
Mathematical Biosciences and Engineering, 2019, 16(4): 24112446. doi: 10.3934/mbe.2019121.
Research article Special Issues
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
Stability of Hopfbifurcating limit cycles in a diffusiondriven preypredator system with Allee effect and time delay
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India.
Received: , Accepted: , Published:
Special Issues: Mathematical Modeling to Solve the Problems in Life Sciences
Keywords: preypredator system; Allee effect; gestation delay; diffusion; stationary patterns; spatiotemporal chaos
Citation: Kalyan Manna, Malay Banerjee. Stability of Hopfbifurcating limit cycles in a diffusiondriven preypredator system with Allee effect and time delay. Mathematical Biosciences and Engineering, 2019, 16(4): 24112446. doi: 10.3934/mbe.2019121
References:
 1. G. F. Gause, The Struggle for Existence, Williams and Wilkins: Baltimore, MD, USA, 1935.
 2. A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond., B, Biol. Sci., 237 (1952), 37–72.
 3. S. A. Levin and L. A. Segel, Hypothesis for origin of planktonic patchiness, Nature, 259 (1976), 659.
 4. C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826–1828.
 5. H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations, Chapman & Hall, London, 2008.
 6. S. V. Petrovskii and H. Malchow, A minimal model of pattern formation in a preypredator system, Math. Comput. Model., 29 (1999), 49–63.
 7. D. Alonso, F. Bartumeus and J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83 (2002), 28–34.
 8. M. Banerjee and S. Petrovskii, Selforganized spatial patterns and chaos in a ratiodependent predatorprey system, Theor. Ecol., 4 (2011), 37–53.
 9. B. Miao, Persistence and Turing instability in a crossdiffusive predatorprey system with generalist predator, Adv. Differ. Equ., 2018 (2018), 260.
 10. A. B. Medvinsky, B. V. Adamovich and A. Chakraborty, et al., Chaos far away from the edge of chaos: A recurrence quantification analysis of plankton time series, Ecol. Complex., 23 (2015), 61–67.
 11. P. Turchin and S. P. Ellner, Living on the edge of chaos: Population dynamics of Fennoscandian voles, Ecology, 81 (2000), 3099–3116.
 12. W. C. Allee, Animal aggregations: A study in general sociology, University of Chicago Press, Chicago, USA, 1931.
 13. B. Dennis, Allee effect: population growth, critical density, and chance of extinction, Nat. Resour. Model., 3 (1989), 481–538.
 14. P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect? Oikos, 87 (1999), 185–190.
 15. M. A. Lewis and P. Kareiva, Allee Dynamics and the Spread of Invading Organisms, Theor. Popul. Biol., 43 (1993), 141–158.
 16. E. Odum and G. W. Barrett, Fundamentals of Ecology, Thomson Brooks/Cole, Belmont, CA, 2004.
 17. J. Wang, J. Shi and J. Wei, Predatorprey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291–331.
 18. P. Aguirre, E. GonzálezOlivares and E. Sáez, Two limit cycles in a LeslieGower predatorprey model with additive Allee effect, Nonlinear Anal. Real World Appl., 10 (2009), 1401–1416.
 19. P. Aguirre, E. GonzálezOlivares and E. Sáez, Three Limit Cycles in a LeslieGower PredatorPrey Model with Additive Allee Effect, SIAM J. Appl. Math., 69 (2009), 1244–1262.
 20. K. Manna and M. Banerjee, Stationary, nonstationary and invasive patterns for a preypredator system with additive Allee effect in prey growth, Ecol. Complex., 36 (2018), 206–217.
 21. Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predatorprey model, Trans. Am. Math. Soc., 359 (2007), 4557–4593.
 22. J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predatorprey system with strong Allee effect in prey, J. Differ. Equ., 251 (2011), 1276–1304.
 23. Y. Cai, W. Wang and J. Wang, Dynamics of a diffusive predatorprey model with additive Allee effect, Int. J. Biomath., 5 (2012), 1250023(11 pages).
 24. Y. Cai, M. Banerjee and Y. Kang, et al., Spatiotemporal complexity in a predatorprey model with weak Allee effects, Math. Biosci. Eng., 11 (2014), 1247–1274.
 25. F. Rao and Y. Kang, The complex dynamics of a diffusive preypredator model with an Allee effect in prey, Ecol. Complex., 28 (2016), 123–144.
 26. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992.
 27. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
 28. R. M. May, Timedelay versus stability in population models with two and three trophic levels, Ecology, 54 (1973), 315–325.
 29. S. Ruan, On nonlinear dynamics of predatorprey models with discrete delay, Math. Model. Nat. Pheno., 4 (2009), 140–188.
 30. P. J. Pal, T. Saha and M. Sen, et al., A delayed predatorprey model with strong Allee effect in prey population growth, Nonlinear Dyn., 68 (2012), 23–42.
 31. S. Roy Choudhury, Turing instability in competition models with delay I: Linear theory, SIAM J. Appl. Math., 54 (1994), 1425–1450.
