Asymptotic Expansions for the Sequences of the Modified Discrete Delay Logistic Model Type

• Dumitru Popa ^[1]
  1. [1] Ovidius University of Constanta
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 3, 2023
• Idioma: inglés
• DOI: 10.1007/s12346-023-00819-7
• Enlaces
  • Texto completo
• Resumen

  • As a consequence of some general results we find the asymptotic expansions for the recurrence xn+1 = xn 1+βxn−k , β > 0. This can be viewed as a modified version of the well-known discrete delay logistic model xn+1 = αxn 1+βxn−k , α ≥ 1, β > 0. We mention that in this case appear a real constant associated with this sequence. Moreover, for various other recursive sequences we find their asymptotic expansions.

• Referencias bibliográficas
  • 1. Arnaudies, J. M., Fraysse, H.: Cours de mathé matiques, tome 2 : Analyse, classes préparatoires 1er cycle universitaire, (1991)
  • 2. Berg, L.: On the asymptotics of nonlinear difference equations. Zeitschrift for Analysis und Ihre Anwendungen 21(4), 1061–1074 (2002)
  • 3. Berg, L.: Inclusion theorems for non-linear difference equations with applications. J. Differ. Equ. Appl. 10(4), 399–408 (2004)
  • 4. Berg, L.: Corrections to ‘Inclusion theorems for non-linear difference equations with applications’. J. Differ. Equ. Appl. 11(2), 181–182...
  • 5. Berg, L.: On the difference equation xn+1 = βxn + γ xn−1 γ xn + βxn−1 . Rostocker Math. Kolloq. 61, 3–11 (2006)
  • 6. Berg, L.: On the asymptotics of the difference equation xn−3 = xn1 + xn−1xn−2 . J. Differ. Equ. Appl. 14(1), 105–108 (2008)
  • 7. Berg, L., Stevi´c, S.: On the asymptotics of the difference equation yn 1 + yn−1 ··· yn−k+1 = yn−k . J. Differ. Equ. Appl....
  • 8. Berg, L., Stevi´c, S.: On the asymptotics of some systems of difference equations. J. Differ. Equ. Appl. 17(9), 1291–1301 (2011)
  • 9. De Bruijn, N.G.: Asymptotic methods in analysis. Dower Publications Inc, New York (1981)
  • 10. Cesáro, E.: Sur une question de limites, Mathesis X, 25-28 (1890) http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN599218835&IDDOC=626221
  • 11. Gutnik, L., Stevi´c, S.: On the behaviour of the solutions of a second-order difference equation. Discrete Dyn. Nat. Soc. 2007, 14 (2007)
  • 12. Hassel, M.P.: Density-dependence in single-species populations. J. Anim. Ecol. 44(1), 283–295 (1975)
  • 13. Kent, C. M., Kosmala, W., Radin, M. A., Stevi´c, S.: On the difference equation xn+1 = xn xn−2 − 1, Abstract and Applied Analysis,...
  • 14. Knopp, K.: Theory and application of infinite series. Springer-Verlag, London (1996)
  • 15. Kocic, V.L., Ladas, G.: Global attractivity in nonlinear delay difference equations. Proc. Amer. Math.Soc. 115(4), 1083–1088 (1992)
  • 16. Kuruklis, S.A., Ladas, G.: Oscillations and global attractivity in a discrete delay logistic model. Quart.Appl. Math. 1(2), 227–233 (1992)
  • 17. Ladas, G.: Open problems and conjectures. J. Diff. Equ. Appl. 4(5), 497–499 (1998)
  • 18. Pielou, E.C.: An Introduction to Mathematical Ecology. Wiley-Interscience (1969)
  • 19. Pielou, E.C.: Population and Community Ecology. Gordon and Breach, New York (1974)
  • 20. Poincaré, H.: Sur les intégrales irré gulières des équations linéaires. Acta Math. 8, 295–344 (1886)
  • 21. Polya, G., Szego, G.: Problems and theorems in analysis I. Springer Verlag, London (1998)
  • 22. Popa, D.: Recurrent sequences and the asymptotic expansion of a function, Gazeta matematic˘a seria A, Nr. 3-4, 1-16. (2019) https://ssmr.ro/gazeta/gma/2019/gma3-4-2019-continut.pdf
  • 23. Popa, D.: Asymptotic expansions for the recurrence xn+1 = 1 n n k=1 f xkk . Math. Methods Appl. Sci. 46(2), 2165–2173 (2023)
  • 24. Stevi´c, S.: Asymptotic behaviour of a sequence defined by iteration. Mat. Vesn. 48(3–4), 99–105 (1996)
  • 25. Stevi´c, S.: Asymptotic behavior of a sequence defined by iteration with applications. Colloq. Math. 93(2), 267–276 (2002)
  • 26. Stevi´c, S.: Asymptotic behavior of a nonlinear difference equation. Indian J. Pure Appl. Math. 34(12), 1681–1687 (2003)
  • 27. Stevi´c, S.: On positive solutions of a (k + 1)-th order difference equation. Appl. Math. Lett. 19(5), 427–431 (2006)
  • 28. Stevi´c, S.: Asymptotic behavior of a class of nonlinear difference equations. Discrete Dyn. Nat. Soc. 2006, 10 (2006)
  • 29. Stevi´c, S.: On monotone solutions of some classes of difference equations. Discrete Dyn. Nat. Soc. 2006, 9 (2006)
  • 30. Stevi´c, S.: Existence of nontrivial solutions of a rational difference equation. Appl. Math. Lett. 20(1), 28–31 (2007)
  • 31. Stevi´c, S.: Asymptotics of some classes of higher-order difference equations. Discrete Dyn. Nat. Soc. 2007, 20 (2007)
  • 32. Stevi´c, S.: On a discrete epidemic model. Discrete Dyn. Nat. Soc. 2007, 10 (2007)
  • 33. Stevi´c, S.: Nontrivial solutions of a higher-order rational difference equation. Math. Notes 84(5), 718–724 (2008)
  • 34. Stieltjes, T.J.: Recherches sur quelques sé ries semiconvergentes. Ann. de l’Éc. Norm. 3, 201–258 (1886)
  • 35. Watkinson, A.R.: Density-dependence in single-species populations of plants. J. Theor. Biol. 83, 345–357 (1980)