RSS feeds
РусскийEnglish
Issue 4, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

1999

 / 

September

←  →

From the current literature


Bifurcations of travelling waves in population taxis models

,
Center of Problems of Forest Ecology and Productivity, Russian Academy of Sciences, Novocheremushkinskaya str. 69, Moscow, 117418, Russian Federation

A penetrating analysis of the wave dynamic modes of a conceptual population system described by the ’reaction-taxis-diffusion’ and ’reaction-autotaxis-cross-diffusion’ polynomial models is carried out for the case of increasing degrees of the reaction and taxis (autotaxis) functions. It is shown that a ’suitable’ nonlinear taxis can affect the wave front sets and generate nonmonotone waves, such as trains and pulses which represent the exact solutions of the model system. Parametric critical points whose neighborhood displays the full spectrum of possible model wave regimes are identified and a wave mode systematization in the form of bifurcation diagrams is given. This enables standard criteria of approach to ’dangerous boundaries’ to be developed. As possible applications, ’pulsing density patches’ in forest insect populations as well as plankton communities and some other examples are discussed.

Fulltext pdf (542 KB)
Fulltext is also available at DOI: 10.1070/PU1999v042n09ABEH000564
PACS: 05.45.−a, 87.10.+e, 87.23.Cc (all)
DOI: 10.1070/PU1999v042n09ABEH000564
URL: https://ufn.ru/en/articles/1999/9/d/
000083399000004
Citation: Berezovskaya F S, Karev G P "Bifurcations of travelling waves in population taxis models" Phys. Usp. 42 917–929 (1999)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Березовская Ф С, Карев Г П «Бифуркации бегущих волн в популяционных моделях с таксисом» УФН 169 1011–1024 (1999); DOI: 10.3367/UFNr.0169.199909d.1011

References (66) ↓ Cited by (31) Similar articles (2)

