Soc. , vol.107 , pp. 1137-1142
    • Hofbauer, J.1    So, J.W.-H.2
  • 37
    • 0001439703 scopus 로고
    • A theorem on average Liapunov functions
    • V. HUTSON, A theorem on average Liapunov functions, Monatsh. Math., 98 (1984), pp. 267-275.
    • (1984) Monatsh. Math. , vol.98 , pp. 267-275
    • Hutson, V.1
  • 38
    • 38249028339 scopus 로고
    • The stability under perturbations of repulsive sets
    • V. HUTSON, The stability under perturbations of repulsive sets, J. Differential Equations, 76 (1988), pp. 77-90.
    • (1988) J. Differential Equations , vol.76 , pp. 77-90
    • Hutson, V.1
  • 39
    • 0242589339 scopus 로고    scopus 로고
    • An approach to practical persistence
    • V. HUTSON AND K. MISCHAIKOW, An approach to practical persistence, J. Math. Biol., 37 (1998), pp. 447-466.
    • (1998) J. Math. Biol. , vol.37 , pp. 447-466
    • Hutson, V.1    Mischaikow, K.2
  • 40
    • 0020357388 scopus 로고
    • Persistence of species obeying difference equations
    • V. HUTSON AND W. MORAN, Persistence of species obeying difference equations, J. Math. Biol., 15 (1982), pp. 203-213.
    • (1982) J. Math. Biol. , vol.15 , pp. 203-213
    • Hutson, V.1    Moran, W.2
  • 41
    • 0001799804 scopus 로고
    • A permanence theorem for replicator and Lotka Volterra systems
    • W. JANSEN, A permanence theorem for replicator and Lotka Volterra systems, J. Math. Biol., 25 (1987), pp. 411-422.
    • (1987) J. Math. Biol. , vol.25 , pp. 411-422
    • Jansen, W.1
  • 43
    • 0000649975 scopus 로고
    • Stable attracting sets in dynamical systems and their one-step discretizations
    • P. E. KLOEDEN AND J. LORENZ, Stable attracting sets in dynamical systems and their one-step discretizations, SIAM J. Numer. Anal., 23 (1986), pp. 986-995.
    • (1986) SIAM J. Numer. Anal. , vol.23 , pp. 986-995
    • Kloeden, P.E.1    Lorenz, J.2
  • 44
    • 0007254737 scopus 로고    scopus 로고
    • Lyapunov functions and attractors under variable time-step discretization
    • P. E. KLOEDEN AND B. SCHMALFUSS, Lyapunov functions and attractors under variable time-step discretization, Discrete Contin. Dynam. Systems, 2 (1996), pp. 163-172.
    • (1996) Discrete Contin. Dynam. Systems , vol.2 , pp. 163-172
    • Kloeden, P.E.1    Schmalfuss, B.2
  • 45
    • 0031527178 scopus 로고    scopus 로고
    • Nonautonomous systems, cocycle attractors and variable time-step discretization
    • P. E. KLOEDEN AND B. SCHMALFUSS, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), pp. 141-154.
    • (1997) Numer. Algorithms , vol.14 , pp. 141-154
    • Kloeden, P.E.1    Schmalfuss, B.2
  • 46
    • 84916155363 scopus 로고
    • Les mesures invariantes et transitives dans la mécanique non linéaire
    • N. KRYLOV AND N. BOGOLIUBOV, Les mesures invariantes et transitives dans la mécanique non linéaire, Mat. Sb. (N.S.), 1 (1936), pp. 707-710.
    • (1936) Mat. Sb. (N.S.) , vol.1 , pp. 707-710
    • Krylov, N.1    Bogoliubov, N.2
  • 47
    • 0242673252 scopus 로고
    • Norm inequalities for derivatives and differences
    • W. N. Everett, ed., Marcel Dekker, Basel
    • M. K. KWONG AND A. ZETTL, Norm inequalities for derivatives and differences, in Inequalities, W. N. Everett, ed., Marcel Dekker, Basel, 1991, pp. 91-121.
    • (1991) Inequalities , pp. 91-121
    • Kwong, M.K.1    Zettl, A.2
  • 48
    • 0031581231 scopus 로고    scopus 로고
    • Structural stability under numerics
    • M. C. LI, Structural stability under numerics, J. Differential Equations, 141 (1997), pp. 1-12.
