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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Analysis of a Prey-predator Model with Prey Refuge in Infected Prey and Strong Allee Effect in Susceptible Prey Population

Discontinuity, Nonlinearity, and Complexity 11(4) (2022) 671--703 | DOI:10.5890/DNC.2022.12.008

Sangeeta Saha$^{1}$, Alakes Maiti$^{2}$, Guruprasad Samanta$^{1}$

$^{1}$ Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah - 711103, India

$^{2}$ Department of Mathematics, Vidyasagar Evening College, Kolkata-700006, India

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An eco-epidemiological predator-prey model with Holling type-II functional response is proposed in this work. In the presence of disease, the prey population has been divided into two subpopulations: susceptible and infected prey. The predator can access the full healthy prey population for hunting but a predator is provided with a fraction of the infected prey as infected prey refuge term is incorporated here. Also, a strong Allee effect in susceptible population is introduced to make the model more realistic. Boundedness and positivity of the system strengthen that the proposed model is well-posed. The strong Allee threshold and the infected refuge parameter have been taken as the key parameters to control the system dynamics. The numerical simulation gives that regulating the refuge parameter can turn an oscillating state into a stable coexistence state. Also, the system changes its dynamics from two interior equilibrium points to no interior point when this refuge parameter crosses the saddle-node bifurcation threshold. Besides, the strong Allee threshold can also change the dynamics of a system from oscillating state to steady-state through Hopf bifurcation.


