Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Stability of Hopf-bifurcating limit cycles in a diffusion-driven prey-predator system with Allee effect and time delay

Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India.

Special Issues: Mathematical Modeling to Solve the Problems in Life Sciences

In this paper, we investigate the effect of the gestation delay on the spatiotemporal pattern formation in a prey-predator system with monotonic functional response with saturation and intraspecific competition among the predator population in presence of the additive Allee effect in prey growth. In this regard, we present rigorous analytical results for determining the delay-induced Hopf bifurcation threshold and the associated properties of the Hopf-bifurcating periodic solutions and verify them with the help of numerical simulations. We derive analytically the delay-induced Hopf bifurcation threshold by employing the linear stability analysis about the unique spatially uniform coexistence steady state. Also, we provide the expressions for determining the direction and stability of the Hopfbifurcating periodic solutions by using the normal form theory and center manifold reduction. The numerical simulation results reveal that the Hopf bifurcation can potentially lead to spatially homogeneous periodic in time distribution of the populations which will eventually settles to chaotic in space and time distribution for sufficiently large value of the time delay. Further, our numerical investigations reveal that the time delay can change one stationary pattern to another through the loss of monotonicity property of the spatially averaged densities.
  Article Metrics

Keywords prey-predator system; Allee effect; gestation delay; diffusion; stationary patterns; spatiotemporal chaos

Citation: Kalyan Manna, Malay Banerjee. Stability of Hopf-bifurcating limit cycles in a diffusion-driven prey-predator system with Allee effect and time delay. Mathematical Biosciences and Engineering, 2019, 16(4): 2411-2446. doi: 10.3934/mbe.2019121


  • 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 prey-predator 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, Self-organized spatial patterns and chaos in a ratio-dependent predator-prey system, Theor. Ecol., 4 (2011), 37–53.
  • 9. B. Miao, Persistence and Turing instability in a cross-diffusive predator-prey 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, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291–331.
  • 18. P. Aguirre, E. González-Olivares and E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Anal. Real World Appl., 10 (2009), 1401–1416.
  • 19. P. Aguirre, E. González-Olivares and E. Sáez, Three Limit Cycles in a Leslie-Gower Predator-Prey Model with Additive Allee Effect, SIAM J. Appl. Math., 69 (2009), 1244–1262.
  • 20. K. Manna and M. Banerjee, Stationary, non-stationary and invasive patterns for a prey-predator 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 predator-prey model, Trans. Am. Math. Soc., 359 (2007), 4557–4593.
  • 22. J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey 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 predator-prey 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 predator-prey model with weak Allee effects, Math. Biosci. Eng., 11 (2014), 1247–1274.
  • 25. F. Rao and Y. Kang, The complex dynamics of a diffusive prey-predator 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, Time-delay versus stability in population models with two and three trophic levels, Ecology, 54 (1973), 315–325.
  • 29. S. Ruan, On nonlinear dynamics of predator-prey 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 predator-prey 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 predator-prey 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., Time-delay-induced instabilities in reaction-diffusion 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, Delay-driven 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 ratio-dependent prey-predator model, Chaos Solitons Fractals, 67 (2014), 73–81.
  • 38. M. Banerjee and L. Zhang, Time delay can enhance spatio-temporal chaos in a prey-predator model, Ecol. Complex., 27 (2016), 17–28.
  • 39. Y. Song, Y. Peng and X. Zou, Persistence, Stability and Hopf bifurcation in a diffusive ratiodependent predator-prey 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 predator-prey system, Int. J. Bifurc. Chaos, 22 (2012), 1250061.
  • 41. H. Fang, L. Hu and Y.Wu, Delay-induced Hopf bifurcation in a diffusive Holling-Tanner predatorprey model with ratio-dependent response and Smith growth, Adv. Differ. Equ., 2018 (2018), 285.
  • 42. T. Faria, Stability and bifurcation for a delayed predator-prey 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 predator-prey system descried by a delayed reaction-diffusion equations, J. Math. Anal. Appl., 339 (2008), 1432–1450.
  • 44. G. P. Hu andW. T. Li, Hopf bifurcation analysis for a delayed predator-prey 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 predator-prey system with additional food, Chaos Solitons Fractals, 91 (2016), 350–359.
  • 46. F. Rao, C. Castillo-Chavez and Y. Kang, Dynamics of a diffusion reaction prey-predator 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 reaction-diffusion 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 predator-prey 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 predator-prey model with delay, Nonlinear Dyn., 62 (2010), 601–608.
  • 50. C. Xu and S. Yuan, Spatial periodic solutions in a delayed diffusive predator-prey model with herd behavior, Int. J. Bifurc. Chaos, 25 (2015), 1550155.
  • 51. X. P. Yan, Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects, Appl. Math. Comput., 192 (2007), 552–566.
  • 52. R. Yang, H. Ren and X. Cheng, A diffusive predator-prey 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 diffusion-competition 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 Beddington-DeAngelis type predator-prey 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 predator-prey 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 three-species 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 food-chain model with time delay, Comput. Math. Appl., 66 (2013), 2139–2153.
  • 60. G. X. Yang and J. Xu, Stability and Hopf bifurcation for a three-species reaction-diffusion predator-prey 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 functional-differential equations and reaction-diffusion systems, Trans. Am. Math. Soc., 321 (1990), 1–44.
  • 67. R. H. Martin and H. L. Smith, Reaction-diffusion 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, Springer-Verlag, 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 two-species 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, Springer-Verlag, 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 trade-off between mutual interference and time lags in predator-prey 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 predator-prey system with the Allee effect, J. Theor. Biol., 238 (2006), 18–35.
  • 79. N. Mukherjee, S. Ghorai and M. Banerjee, Effects of density dependent cross-diffusion on the chaotic patterns in a ratio-dependent prey-predator model, Ecol. Complex., 36 (2018), 276–289.
  • 80. M. Pascual, Diffusion-induced chaos in a spatial predator-prey 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 predator-prey systems: Spatio-temporal paterns in the neighborhood of Turing-Hopf 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, Springer-Verlag, 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, Prey-taxis destabilizes homogeneous stationary state in spatial Gause-Kolmogorov-type model for predator-prey system, Ecol. Complex., 31 (2017), 170–180.
  • 93. J. D. Murray, Mathematical Biology, Springer, Heidelberg, 1989.


Reader Comments

your name: *   your email: *  

© 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)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved