Systems of differential equations which are competitive or cooperative: III. Competing species

Nonlinearity

Volumn 1, Issue 1, 1988, Pages 51-71

  Hirsch, Morris W ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84956106116     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/1/1/003     Document Type: Article

References (42)

1. Albrecht F, Gatzke H, Haddad A and Wax I 1974 The dynamics of two interacting populations J. Math. Anal. Appl. 46 658-70
2. Arnedo A, Coullet P, Peyraud J and Tresser C 1982 Strange attractors in Volterra equations for species in competition J. Math. Biol. 14 3-157
3. Brown M 1961 The monotone union of open «-cells is an open «-cell Proc. Am. Math. Soc. 12 812-4
4. Butler G, Freedman H I and Waltman P 1986 Uniformly persistent systems Emory University Dept. Math. Computer Sei. Technical Report No 17
5. Chenciner A 1977 Comportement asymptotique de systèmes différentiels du type ‘compétition d’espèces’ C.R. Acad. Sei., Paris 284 313-5
6. Cohen M A and Grossberg S 1983 Absolute stability of global pattern formation and parallel memory storage by competitive neural networks IEEE Trans. Syst. Man Cybernet. SMC-13 815-26
7. Coste J 1985 Dynamical repression in many-species ecosystems: the case of many predators competing for several prey SIAM J. Appl. Math. 45 559-64
8. Coste J, Peyraud J and Coullet P 1979 Asymptotic behaviors in the dynamics of competing species SIAM J. Appl. Math. 36 516-43
9. Coste J, Peyraud J, Coullet P and Chenciner A 1978 About the theory of competing species Theor. Popul. Biol. 14 165-84
10. Freedman H I 1980 Deterministic Mathematical Models in Population Ecology (New York: Marcel Dekker)
11. Freedman H I and Waltman P 1985 Persistence in a model of three competitive populations Math. Biosci. 73 89-101
12. Gilpin M E 1975 Limit cycles in competition communities Am. Natural. 109 51-60
13. Grossberg S 1978 Competition, decision and consensus J. Math. Anal. Appl. 66 470-93
14. Hadeler K and Glas D 1983 Quasimonotone systems and convergence to equilibrium in a population genetic model J. Math. Anal. Appl. 95 297-303
15. Hale J and Somolinos A 1983 Competition for fluctuating nutrient J. Math. Biol. 18 225-80
16. Hallam T G, Svoboda L J and Gard T C 1979 Persistence and extinction in three species Lotka-Volterra competitive systems Math. Biosci. 46 117-24
17. Hirsch M W 1982a Systems of differential equations that are competitive or cooperative. I: Limit sets SIAM], Math. Anal. 13 167-79
18. Hirsch M W 1982b Lecture Notes for Colloq. Lectures at University of Toronto, August 1982 (Providence, R I:American Mathematical Society)
19. Hirsch M W 1984 The dynamical systems approach to differential equations Bull. Am. Math. Soc. 111-64
20. Hirsch M W 1985 Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere SIAM J. Math. Anal. 16 423-39
21. Hirsch M W 1987 Stability and convergence in strongly monotone dynamical systems J. Reine Angew. Math, in press
22. Hirsch M W and Pugh C 1987 Cohomology of chain recurrent sets Ergodic Theory in press
23. Holtz M 1987 Dissertation University of California at Berkeley
24. Hutson V and Law R 1985 Permanent coexistence in models of three interacting species J. Math. Biol. 21 285-98
25. Kamke E 1932 Zur Theorie der Systeme gewöhnlicher differential Gleichungen II Acta Math. 58 57-85
26. Kerner E H 1961 On the Volterra-Lotka principle Bull. Math. Biophys. 23 141-57
27. Levin S A 1970 Community equilibria and stability, and an extension of the competitive exclusion principle Ann. Nat. 104 413-23
28. MacArthur R H 1969 Species packing, or what competition minimizes Proc. Natl. Acad. Sei. USA 64 1369-75
29. May R M and Leonard W J 1975 Nonlinear aspects of competition between three species SIAM J. Appl. Math. 29 243-53
30. Midler M 1926 Uber das Fundamentaltheorem in der Theorie der gewöhnlichen Differentialgleichungen Math. Z. 26 619-45
31. Othmer H 1976 The qualitative dynamics of a class of biochemical control circuits J. Math. Biol. 3 53-78
32. Phillipson P E, Schuster P and Johnson R G 1985 An analytic study of the May-Leonard equation SIAM J. Appl. Math. 45 541-54
33. Rescigno A 1968 The struggle for life: II Three competitors Bull. Math. Biophys. 30 291-8
34. Schuster P, Sigmund K and Wolff R 1979 On cu-limits for competition between three species SIAM J. Appl. Math. 37 49-54
35. Smale S 1976 On the differential equations of species in competition J. Math. Biol. 3 5-7
36. Smillie J 1984 Competitive and cooperative tridiagonal systems of differential equations SIAM J. Math. Anal. 15 530-4
37. Smith H 1986a Periodic orbits of competitive and cooperative systems J. Diff. Eq. in press
38. Smith H 1986b On the asymptotic behavior of a class of deterministic models of cooperating species SIAM J. Appl. Math. 368-75
39. Smith H 1986c Periodic solutions of periodic competitive and cooperative systems SIAM J. Math. Anal. 17 1289-318
40. Smith H 1986d Systems of differential equations which generate a monotone flow. A survey of results SIAM Rev. in press
41. Smith H and Waltman P 1988 A classification theorem for three dimensional competitive systems (in preparation)
42. Tyson J and Othmer H 1978 The dynamics of feedback control circuits in biochemical pathways Prog. Theor. Biol. 5 1-62