  1. [1]  Lotka, A. (1925), Elements of Physical Biology, Williams and Wilkins, Baltimore.
  2. [2]  Volterra, V. (1926), Variazioni e fluttuazioni del numero di individui in specie animali conviventi, Mem. Accl. Lincei., 2, 31-113.
  3. [3]  Baleanu, B., Jajarmi, A., Sajjadi, S.S., and Asad, J.H. (2020), The fractional features of a harmonic oscillator with position-dependent mass, Commun. Theor. Phys., 72, 055002. DOI:
  4. [4]  Baleanu, D., Jajarmi, A., Mohammadi, H., and Rezapour, S. (2020), A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos, Solitons \& Fractals, 134, 109705. https://doi:10.1016/ j.chaos.2020.109705
  5. [5]  Ghanbari, B. and Kumar, D. (2019), Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(6), 063103. https://doi:10.1063/1.5094546
  6. [6]  Jajarmi, A., Yusuf, A., Baleanu, D., and Inc, M. (2019), A new fractional HRSV model and its optimal control: A non-singular operator approach, Physica A: Statistical Mechanics and Its Applications, 547, 123860. https://doi:10.1016/ j.physa.2019.123860
  7. [7]  Jajarmi, A., Baleanu, D., Sajjadi, S., and Asad, J.H. (2019), A new feature of the fractional Euler-lagrange equations for a coupled oscillator using a nonsingular operator approach, Front. Phys., 7, 196. https://doi:10.3389/ fphy.2019.00196.
  8. [8]  Singh, J., Kumar, D., and Baleanu, D. (2020), A new analysis of fractional fish farm model associated with Mittag-Leffler type kernel, International Journal of Biomathematics, 13(2), 2050010 (17 pages). https://doi:10.1142/ s1793524520500102.
  9. [9]  Srivastava, H.M., Dubey, V.P., Kumar, R., Singh, J., Kumar, D., and Baleanu, D. (2020), An efficient computational approach for a fractional-order biological population model with carrying capacity, Chaos, Solitons \& Fractals, 138, 109880. https://doi:10.1016/j.chaos.2020.109880
  10. [10]  Holling, C.S. (1959a), The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Canadian Entomologist, 91, 293-320.
  11. [11]  Murray, J.D. (1993) Mathematical Biology, Springer-Verlag: New York.
  12. [12]  Maiti, A. and Samanta, G.P. (2006), Deterministic and stochastic analysis of a preydependent predator-prey system, Int J Math Educ Sci Technol, 36, 65-83.
  13. [13]  Brown, J.H. (1991), Methodological advances: new approaches and methods in ecology, Foundations of Ecology: Classic Papers with Commentaries, 445-455 in L. A. Real and J. H. Brown, editors. University of Chicago Press, Chicago, Illinois, USA.
  14. [14]  Huang, Y., Chen, F., and Zhong, L. (2006), Stability analysis of a prey-predator model with Holling type-II response function incorporating a prey refuge, Appl. Math. Comput., 182, 672-683.
  15. [15]  Ruan, S. and Xiao, D. (2001), Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61, 1445-1472.
  16. [16]  Chen, F., Chen, L., and Xie, X. (2009), On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal: Real World Appl., 10, 2905-2908.
  17. [17]  Du, Y. and Shi, J. (2006), A diffusive predator-prey model with a protection zone, J. Differ. Equ., 229, 63-91.
  18. [18]  Lv, Y., Yuan, R., and Pei, Y. (2013), A prey-predator model with harvesting for fishery resource with reserve area, Appl. Math. Model., 37, 3048-3062.
  19. [19]  Hoy, M.A. Almonds (California). (1985), In: Helle, W., Sabelis, M.W. (eds.) Spider Mites: Their Biology, Natural Enemies and Control, World Crop Pests, 1B, 229-310, Elsevier: Amsterdam.
  20. [20]  Courchamp, F., Berec, L., and Gascoigne, J. (2008), Allee Effects in Ecology and Conservation, Oxford University Press, Oxford.
  21. [21]  Allee, W.C. (1931), Animal Aggregations. A study in general sociology, Univ. of Chicago Press, Chicago.
  22. [22]  van, Voorn, G.A.K., Hemerik, L., Boer, M.P., and Kooi, B.W. (2007), Heteroclinic orbits indicate over-exploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209, 451-469.
  23. [23]  Wang, J., Shi, J., and Wei, J. (2011), Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62, 291-331.
  24. [24]  Wang, G., Liang, X.G., and Wang, F.Z. (1999), The competitive dynamics of populations subject to an Allee effect, Ecol. Model., 124, 183-192.
  25. [25]  Kot, M. (2001), Elements of Mathematical Biology, Cambridge University Press, Cambridge.
  26. [26]  Bazykin, A.D., Berezovskaya, F.S., Isaev, A.S., and Khlebopros, R.G. (1997), Dynamics of forest insect density: bifurcation approach, J. Theor. Biol., 186, 267-278.
  27. [27]  Sharma, S. and Samanta, G.P. (2014), Dynamical behaviour of an HIV/AIDS epidemic model, Differ Eqn Dyn Syst, 22(4), 369-395.
  28. [28]  Van, Dobben, W.H. (1952), The food of cormorants in the Netherlands, Ardea., 40, 1-63.
  29. [29]  Vaughn, G.E. and Coble, P.W. (1975), Sublethal effects of three ectoparasites on fish, J. Fisheries Biol., 7, 283-294.
  30. [30]  Temple, S.A. (1987), Do predators always capture substandard individuals disproportionately from prey populations?, Ecology, 68, 669-674.
  31. [31]  Holmes, J.C. and Bethel, W.M. (1972), Modifications of intermediate host behaviour by parasites. In: Canning, E.V., Wright, C.A. (Eds.), Behavioural Aspects of Parasite Transmission, Suppl I to Zool. f. Linnean Soc., 51, 123-149.
  32. [32]  Dobson, A.P. (1988), The population biology of parasite-induced changes in host behavior, Q. Rev. Biol., 63, 139-165.
  33. [33]  Moore, J. (2002), Parasites and the Behaviour of the Animals, Oxford University Press, Oxford.
  34. [34]  Krebs, J.R. (1978), Optimal foraging: decision rules for predators. In: Krebs, J.R., Davies, N.B. (Eds.) Behavioural Ecology: an Evolutionary approach, First ed. Blackwell Scientific Publishers, Oxford, 23-63.
  35. [35]  Peterson, R.O. and Page, R.E. (1988), The rise and fall of Isle royale wolves, 1975-1986, Journal of Mammalogy, 69(1), 89-99.
  36. [36]  Mech, L.D. (1970), The Wolf, Natural History Press: New York.
  37. [37]  Schaller, G.B. (1972), The Serengeti Lion: A Study of Predator Prey Relations, University of Chicago Press: Chicago.
