354
Views
10
CrossRef citations to date
0
Altmetric
Articles

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

Pages 687-723 | Received 21 May 2015, Accepted 17 Dec 2015, Published online: 22 Jan 2016

References

  • A.S. Ackleh, L.J.S. Allen, and J. Carter, Establishing a beachhead: A stochastic population model with an Allee effect applied to species invasion, Theor. Population Biol. 71 (2007), pp. 290–300.
  • P. Amarasekare, Allee effects in metapopulation dynamics, Am. Nat. 152 (1998), pp. 298–302.
  • P. Amarasekare, Interactions between local dynamics and dispersal: Insights from single species models, Theor. Population Biol. 53 (1998), pp. 44–59.
  • M. Begon, J.L. Harper, and C.R. Townsend, Ecology: Individuals, Populations and Communities, Blackwell Science Ltd., Oxford, 1996.
  • A. Brett and M.R.S. Kulenović, Two species competitive model with the Allee effect, Adv. Differ. Equ. 2014 (2014), pp. 307–328.
  • A. Brett and M.R.S. Kulenović, Basins of attraction for two species competitive model with quadratic terms and the singular Allee effect, Discrete Dyn. Nat. Soc. (2015), pp. 16p.
  • Y. Cai, M. Banerjee, Y. Kang, and W. Wang, Spatiotemporal complexity in a predator-prey model with weak Allee effects, Math. Biosci. Eng. 11(6) (2014), pp. 1247–1274.
  • P. Calow, D.A. Falk, J. Grace, and P.D. Moore, The Encyclopedia of Ecology and Environmental Management, Blackwell Science, Oxford, 1998.
  • P.L. Chesson and S. Ellner, Invasibility and stochastic boundedness in monotonic competition models, J. Math. Biol. 27 (1989), pp. 117–138.
  • S.N. Chow and J.K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.
  • D. Clark, M.R.S. Kulenovic, and J.F. Selgrade, Global asymptotic behavior of a two-dimensional difference equation modelling competition, Nonlinear Anal. 52 (2003), pp. 1765–1776.
  • F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, USA, 2009.
  • J. Cushing and J. Hudson, Evolutionary dynamics and strong Allee effects, J. Biol. Dyn. 6(2) (2012), pp. 941–958.
  • J. Cushing, The evolutionary dynamics of a population model with a strong Allee effect, Math. Biosci. Eng. 12(4) (2015), pp. 643–660.
  • E.N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine. Angew. Math. 419 (1991), pp. 125–139.
  • B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction, Nat. Res. Model. 3 (1989), pp. 481–538.
  • B. Dennis, Allee effects in stochastic populations, Oikos 96 (2002), pp. 389–401.
  • J.M. Drake, Allee effects and the risk of biological invasion, Risk Anal. 24 (2004), pp. 795–802.
  • C. Egami, Permanence of delay competitive systems with weak Allee effects, Nonlinear Anal.: Real World Appl. 11 (2010), pp. 3936–3945.
  • S.N. Elaydi and R.J. Sacker, Population models with Allee effect: A new model, J. Biol. Dyn. 4 (2010), pp. 397–408.
  • R. Etiemme, B. Werthei, L. Hemerik, P. Schneider, and J. Powell, The interaction between dispersal, the Allee effect and scramble competition affects population dynamics, Ecol. Model. 148 (2002), pp. 153–168.
  • W.F. Fagan, M.A. Lewis, M.G. Neubert, and P. van den Driessche, Invasion theory and biological control, Ecol. Lett. 5 (2002), pp. 148–157.
  • G.R.J. Gaut, K. Goldring, F. Grogan, C. Haskell, and R.J. Sacker, Difference equations with the Allee effect and the periodic Sigmoid Beverton--Holt equation revisited, J. Biol. Dyn. 6(2) (2012), pp. 1019–1033.
  • C. Greene and J.A. Stamps, Habitat selection at low population densities, Ecology 82 (2001), pp. 2091–2100.
  • M. Gyllenberg, J. Hemminki, and T. Tammaru, Allee effects can both conserve and create spatial heterogeneity in population densities, Theor. Population Biol. 56 (1999), pp. 231–242.
  • F.A. Hopf, T.J. Valone, and J.H. Brown, Competition theory and the structure of ecological communities, Evol. Ecol. 7 (1993), pp. 142–154.
