Issue
Math. Model. Nat. Phenom.
Volume 16, 2021
Control of instabilities and patterns in extended systems
Article Number 16
Number of page(s) 22
DOI https://doi.org/10.1051/mmnp/2021013
Published online 23 March 2021
  1. Q. An, E. Beretta, Y. Kuang, C. Wang and H. Wang, Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters. J. Differ. Equ. 266 (2019) 7073–7100. [Google Scholar]
  2. E. Beretta and Y. Kuang, Geometric, stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33 (2002) 1144–1165. [CrossRef] [MathSciNet] [Google Scholar]
  3. E. Beretta and Y. Kuang, Global analysis in some delayed ratio-dependent predator-prey systems. Nonlinear Anal. Theory Methods Appl. 32 (1998) 381–408. [Google Scholar]
  4. E. Buskey and D. Stockwell, Effects of a persistent brown tide on zooplankton population in the Laguno Madre of Southern Texas, in: T. J. Smayda, Shimuzu, Toxic Phytoplankton Blooms in the Sea. Elsevier, Amsterdam (1993). [Google Scholar]
  5. S. Chakarborty, S. Roy and J. Chattopadhyay, Nutrientlimiting toxin producing and the dynamics of two phytoplankton in culture media: a mathematical model. J. Ecol. Model. 213 (2008) 191–201. [CrossRef] [Google Scholar]
  6. J. Chattopadhyay, R. Sarkar and S. Mandal, Toxin producing plankton may act as a biological control for planktonic blooms: A field study and mathematical modelling. J. Theoret. Biol. 215 (2002) 333–344. [CrossRef] [PubMed] [Google Scholar]
  7. J. Chattopadhyay, R. Sarkar and A. Abdllaoui, A delay differential equation model on harmful algal blooms in the presence of toxic substances. IMA J. Math. Appl. Med. Biol. 19 (2002) 137–161. [Google Scholar]
  8. Z. Cheng, Anti-control of Hopf bifurcation for Chen’s system through washout filters. Neurocomputing 73 (2010) 3139–3146. [Google Scholar]
  9. R. Etoua and C. Rousseau, Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III. J. Differ. Equ. 249 (2010) 2316–356. [Google Scholar]
  10. R. Fleming, The control of diatom populations by grazing. J. Cons. Perm Expl. Mer. 14 (1939) 210–227. [CrossRef] [Google Scholar]
  11. H. Freedman and R. Mathse, Persistence in predator-prey systems with ratio-dependent predator influence. Bull Math. Biol. 55 (1993) 817–827. [Google Scholar]
  12. K. Gu, S. Niculescu and J. Chen, On stability crossing curves for general systems with two delays. J. Math. Anal. Appl. 311 (2005) 231–253. [Google Scholar]
  13. J. Hale and S. Lunel, Introduction to Functional Differential Equations. Springer-Verlag, New York (1993). [Google Scholar]
  14. B. Hassard, N. Kazarinoff and Y. Wan, Theory and application of Hopf bifurcation. Cambridge University Press, Cambridge (1981). [Google Scholar]
  15. C. Holling, The functional response of predator to prey density and its role in mimicry and population regulation. Men. Ent. Sec. Can. 45 (1965) 1–60. [Google Scholar]
  16. S. Hsu and T. Huang, Global stability for a class of predator–prey system. SIAM J. Appl. Math. 55 (1995) 763–783. [Google Scholar]
  17. V. Ivlev, Biologicheskaya produktivnost’vodoemov. Uspekhi Sovremennoi Biologii. 19 (1945) 98–120. [Google Scholar]
  18. Z. Jiang and T. Zhang, Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay. Chaos Solitons Fractals 104 (2017) 693–704. [Google Scholar]
  19. Z. Jiang, W. Zhang, J. Zhang and T. Zhang, Dynamical analysis of a phytoplankton-zooplankton system with harvesting term and holling III functional response. Internat. J. Bifur. Chaos 28 (2018) 1850162. [CrossRef] [Google Scholar]
  20. Z. Jiang, Y. Guo and T. Zhang, Double delayed feedback control of a nonlinear finance system. Discrete Dyn. Nat. Soc. 2019 (2019) 7254121. [Google Scholar]
  21. Z. Jiang, J. Dai and T. Zhang, Bifurcation analysis of phytoplankton and zooplankton interaction system with two delays. Inter. J. Bifur. Chaos 30 (2020) 2050039. [CrossRef] [Google Scholar]
  22. Z. Jiang and Y. Guo, Hopf bifurcation and stability crossing curve in a planktonic resource-consumer system with double delays. Internat. J. Bifur. Chaos 30 (2020) 2050190. [CrossRef] [Google Scholar]
  23. T. Kar and A. Ghorai, Dynamic behaviour of a delayed predator–prey model with harvesting. Appl. Math. Comput. 217 (2011) 9085–9104. [Google Scholar]
  24. P. Leslie, Some further notes on the use of matrics in the population mathematics. Biomatrika 35 (1948) 213–245. [CrossRef] [Google Scholar]
  25. P. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods. Biometrika 45 (1958) 16–31. [Google Scholar]
  26. P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika 47 (1960) 219–234. [Google Scholar]
  27. A. Lotka, Elements of Physical Biology. Williams and Wilkins, Baltimore (1925). [Google Scholar]
  28. X. Luo, G. Chen, B. Wang and J. Fang, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems. Chaos Solitons Fractals 18 (2003) 775–783. [Google Scholar]
  29. Y. Ma, Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays. Nonlinear Anal. Real. World Appl. 13 (2012) 370–375. [Google Scholar]
  30. A. Nindjin, M. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay. Nonlinear Anal. Real World Appl. 7 (2006) 1104–118. [Google Scholar]
  31. H. Odum, Primary production in flowing waters. Limnol Oceanogr. 1 (1956) 102–117. [Google Scholar]
  32. S. Pal, S. Chatterjee and J. Chattopadhyay, Role of toxin and nutrient for the occurrence and termination of plankton bloom-results drawn from field observations and a mathematical model. J. Biosyst. 90 (2007) 87–100. [CrossRef] [Google Scholar]
  33. K. Pyragas, Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170 (1992) 421–428. [Google Scholar]
  34. G. Riley, Factors controlling phytoplankton populations on Georges Bank. J. Mar. Res. 6 (1946) 54–73. [Google Scholar]
  35. S. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling. J. Math. Biol. 31 (1993) 633–654. [Google Scholar]
  36. R. Sarkar and J. Chattopadhyay, Occurrence of planktonic blooms under environmental fluctuations and its possible control mechanism-mathematical models and experimental observations. J. Theor. Biol. 224 (2003) 501–516. [CrossRef] [PubMed] [Google Scholar]
  37. Y. Tian and P. Weng, Stability analysis of diffusive predator- prey model with modified Leslie-Gower and Holling-type III schemes. Appl. Math. Comput. 218 (2011) 3733–745. [Google Scholar]
  38. V. Volterra, Variations and fluctuations of the number of individuals in animal species living together. J. Conseil. 3 (1928) 3–51. [CrossRef] [Google Scholar]
  39. R. Yafia, F. El Adnani and H. Alaoui, Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Nonlinear Anal. Real World Appl. 9 (2008) 2055–067. [Google Scholar]
  40. Y. Yuan, A coupled plankton system with instantaneous and delayed predation. J. Biol. Dyn. 6 (2012) 148–165. [CrossRef] [PubMed] [Google Scholar]
  41. H. Zhao, Y. Lin and Y. Dai, Bifurcation analysis and control of chaos for a hybrid ratio-dependent three species food chain. Appl. Math. Comput. 218 (2011) 1533–1546. [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.