    • (1997) J. Differential Equations , vol.141 , pp. 1-12
    • Li, M.C.1
  • 49
    • 0010403673 scopus 로고
    • Dynamics of games and genes: Discrete versus continuous time
    • V. LOSERT AND E. AKIN, Dynamics of games and genes: Discrete versus continuous time, J. Math. Biol., 17 (1983), pp. 241-251.
    • (1983) J. Math. Biol. , vol.17 , pp. 241-251
    • Losert, V.1    Akin, E.2
  • 50
    • 0033192703 scopus 로고    scopus 로고
    • Permanence and global attractivity for Lotka-Volterra difference systems
    • Z. LU AND W. WANG, Permanence and global attractivity for Lotka-Volterra difference systems, J. Math. Biol., 39 (1999), pp. 269-282.
    • (1999) J. Math. Biol. , vol.39 , pp. 269-282
    • Lu, Z.1    Wang, W.2
  • 52
    • 0036501742 scopus 로고    scopus 로고
    • Kolmogorov vector fields with robustly permanent subsystems
    • J. MIERCZYŃSKI AND S. J. SCHREIBER, Kolmogorov vector fields with robustly permanent subsystems, J. Math. Anal. Appl., 267 (2002), pp. 329-337.
    • (2002) J. Math. Anal. Appl. , vol.267 , pp. 329-337
    • Mierczyński, J.1    Schreiber, S.J.2
  • 53
    • 0040232025 scopus 로고
    • On time-duality of the Conley index
    • M. MROZEK AND R. SRZEDNICKI, On time-duality of the Conley index, Results Math., 24 (1993), pp. 161-167.
    • (1993) Results Math. , vol.24 , pp. 161-167
    • Mrozek, M.1    Srzednicki, R.2
  • 54
    • 0002432742 scopus 로고
    • Discretized ordinary differential equations and the Conley index
    • M. MROZEK AND K. P. RYBAKOWSKI, Discretized ordinary differential equations and the Conley index, J. Dynam. Differential Equations, 4 (1992), pp. 57-63.
    • (1992) J. Dynam. Differential Equations , vol.4 , pp. 57-63
    • Mrozek, M.1    Rybakowski, K.P.2
  • 56
    • 0002610570 scopus 로고    scopus 로고
    • On growth rates of subadditive functions for semiflows
    • S. SCHREIBER, On growth rates of subadditive functions for semiflows, J. Differential Equations, 148 (1998), pp. 334-350.
    • (1998) J. Differential Equations , vol.148 , pp. 334-350
    • Schreiber, S.1
  • 58
    • 0000491205 scopus 로고
    • Dynamical systems under constant organization. III. Cooperative and competitive behaviour of hypercycles
    • P. SCHUSTER, K. SIGMUND, AND R. WOLFF, Dynamical systems under constant organization. III. Cooperative and competitive behaviour of hypercycles, J. Differential Equations, 32 (1979), pp. 357-368.
    • (1979) J. Differential Equations , vol.32 , pp. 357-368
    • Schuster, P.1    Sigmund, K.2    Wolff, R.3
  • 63
    • 0242673257 scopus 로고    scopus 로고
    • On the discrete Conley index in the invariant subspace
    • A. SZYMCZAK, K. WOJCIK, AND P. ZGLICZYNSKI, On the discrete Conley index in the invariant subspace, Topology Appl., 87 (1998), pp. 105-115.
    • (1998) Topology Appl. , vol.87 , pp. 105-115
    • Szymczak, A.1    Wojcik, K.2    Zgliczynski, P.3
  • 64
    • 0242589358 scopus 로고
    • Sur le courant extérieur à une région invariante. Théorème de Bendixson
    • T. URA AND I. KIMURA, Sur le courant extérieur à une région invariante. Théorème de Bendixson, Comm. Math. Univ. Sanctii Pauli, 8 (1960), pp. 23-39.
    • (1960) Comm. Math. Univ. Sanctii Pauli , vol.8 , pp. 23-39
    • Ura, T.1    Kimura, I.2
  • 65
    • 0242504710 scopus 로고    scopus 로고
    • n × [0, ∞)
    • n × [0, ∞), Ann. Polon. Math., 65 (1997), pp. 203-211.
    • (1997) Ann. Polon. Math. , vol.65 , pp. 203-211
    • Wojcik, K.1
  • 66
    • 0010476207 scopus 로고
    • Invasion of a persistent system
    • G. WOLKOWICZ, Invasion of a persistent system, Rocky Mountains J. Math., 20 (1990), pp. 1217-1234.
    • (1990) Rocky Mountains J. Math. , vol.20 , pp. 1217-1234
    • Wolkowicz, G.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.