Ecology and evolution
Open Access
Math. Model. Nat. Phenom.
Volume 15, 2020
Ecology and evolution
Article Number 38
Number of page(s) 16
Published online 14 August 2020
  1. E. Ahmed and A. Elgazzar, On fractional order differential equations model for nonlocal epidemics. Physica A 379 (2007) 607–614. [CrossRef] [PubMed] [Google Scholar]
  2. E. Ahmed, E.-S. Ama, El-Saka and A.A. Hala, On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rässler, Chua and Chen systems. Phys. Lett. A 358 (2006) 1–4. [Google Scholar]
  3. A.A. Berryman, The origins and evolutions of predator-prey theory. Ecology 73 (1992) 1530–1535. [Google Scholar]
  4. K. Chakraborty and K. Das, Modeling and analysis of a two-zooplankton one-phytoplankton system in the presence of toxicity. Appl. Math. Model. 39 (2015) 1241–1265. [Google Scholar]
  5. D.R. Curtiss, Recent extentions of descartes rule of signs. Ann. Math. (1918) 251–278. [Google Scholar]
  6. T. Das, R.N. Mukherjee and K.S. Chaudhuri, Harvesting of a prey–predator fishery in the presence of toxicity. Appl. Math. Model. 33 (2009) 2282–2292. [Google Scholar]
  7. B. Dubey, A prey–predator model with a reserved area. Nonlinear Anal. Model. Control. 12 (2007) 479–494. [CrossRef] [Google Scholar]
  8. B. Dubey, P. Chandra and P. Sinha, A model for fishery resource with reserve area. Nonlinear Anal. Real World Appl. 4 (2003) 625–637. [Google Scholar]
  9. M. Edelman, Fractional maps as maps with power-law memory. Nonlinear dynamics and complexity. Springer, Cham (2014) 79–120. [CrossRef] [Google Scholar]
  10. A. Elsadany and A. Matouk, Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization. J. Appl. Math. Comput. 49 (2015) 269–283. [Google Scholar]
  11. S. Jana, A. Ghorai, S. Guria and T.K. Kar, Global dynamics of a predator weaker prey and stronger prey system. Appl. Math. Comput. 250 (2015) 235–248. [Google Scholar]
  12. T.K. Kar, A model for fishery resource with reserve area and facing prey predator interactions. Can. Appl. Math. Quart. 14 (2006) 385–399. [Google Scholar]
  13. Y. Li, Y. Chen and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comp. Math. Appl. 59 (2010) 1810–1821. [CrossRef] [Google Scholar]
  14. Y. Louartassi, A. Alla, K. Hattaf and A. Nabil, Dynamics of a predator–prey model with harvesting and reserve area for prey in the presence of competition and toxicity. J. Appl. Math. Comput. 1 (2018) 305–321. [Google Scholar]
  15. Y. Louartassi, E. El Mazoudi and N. Elalami, A new generalization of lemma Gronwall-Bellman. Appl. Math. Sci. 6 (2012) 621–628. [Google Scholar]
  16. R.L. Magin, Fractional calculus in bioengineering. CRC Crit. Rev. Biomed. Eng. 32 (2004) 1–377. [CrossRef] [Google Scholar]
  17. D. Matignon, Stability results for fractional differential equations with applications to control processing. Proc. Comput. Eng. Syst. Appl. Multiconf . 2 (1996) 963–968. [Google Scholar]
  18. R.M. May, Stability and complexity in model ecosystems. Princeton University Press, Princeton, New Jersey (1973). [Google Scholar]
  19. T.M. Michelitsch, G.A. Maugin, F.C.G.A. Nicolleau, A.F. Nowakowski and S. Derogar, Dispersion relations and wave operators in self-similar quasicontinuous linear chains. Phys. Rev. E 80 (2009) 011135. [Google Scholar]
  20. A. Mouaouine, A. Boukhouima, K. Hattaf and N. Yousfi, A fractional order SIR epidemic model with nonlinear incidence rate. Adv. Differ. Equ. 2018 (2018) 160. [Google Scholar]
  21. K. Oldham and J. Spanier, Vol. 111 of The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elseiver, Amsterdam (1974). [Google Scholar]
  22. I. Podlubny, Vol. 198 of Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to methods of their Solution and Some of their applications. Elsevier, Amsterdam (1998). [Google Scholar]
  23. M. Riesz, L’intégrale de Riemann-Liouville et le probléme de Cauchy. Acta Math. 81 (1949) 1–222. [CrossRef] [MathSciNet] [Google Scholar]
  24. B. Ross, S.G. Samko and E. Russel Love, Functions that have no First Order Derivative might have fractional derivatives of all ordersless than one. Real Anal. Exchange 20 (1994) 140–157. [CrossRef] [Google Scholar]
  25. M. Sambath, P. Ramesh and K. Balachandran, Asymptotic behavior of the fractional order three species prey–predator model. Int. J. Nonlinear Sci. Numer. Simul. 19 (2018) 721–733. [CrossRef] [Google Scholar]
  26. S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications (1993). [Google Scholar]
  27. J.B. Shukla, A.K. Agrawal, B. Dubey and P. Sinha, Existence and survival of two competing species in a polluted environment: a mathematical model. J. Biol. Syst. 9 (2001) 89–103. [Google Scholar]
  28. C. Vargas De-León Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24 (2015) 75–85. [Google Scholar]
  29. H. Yang and J. Jia, Harvesting of a predator–prey model with reserve area for prey and in the presence of toxicity. J. Appl. Math. Comput. 53 (2017) 693–708. [Google Scholar]
  30. X.Q. Zhao. Dynamical Systems in Population Biology. Springer New York (2000). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.