  1. Semevskiî F N, Semenov S M Matematicheskoe Modelirovanie Ekologicheskikh Protsessov (Mathematical Modelling of Ecological Processes, Leningrad: Gidrometeoizdat, 1982)
  2. Tinbergen N Social Behavior in Animals, with Special Reference to Vertebrates (London: Chapman and Hall, 1990) [Translated into Russian (Moscow: Mir, 1993)]
  3. Patlak C S Bull. Math. Biophys. 15 311 (1953)
  4. Adler J Science 153 708 (1966)
  5. Keller E F, Segel L A J. Theor. Biol. 30 235 (1971)
  6. Keller E F, in Chemotaxis: Its Biology and Biochemistry (Antibiotics and Chemotherapy, Vol. 19, Ed. E Sorkin, Basel: S Karger, 1974)
  7. Monk A, Othmer H G Philos. Trans. R. Soc. London 323 185 (1989)
  8. Murray J D Mathematical Biology ( Biomathematics, Vol. 19) 2nd corr. ed. (Berlin: Springer-Verlag, 1993)
  9. Ivanitskiî G R, Medvinskiî A B, Tsyganov M A Usp. Fiz. Nauk 161 (4) 13 (1991) [Sov. Phys. Usp. 34 289 (1991)]
  10. Ivanitskiî G R, Medvinskiî A B, Tsyganov M A Usp. Fiz. Nauk 164 1041 (1994) [Phys. Usp. 37 961 (1994)]
  11. Svirezhev Yu M Nelineînye Volny, Dissipativnye Struktury i Katastrofy v Ekologii (Nonlinear Waves, Dissipative Structures and Catastrophes in Ecology, Moscow: Nauka, 1987)
  12. Berezovskaya F S et al. Dokl. Akad. Nauk (Russia) 365 (3) 1 (1999)
  13. Alle W C Animal Aggregation: A Study in General Sociology (Chicago: Chicago Press, 1931)
  14. Gorban’ A N et al., in Problemy Ekologicheskogo Monitoringa i Modelirovaniya Ekosistem Vol. 6 (Problems of Ecological Monitoring and Modelling of Ecosystems, Leningrad: Gidrometeoizdat, 1984) p. 161
  15. Isaev A S et al. Dinamika Chislennosti Lesnykh Nasekomykh (Dynamics of Forest Insect Populations, Novosibirsk: Nauka, 1984)
  16. Ludwig D, Jones D D, Holling C S J. Animal Ecology 47 315 (1978)
  17. Davis B Probab. Th. Rel. Fields 84 203 (1990)
  18. Othmer H G, Stevens A SIAM J. Appl. Math. 57 1044 (1997)
  19. Kolmogorov A N, Petrovskiî I G, Piskunov N S Byull. Mosk. Gos. Univ. 1 (6) 1 (1937)
  20. Turing A M Philos. Trans. R. Soc. London Ser. B 237 37 (1952)
  21. Romanovskiî Yu M, Stepanova N S, Chernavskiî D S Matematicheskoe Modelirovanie v Biofizike (Mathematical Modelling in Biophysics, Moscow: Nauka, 1975)
  22. Jorne J J. Theor. Biol. 65 133 (1977)
  23. Vasil’ev V A, Romanovskiî Yu M, Yakhno V G Usp. Fiz. Nauk 128 625 (1979) [Sov. Phys. Usp. 22 615 (1979)]
  24. Okubo A Diffusion and Ecological Problems: Mathematical Models (Biomathematics, Vol. 10, Berlin: Springer-Verlag, 1980)
  25. Krinskiî V I, Zhabotinskiî A M, in Avtovolnovye Protsessy v Sistemakh s Diffuzieî (Autowave Processes in Systems with Diffusion, Ed. M T Grekhova et al., Gor’kiî: IPF AN SSSR, 1981) p. 6
  26. Levin S A, in Proc. Conf. on Differential Equations and Applications in Ecology, Epidemics, and Population Problems: Claremont, Calif., 1981 (Eds S N Busenberg, K L Cooke, New York: Academic Press, 1981) p. 1
  27. Aronson D G, in Mathematical Biology and Medicine (Eds V Capasso, E Grosso, S L Paveri-Fontana, 1985) p. 2
  28. Ivanitskiî G R et al. Usp. Fiz. Nauk 168 1221 (1998) [Phys. Usp. 41 1115 (1998)]
  29. Kuznetsov Yu A et al., in Problemy Ekologicheskogo Monitoringa i Modelirovaniya Ekosistem Vol. 14 (Problems of Ecological Monitoring and Modelling of Ecosystems, Leningrad: Gidrometeoizdat, 1993) p. 198
  30. Wei-Ming Ni Notices 45 9 (1998)
  31. Samarskiî A A, Mikhaîlov A P Matematicheskoe Modelirovanie (Mathematical Modelling, Moscow: Nauka, Fizmatlit, 1997)
  32. Kopell N, Howard L N Stud. Appl. Math. LII 291 (1973)
  33. Fife P, McLeod J B Arch. Rat. Mech. Anal. 65 333 (1977)
  34. Levin S A, Segel L A SIAM Rev. 27 45 (1985)
  35. Jager W, Luckhaus S Trans. AMS 329 819 (1992)
  36. Agladze K L et al. Proc. R. Soc. London B: Biol. Sci. 253 131 (1993)
  37. Tsyganov M A et al. Dokl. Akad. Nauk (Russia) 333 532 (1993)
  38. Stevens A J. Biol. Sci. 3 1059 (1995)
  39. Kopell N, Howard L N Adv. Math. 18 306 (1975)
  40. Arnol’d V I Dopolnitel’nye Glavy Teorii Obyknovennykh Differentsial’nykh Uravneniî (Additional Chapters to the Theory of Ordinary Differential Equations, Moscow: Nauka, 1978)
  41. Bogdanov R I, in Tr. Seminara im. I G Petrovskogo Vol. 2 (Transactions of I G Petrovskiî Seminar, Moscow: Izd. MGU, 1976) p. 37
  42. Berezovskaya F S, Khibnik A I, in Metody Kachestvennoî Teorii Differentsial’nykh Uravneniî (Methods of Qualitative Theory of Differential Equations, Gor’kiî: Izd. GGU, 1985) p. 128
  43. Bazykin A D, Kuznetsov Yu A, Khibnik A I Portrety Bifurkatsiî (Portraits of Bifurcations, Moscow: Znanie, 1989)
  44. Dumortier F, Rossarie R, Sotomayor J, in Lect. Notes Math. Vol. 1480 (Berlin: Springer-Verlag, 1991)
  45. Kuznetsov Yu A Elements of Applied Bifurcation Theory (Applied Mathematics Science, Vol. 112, New York: Springer-Verlag, 1995)
  46. Khibnik A, Krauskopf B, Rousseau C Nonlinearity 11 1505 (1998)
  47. Berezovskaya F S, in Matematika, Komp’yuter, Obrazovanie Vol. 5 (Mathematics, Computer, Education, Ed. G Yu Riznichenko, Moscow: Progress-Traditsiya, 1998) p. 22
  48. Berezovskaya F S, Karev G P Dokl. Akad. Nauk (Russia) 368 (3) 5 (1999)
  49. Marsden J E, McCracken M The Hopf Bifurcation and Its Applications (New York: Springer-Verlag, 1976) [Translated into Russian (Moscow: Mir, 1980)]
  50. Fisher R A Ann. Eugenics 7 355 (1937)
  51. Zel’dovich Ya B et al. Matematicheskaya Teoriya Goreniya i Vzryva (Mathematical Theory of Combustion and Explosion, Moscow: Nauka, 1980)
  52. Nagumo J, Yoshikawa S, Arimoto S IEEE Trans. Commun. Technol. 12 400 (1965)
  53. Dumortier F, Rousseau C Nonlinearity 3 1015 (1990)
  54. Molchanov A M Nelineînost’ v Biologii (Nonlinearity in Biology, Pushchino: ONTI NTsBI, 1992)
  55. Keller E F, Segel L A Theor. Biol. 26 399 (1970)
  56. Budrene E O, Berg H C Nature (London) 349 630 (1991)
  57. Henry D Geometric Theory of Semilinear Parabolic Equations (New York: Springer-Verlag, 1981) [Translated into Russian (Moscow: Mir, 1984)]
  58. Vol’pert A I, Ivanova A N, in Avtovolnovye Protsessy v Sistemakh s Diffuzieî (Autowave Processes in Systems with Diffusion, Ed. M T Grekhova, Gor’kiî: IPF AN SSSR, 1981) p. 33
  59. Maini P K, Murray J D, Oster G F, in Lecture Notes in Mathematics Vol. 1151 (Berlin: Springer-Verlag, 1985) p. 252
  60. Maginu K J. Diff. Eq. 139 73 (1981)
  61. Sattinger D H Adv. Math. 22 312 (1976)
  62. Evanse J W, Feroe J Math. Biosci. 37 23 (1977)
  63. Conley C C, Smoller J, in Lecture Notes in Mathematics Vol. 525 (Berlin: Springer-Verlag, 1976) p. 77
  64. Cartwright J H E, Hermandez-Garcia E, Piro O Phys. Rev. Lett. 79 527 (1997)
  65. Akhromeeva T S et al. Nestatsionarnye Struktury i Diffuzionnyî Khaos (Nonstationary Structures and Diffusive Chaos, Moscow: Nauka, 1992)
  66. Gueron S, Liron N J. Math. Biol. 27 595 (1989)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions