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Original Articles

Bifurcation and temporal periodic patterns in a plant–pollinator model with diffusion and time delay effects

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Pages 138-159 | Received 15 Oct 2015, Accepted 19 Apr 2016, Published online: 17 May 2016

References

  • V.D. Adams, D.L. DeAngelis, and R.A. Goldstein, Stability analysis of the time delay in a host-parasitoid model, J. Theor. Biol. 83 (1980), pp. 43–62. doi: 10.1016/0022-5193(80)90371-9
  • J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol. 44 (1975), pp. 331–340. doi: 10.2307/3866
  • S.A. Chamberlain and J.N. Holland, Density-mediated, context-dependent consumer–resource interactions between ants and extrafloral nectar plants, Ecology 89 (2008), pp. 1364–1374. doi: 10.1890/07-1139.1
  • S. Chen, J. Shi, and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie–Gower predator–prey system, Inter. J. Bifur. Chaos 22 (2012), pp. 1250061-1-11.
  • J.M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, Vol. 20, Springer-Verlag, Heidelberg, 1977.
  • D.L. DeAngelis, R.A. Goldstein, and R.V. O'Neill, A model for trophic interaction, Ecology 56 (1975), pp. 881–892. doi: 10.2307/1936298
  • T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc. 352 (2000), pp. 2217–2238. doi: 10.1090/S0002-9947-00-02280-7
  • M.A. Fishman and L. Hadany, Plant-pollinator population dynamics, Theor. Popul. Biol. 78 (2010), pp. 270–277. doi: 10.1016/j.tpb.2010.08.002
  • J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin, 1977.
  • B.D. Hassard, N.D. Kazarinoff, and Y.-H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
  • J.N. Holland and D.L. DeAngelis, Consumer–resource theory predicts dynamic transitions between outcomes of interspecific interactions, Ecol. Lett. 12 (2009), pp. 1357–1366. doi: 10.1111/j.1461-0248.2009.01390.x
  • G. Hu and W.-T. Li, Hopf bifurcation analysis for a delayed predator–prey system with diffusion effects, Nonlinear Anal. Real World Appl. 11 (2010), pp. 819–826. doi: 10.1016/j.nonrwa.2009.01.027
  • S.S. Hu, D.L. Dilcher, D.M. Jarzen, and D.W. Taylor, Early steps of angiosperm pollinator coevolution, Proc. Natl. Acad. Sci. USA 105 (2008), pp. 240–245. doi: 10.1073/pnas.0707989105
  • S.R.-J. Jang, Dynamics of herbivore–plant–pollinator models, J. Math. Biol. 44 (2002), pp. 129–149. doi: 10.1007/s002850100117
  • C.A. Kearns, D.W. Inouye, and N.M. Waser, Endangered mutualisms: The conservation of plant–pollinator interactions, Annu. Rev. Ecol. Syst. 29 (1998), pp. 83–112. doi: 10.1146/annurev.ecolsys.29.1.83
  • Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
  • X. Li, H. Wang, and Y. Kuang, Global analysis of a stoichiometric producer–grazer model with Holling type functional responses, J. Math. Biol. 63 (2011), pp. 901–932. doi: 10.1007/s00285-010-0392-2
  • Z. Liu, P. Magal, and S. Ruan, Oscillations in age-structured models of consumer–resource mutualisms, Discret Contin. Dyn. Syst. B 21 (2016), pp. 537–555. doi: 10.3934/dcdsb.2016.21.537
  • S. Lundberg and P. Ingvarsson, Population dynamics of resource limited plants and their pollinators, Theor. Popul. Biol. 54 (1998), pp. 44–49. doi: 10.1006/tpbi.1997.1349
  • C. Neuhauser and J.E. Fargione, A mutualism–parasitism continuum model and its application to plant–mycorrhizae interactions, Ecol. Model. 177 (2004), pp. 337–352. doi: 10.1016/j.ecolmodel.2004.02.010
  • E.R. Pianka, Evolutionary Ecology, Harper and Row, New York, 1974.
  • S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator–prey systems with discrete delays, Quart. Appl. Math. 59 (2001), pp. 159–173.
  • S. Ruan and J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. Math. Appl. Med. Biol. 18 (2001), pp. 41–52. doi: 10.1093/imammb/18.1.41
  • J .M. Soberon and C.M. Del Rio, The dynamics of a plant-pollinator interaction, J. Theor. Biol. 91 (1981), pp. 363–378. doi: 10.1016/0022-5193(81)90238-1
  • L. Wang, H. Jiang and Y. Li, Positive steady state solutions of a plant-pollinator model with diffusion, Discret. Contin. Dyn. Syst. B 20 (2015), pp. 1805–1819. doi: 10.3934/dcdsb.2015.20.1805
  • Y. Wang, Dynamics of plant-pollinator-robber systems, J. Math. Biol. 66 (2013), pp. 1155–1177. doi: 10.1007/s00285-012-0527-8
  • Y. Wang and D.L. DeAngelis, Transitions of interaction outcomes in a uni-directional consumer–resource system, J. Theor. Biol. 208 (2011), pp. 43–49. doi: 10.1016/j.jtbi.2011.03.038
  • Y. Wang, D.L. DeAngelis, and J.N. Holland, Uni-directional consumer–resource theory characterizing transitions of interaction outcomes, Ecol. Complex. 8 (2011), pp. 249–257. doi: 10.1016/j.ecocom.2011.04.002
  • Y. Wang, D.L. DeAngelis, and J.N. Holland, Uni-direction interation and plant-pollinator-robber coexistence, Bull. Math. Biol. 74 (2012), pp. 2142–2164. doi: 10.1007/s11538-012-9750-0
  • J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
  • W. Zuo and J. Wei, Stability and Hopf bifurcation in a diffusive predator–prey system with delay effect, Nonlinear Anal. Real World Appl. 12 (2011), pp. 1998–2011. doi: 10.1016/j.nonrwa.2010.12.016