 32. S. Roy Choudhury, Analysis of spatial structure in a predatorprey model with delay II: Nonlinear theory, SIAM J. Appl. Math., 54 (1994), 1451–1467.
 33. K. P. Hadeler and S. Ruan, Interaction of diffusion and delay, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 95–105.
 34. S. Sen, P. Ghosh and S. S. Riaz, et al., Timedelayinduced instabilities in reactiondiffusion systems, Phys. Rev. E, 80 (2009), 046212.
 35. S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlcal delay effect, J. Differ. Equ., 253 (2012), 3440–3470.
 36. C. Tian and L. Zhang, Delaydriven irregular spatiotemporal patterns in a plankton system, Phys. Rev. E, 88 (2013), 012713.
 37. M. Banerjee and L. Zhang, Influence of discrete delay on pattern formation in a ratiodependent preypredator model, Chaos Solitons Fractals, 67 (2014), 73–81.
 38. M. Banerjee and L. Zhang, Time delay can enhance spatiotemporal chaos in a preypredator model, Ecol. Complex., 27 (2016), 17–28.
 39. Y. Song, Y. Peng and X. Zou, Persistence, Stability and Hopf bifurcation in a diffusive ratiodependent predatorprey model with delay, Int. J. Bifurc. Chaos, 24 (2014), 1450093.
 40. S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie Gower predatorprey system, Int. J. Bifurc. Chaos, 22 (2012), 1250061.
 41. H. Fang, L. Hu and Y.Wu, Delayinduced Hopf bifurcation in a diffusive HollingTanner predatorprey model with ratiodependent response and Smith growth, Adv. Differ. Equ., 2018 (2018), 285.
 42. T. Faria, Stability and bifurcation for a delayed predatorprey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433–463.
 43. Z. Ge and Y. He, Diffusion effect and stability analysis of a predatorprey system descried by a delayed reactiondiffusion equations, J. Math. Anal. Appl., 339 (2008), 1432–1450.
 44. G. P. Hu andW. T. Li, Hopf bifurcation analysis for a delayed predatorprey system with diffusion effects, Nonlinear Anal. Real World Appl., 11 (2010), 819–826.
 45. J. Li, Z. Jin and G. Q. Sun, Periodic solutions of a spatiotemporal predatorprey system with additional food, Chaos Solitons Fractals, 91 (2016), 350–359.
 46. F. Rao, C. CastilloChavez and Y. Kang, Dynamics of a diffusion reaction preypredator model with delay in prey: Effects of delay and spatial components, J. Math. Anal. Appl., 461 (2018), 1177–1214.
 47. S. Ruan and X. Q. Zhao, Persistence and Extinction in two species reactiondiffusion systems with delays, J. Differ. Equ., 156 (1999), 71–92.
 48. X. Tang and Y. Song, Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predatorprey model with herd behavior, Appl. Math. Comput., 254 (2015), 375–391.
 49. B. Wang, A. L. Wang and Y. J. Liu, et al., Analysis of a spatial predatorprey model with delay, Nonlinear Dyn., 62 (2010), 601–608.
 50. C. Xu and S. Yuan, Spatial periodic solutions in a delayed diffusive predatorprey model with herd behavior, Int. J. Bifurc. Chaos, 25 (2015), 1550155.
 51. X. P. Yan, Stability and Hopf bifurcation for a delayed preypredator system with diffusion effects, Appl. Math. Comput., 192 (2007), 552–566.
 52. R. Yang, H. Ren and X. Cheng, A diffusive predatorprey system with prey refuge and gestation delay, Adv. Differ. Equ., 2017 (2017), 158.
 53. J. F. Zhang, W. T. Li and X. P. Yan, Bifurcation and spatiotemporal patterns in a homogeneous diffusioncompetition system with delays, Int. J. Biomath., 5 (2012), 1250049.
 54. J. Zhao and J. Wei, Persistence, Turing instability and Hopf bifurcation in a diffusive plankton system with delay and quadratic closure, Int. J. Bifurc. Chaos, 26 (2016), 1650047.
 55. W. Zuo, Global stability and Hopf bifurcations of a BeddingtonDeAngelis type predatorprey system with diffusion and delays, Appl. Math. Comput., 223 (2013), 423–435.
 56. W. Zuo and J. Wei, Stability and Hopf bifurcation in a diffusive predatorprey system with delay effect, Nonlinear Anal. Real World Appl., 12 (2011), 1998–2011.
 57. M. Jankovic, S. Petrovskii and M. Banerjee, Delay driven spatiotemporal chaos in single species population dynamics models, Theor. Popul. Biol., 110 (2016), 51–62.
 58. Z. P. Ma, W. T. Li and X. P. Yan, Stability and Hopf bifurcation for a threespecies food chain model with time delay and spatial diffusion, Appl. Math. Comput., 219 (2012), 2713–2731.
 59. C. Tian and L. Zhang, Hopf bifurcation analysis in a diffusive foodchain model with time delay, Comput. Math. Appl., 66 (2013), 2139–2153.