  • A.J. Harry, C.M. Kent, and V.L. Kocic, Global behavior of solutions of a periodically forced Sigmoid Beverton--Holt model, J. Biol. Dyn. 6 (2012), pp. 212–234.
  • P. Hess and A.C. Lazer, On an abstract competition model and applications, Nonlinear Anal: TMA 16 (1991), pp. 917–940.
  • S.R.J. Jang, Allee effects in a discrete-time host-parasitoid model, J. Differ. Equ. Appl. 12 (2006), pp. 165–181.
  • J. Gascoigne and R.N. Lipcius, Allee effects in marine systems, Mar. Ecol. Prog. Ser. 269 (2004), pp. 49–59.
  • Y. Kang and P. Chesson, Relative nonlinearity and permanence, Theor. Population Biol. 78 (2010), pp. 26–35.
  • Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects, J. Math. Biol. 62 (2011), pp. 925–973.
  • Y. Kang and D. Armbruster, Dispersal effects on a two-patch discrete model for plant-herbivore interactions, J. Theor. Biol. 268 (2011), pp. 84–97.
  • Y. Kang and A.A. Yakubu, Weak Allee effects and species coexistence, Nonlinear Anal.: Real World Appl.. 12 (2011), pp. 3329–3345.
  • Y. Kang and C. Castillo-Chavez, Multiscale analysis of compartment models with dispersal, J. Biol. Dyn. 6(2) (2012), pp. 50–79.
  • Y.Kang, Permanence of a general discrete-time two-species interaction model with nonlinear per-capita growth rates, Discrete Continuous Dyn. Syst. - Ser. B. 18 8pp. 2123–2142.
  • Y. Kang, Scramble competitions can rescue endangered species subject to strong Allee effects, Math. Biosci. 241(1) (2013), pp. 75–87.
  • Y. Kang, R.A. Bhowmick, K.S. Sasmal, J. Chattopadhyay, Host-parasitoid systems with predation-driven Allee effects in host population, J. Biol. Dyn. (2014), [Epub ahead of print]. doi:10.108017513758.2014.972473
  • Y. Kang and O. Udiani, Dynamics of a single species evolutionary model with Allee effects, J. Math. Anal. Appl. 418(1) (2014), pp. 492–515.
  • Y. Kang, K.S. Sasmal, R.A. Bhowmick, and J. Chattopadhyay, Dynamics of a predator-prey system with prey subject to Allee effects and disease, J. Math. Biosci. Eng. 11(4) (2014), pp. 877–918.
  • Y. Kang and C. Castillo-Chavez, Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects, J. Math. Biosci. 248 (2014), pp. 97–116.
  • Y. Kang and C. Castillo-Chavez, A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness, Discrete Continuous Dyn. Syst. - Ser. B. 19 1(2014), pp. 89–130.
  • Y. Kang and C. Castillo-Chavez, A simple two-patch epidemiological model with Allee effects and disease-modified fitness, in Mathematics of Continuous and Discrete Dynamical Systems, A.B.Gumel, ed. Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, 2014, pp. 618.
  • T.H. Keitt, M.A. Lewis, and R.D. Holt, Allee effects, invasion pinning, and species’ borders, Am. Nat. 157 (2001), pp. 203–216.
  • M.R.S. Kulenović and M. Nurkanović, Global behavior of a two-dimensional competitive system of difference equations with stocking, Math. Comput. Model. 55 (2012), pp. 1998–2011.
  • M.R.S. Kulenović, O. Merinoa, and M. Nurkanović, Global dynamics of certain competitive system in the plane, J. Differ. Equ. Appl. 18(12) (2012), pp. 1951–1966.
  • A. Liebhold and J. Bascompte, The Allee effect, stochastic dynamics and the eradication of alien species, Ecol. Lett. 6 (2003), pp. 133–140.
  • W.Z. Lidicker, The Allee effect: Its history and future importance, Open Ecol, J. 3 (2010), pp. 71–82.
  • G. Livadiotis and S. Elaydi, General Allee effect in two-species population biology, J. Biol. Dyn. 6(2) (2012), pp. 959–973.
  • G. Livadiotis, L. Assas, B. Dennis, S. Elaydid, and E. Kwessi, discrete-time host-parasitoid model with an Allee effect, J. Biol. Dyn. 9(1) (2015) doi:10.108017513758.2014.982219.
  • M.A. McCarthy, The Allee effect, finding mates and theoretical models, Ecol. Model. 103 (1997), pp. 99–102.
  • B.E. Meserve, Fundamental Concepts of Algebra, Dover Publications, New York, 1982.
  • C.Mira, L.Gardini and A.Barugola, Nonlinear Sciences, Series A, in Chaotic Dynamics in Two-dimensional Non-invertible Maps20World Scientific Publishing Co., Pte. Ltd, Singapore, 1996.
  • R.A. Myers, N.J. Barrowman, J.A. Hutchings, and A.A. Rosenberg, Population dynamics of exploited fish stocks at low population levels, Science 269 (1995), pp. 1106–1108.
  • J.A. Myers and G. Mertz, Reducing uncertainty in the biological basis of fisheries management by meta-analysis of data from many population: A synthesis, Fish. Res. 37 (1998), pp. 51–60.
  • R.A. Myers, Stock and recruitment: Generalizations about maximum reproductive rate, density dependence, and variability using meta-analytic approaches, ICES, J. Mar. Sci. 58 (2001), pp. 937–951.
  • P. Feng and K. Yun, Dynamics of a modified Leslie-Gower model with double Allee effects, Nonlinear Dyn. 80(1–2) (2015), pp. 1051–1062.
  • S. Petrovskii, A. Morozov, and B.-L. Li, Regimes of biological invasion in a predator-prey system with the Allee effect, Bull. Math. Biol. 67 (2005), pp. 637–661.
  • S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Population Biol. 64 (2003), pp. 201–209.
  • J.F. Selgrade and M. Ziehe, Convergence to equilibrium in a genetic model with differential viability between the sexes, J. Math. Biol. 25 (1987), pp. 477–490.
  • J.F. Selgrade and G. Namkoong, Dynamical behavior for population genetics models of differential and difference equations with nonmonotone fitnesses, J. Math. Biol. 30 (1992), pp. 815–826.
  • N. Shigesada and K. Kawasaki, Introduction, in Biological Invasions: Theory and Practice, N. Shigesada and K. Kawasaki, eds., Oxford University Press, New York, USA, 1997, pp. 1–5.
  • D.E. Smith and M.L. Latham, The Geometry of Rene Descartes with a facsimile of the first editionDover Publications, New York, 1954; translated by D.E. Smith and M.L Latham.
  • H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.
  • H.L. Smith, Planar competitive and cooperative difference equations, J. Differ. Equ. Appl. 3 (1998), pp. 335–357.
  • P.A. Stephens, W.J. Sutherland, and R.P. Freckleton, What is the Allee effect?, Oikos 87 (1999), pp. 185–190.
  • A.W. Stoner and M. Ray-Culp, Evidence for Allee effects in an over-harvested marine gastropod: Density-dependent mating and egg production, Mar. Ecol. Prog. Ser. 202 (2000), pp. 297–302.
  • C.M. Taylor and A. Hastings, Allee effects in biological invasions, Ecol. Lett. 8 (2005), pp. 895–908.
  • I. Terescák, Dynamics of C1 smooth strongly monotone discrete-time dynamical systems, Technical report, Comenius University, Bratislava, (1996).
  • H. Thieme, T. Dhirasakdanon, Z. Han, and R. Trevino, Species decline and extinction: Synergy of infectious disease and Allee effect?, J. Biol. Dyn. 3 (2009), pp. 305–323.
  • G.G.Thomson, A proposal for a threshold stock size and maximum fishing mortality rate, Risk Evaluation and Biological Reference Points for Fisheries Management,, Canadian Special Publication of Fisheries and Aquatic Sciences, Vol. 120S.J. Smith, J.J. Hunt, D. Rivard, eds., NRC Research Press, Ottawa, Canada, 1993, pp. 303–320.
  • Y. Wang and J. Jifa, The general properties of discrete-time competitive dynamical systems, J. Differ. Equ. 176 (2001), pp. 470–493.
  • M.H. Wang, M. Kot, and M.G. Neubert, Integrodifference equations, Allee effects, and invasions, J. Math. Biol. 44 (2002), pp. 150–168.
  • S.R. Zhou, C.Z. Liu, and G. Wang, The competitive dynamics of metapopulation subject to the Allee-like effect, Theor. Population Biol. 65 (2004), pp. 29–37.
  • Y. Kang, and H. Smith, Global dynamics of a discrete-time two-species Lottery-Ricker competition model, Journal of Biological Dynamics. 6 (2012), pp. 358–376.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.