  38. [38]  Lafferty, K.D. and Morris, A.K. (1996), Altered behavior of parasitized killifish increases susceptibility to predation by birf final hosts, Ecology, 77(5), 1390-1397.
  39. [39]  Uhlig, G. and Sahling, G. (1990), Long-term studies on Noctiluca scintillans in the German Bight population dynamics and red tide phenomena 1968-1988, Netherlands Journal of Sea Research, 25(1-2), 101-112.
  40. [40]  Gilpin, M.E. and Rosenzweig, M.L. (1972), Enriched predator-prey systems: Theoretical stability, Science, 177(4052), 902-904. doi:10.1126/science.177.4052.902.
  41. [41]  Kuznetsov, Y. and Rinaldi, S. (1996), Remarks on food chain dynamics, Mathematical Biosciences, 134(1), 1-33. https://doi:10.1016/0025-5564(95)00104-2.
  42. [42]  Li, Y. and Li, C. (2013), Stability and Hopf bifurcation analysis on a delayed Leslie-Gower predator-prey system incorporating a prey refuge, Applied Mathematics and Computation, 219(9), 4576-4589. https://doi:10.1016/ j.amc.2012.10.069.
  43. [43]  Hethcote, H. W., Wang, W., Han, L., and Ma, Z. (2004), A predator-prey model with infected prey. Theoretical Population Biology, 66(3), 259-268. https://doi:10.1016/j.tpb.2004.06.010.
  44. [44]  Tewa, J.J., Djeumen, V.Y., and Bowong, S. (2013), Predator-Prey model with Holling response function of type II and SIS infectious disease, Applied Mathematical Modelling, 37(7), 4825-4841. https://doi:10.1016/j.apm.2012.10.003.
  45. [45]  Zhou, X., Shi, X., and Song, X. (2009), Analysis of a delay prey-predator model with disease in the prey species only, J Korean Math Soc, 46(4), 713-731. https://10.4134/JKMS.2009.46.4.713.
  46. [46]  Haque, M., Zhen, J., and Venturino, E. (2009), An eco-epidemiological predator-prey model with standard disease incidence, Mathematical Methods in the Applied Sciences, 32(7), 875-898. https://doi:10.1002/mma.1071.
  47. [47]  Haque, M. and Venturino, E. (2009), Modelling disease spreading in symbiotic communities, wildlife destruction, conservation and biodiversity, Chapter: 5, Nova Science Publishers.
  48. [48]  Haque, M. (2010), A prey-predator model with disease in the predator species only, Nonlinear Anal. Real World Appl., 11, 2224-2236.
  49. [49]  Haque, M. and Greenhalgh D. (2010), When a predator avoids infected prey: a model-based theoretical study, Math Med Biol., 27(1), 75-94. https://doi:10.1093/imammb/dqp007.
  50. [50]  Saha, S., Maiti, A., and Samanta, G.P. (2018), A michaelis-menten predator-prey model with strong Allee effect and disease in prey incorporating prey refuge, International Journal of Bifurcation and Chaos, 28(6), 1850073 (21 pages).
  51. [51]  Saha, S. and Samanta, G.P. (2019), Analysis of a predator-prey model with herd behavior and disease in prey incorporating prey refuge, International Journal of Biomathematics, 12(01), 1950007 (39 pages). S1793524519500074.
  52. [52]  Xiao, Y. and Chen, L. (2001), Modeling and analysis of a predator-prey model with disease in the prey, Mathematical Biosciences, 171(1), 59-82. https://10.1016/s0025-5564(01)00049-9.
  53. [53]  Haque, M., Sarwardi, S., Preston, S., and Venturino, E. (2011), Effect of delay in a Lotka-Volterra type predator-prey model with a transmissible disease in the predator species, Mathematical Biosciences, 234(1), 47-57. https://doi:10.1016/j.mbs.2011.06.009.
  54. [54]  Venturino, E. (2002), Epidemics in predator-prey models: disease in the predators, IMA J Math Appl Med Biol., 19(3), 185-205.
  55. [55]  Anderson, R.M. and May, R.M. (1986), The invasion, persistence and spread of infectious diseases within animal and plant communities, Philos Trans R Soc Lond B Biol Sci., 314(1167), 533-570. https://doi:10.1098/rstb.1986.0072.
  56. [56]  Hadeler, K.P. and Freedman, H.I. (1989), Predator-prey populations with parasitic infection, J. Math. Biology, 27, 609-631.
  57. [57]  Wang, S. and Ma, Z. (2012), Analysis of an ecoepidemiological model with prey refuges, Journal of Applied Mathematics, 2012, 1-16. https://doi:10.1155/2012/371685
  58. [58]  Pal, A.K. and Samanta, G.P. (2010), Stability analysis of an eco-epidemiological model incorporating a prey refuge, Nonlinear Analysis: Modelling and Control, 15(4), 473-491. https://doi:10.1016/j.mbs.2011.06.009.
  59. [59]  Sharma, S. and Samanta, G.P. (2015), A Leslie-Gower predator-prey model with disease in prey incorporating a prey refuge, Chaos, Solitons \& Fractals, 70, 69-84. https://doi:10.1016/j.chaos.2014.11.010.
  60. [60]  Perrot-Minnot, M.J., Kaldonski, N., and CĂ©zilly, F., (2007), Icreased susceptibility to predation and alteredanti-predator behaviour in an acanthosephalan-infected amphipod, International Journal of Parasitology, 37, 645-651.
  61. [61]  Freedman, H.I., Addicott, J.F., and Rai, B. (1987), Obligate mutualism with a predator: persistence of three-species models, Theoretical Population Biology, 32, 157-175.
  62. [62]  Murray, D.L., Carry, J.R., and Keith, L.B. (1997), Interactive effects of sublethal nematodes and nutritional status on snowshoe hare vulnerability to predation?, J. Anim. Ecol., 66, 250-264.
  63. [63]  Perko, L. (2001), Differential Equations and Dynamical Systems, Springer-Verlag: New York.
  64. [64]  LaSalle, J.P. (1976), The Stability of Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, Penn, USA.
  65. [65]  Hudson, P.J., Dobson, A.P., and Newborn, D. (1992), Do parasites make prey vulnerable to predation? Red grouse and parasites, J. Anim. Ecol., 61, 681-692.
  66. [66]  Hudson, P.J., Newborn, D., and Dobson, A.P. (1992), Regulation and stability of a free-leaving host-parasite system, Trichostrongylus tenuis in red grouse. I. Monitoring and parasite reduction experiment, J. Anim. Ecol., 61, 477-486.
  67. [67]  Berezovskaya, F.S., Song, B., and Castillo-Chavez, C. (2010), Role of prey dispersal and refuges on predator-prey dynamics, SIAM J. Appl. Math., 70(6), 1821-1839.