 60. G. X. Yang and J. Xu, Stability and Hopf bifurcation for a threespecies reactiondiffusion predatorprey system with two delays, Int. J. Bifurc. Chaos, 23 (2013), 1350194.
 61. R. Arditi, L. F. Bersier and R. P. Rohr, The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka, Ecosphere, 7 (2016), e01599.
 62. J. P. Gabriel, F. Saucy and L. F. Bersier, Paradoxes in the logistic equation? Ecol. Model., 185 (2005), 147–151.
 63. L. R. Ginzburg, Evolutionary consequences of basic growth equations, TREE, 7 (1992), 133.
 64. J. Mallet, The struggle for existence: How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution, and speciation, Evol. Ecol. Res., 14 (2012), 627–665.
 65. W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differ. Equ., 29 (1978), 1–14.
 66. R. H. Martin and H. L. Smith, Abstract functionaldifferential equations and reactiondiffusion systems, Trans. Am. Math. Soc., 321 (1990), 1–44.
 67. R. H. Martin and H. L. Smith, Reactiondiffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. reine angew. Math., 413 (1991), 1–35.
 68. C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Am. Math. Soc., 200 (1974), 395–418.
 69. J. Wu, Theory and Applications of Partial Functional Differential Equations, SpringerVerlag, New York, 1996.
 70. Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, 2006.
 71. M. H. Protter and H. F.Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, 1967.
 72. M. Sen, M. Banerjee and E. Venturino, A model for biological control in agriculture, Math. Comput. Simul., 87 (2013), 30–44.
 73. X. Song and L. Chen, Optimal harvesting and stability for a twospecies competitive system with stage structure, Math. Biosci., 170 (2001), 173–186.
 74. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, SpringerVerlag, Berlin, New York, 1981.
 75. B. Hassard, N. Kazarinoff and Y. Wan, Theory and applications of Hopf bifurcation, Cambridge University Press, Cambridge, 1981.
 76. H. I. Freedman and V. S. H. Rao, The tradeoff between mutual interference and time lags in predatorprey system, Bull. Math. Biol., 45 (1983), 991–1004.
 77. T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Am. Math. Soc., 352 (2000), 2217–2238.
 78. A. Morozov, S. Petrovskii and B. L. Li, Spatiotemporal complexity of patchy invasion in a predatorprey system with the Allee effect, J. Theor. Biol., 238 (2006), 18–35.
 79. N. Mukherjee, S. Ghorai and M. Banerjee, Effects of density dependent crossdiffusion on the chaotic patterns in a ratiodependent preypredator model, Ecol. Complex., 36 (2018), 276–289.
 80. M. Pascual, Diffusioninduced chaos in a spatial predatorprey system, Proc. R. Soc. Lond., B, Biol. Sci., 251 (1993), 1–7.
 81. A. Wolf, J. B. Swift and H. L. Swinney, et al., Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285–317.
 82. M. Baurmann, T. Gross and U. Feudel, Instabilities in spatially extended predatorprey systems: Spatiotemporal paterns in the neighborhood of TuringHopf bifurcations, J. Theor. Biol., 245 (2007), 220–229.
 83. S. V. Petrovskii and H. Malchow, Wave of chaos: New mechanism of pattern formation in spatiotemporal population dynamics, Theor. Popul. Biol., 59 (2001), 157–174.
 84. E. Ranta, V. Kaitala and J. Lindström, et al., Synchrony in population dynamics, Proc. R. Soc. Lond., B, Biol. Sci., 262 (1995), 113–118.
 85. M. Rietkerk and J. van de Koppel, Regular pattern formation in real ecosystems, Trends Ecol. Evol., 23 (2008), 169–175.
 86. P. Kareiva, A. Mullen and R. Southwood, Population dynamics in spatially complex environments: Theory and data (and discussion), Philos. Trans. R. Soc. Lond., B, Biol. Sci., 330 (1990), 175–190.
 87. T. M. Powell, P. J. Richerson and T. M. Dillon, et al., Spatial scales of current speed and phytoplankton biomass fluctuations in Lake Tahoe, Science, 189 (1975), 1088–1090.
 88. A. A. Sharov, A. M. Liebhold and E. A. Roberts, Correlation of counts of gypsy moths (Lepidoptera: Lymantriidae) in pheromone traps with landscape characteristics, Forest Sci., 43 (1997), 483–490.
 89. A. Okubo, Diffusion and Ecological Problems: Mathematical Models, SpringerVerlag, Berlin, 1980.
 90. A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, Berlin, 2001.
 91. L. A. Segel and J. L. Jackson, Dissipative structure: An explanation and an ecological example, J. Theor. Biol., 37 (1972), 545–559.
 92. Y. V. Tyutyunov, L. I. Titova and I. N. Senina, Preytaxis destabilizes homogeneous stationary state in spatial GauseKolmogorovtype model for predatorprey system, Ecol. Complex., 31 (2017), 170–180.
 93. J. D. Murray, Mathematical Biology, Springer, Heidelberg, 1989.
Reader Comments
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)
Associated material
Metrics
Other articles by authors
Related pages
Tools
your name: * your email: *