Fear and Hunting Cooperation's Impact on the Eco-Epidemiological Model's Dynamics

Main Article Content

Nabaa Hassain Fakhry, Raid Kamel Naji


Due to the fact that living organisms do not exist individually, but rather exist in clusters interacting with each other, which helps to spread epidemics among them. Therefore, the study of the prey-predator system in the presence of an infectious disease is an important topic because the disease affects the system's dynamics and its existence. The presence of the hunting cooperation characteristic and the induced fear in the prey community impairs the growth rate of the prey and therefore affects the presence of the predator as well. Therefore, this research is interested in studying an eco-epidemiological system that includes the above factors. Therefore, an eco-epidemiological prey-predator model incorporating predation fear and cooperative hunting is built and examined. It is considered that the disease in the predator is of the SIS kind, which means that the infected predator can recover and become susceptible through medical treatment. All possible equilibrium points have been found. The solution's positivity and boundedness are examined. Local and global stability analyses are performed. The uniform persistence conditions are established. The local bifurcation around the equilibrium points is studied. Finally, numerical simulation is performed to validate the obtained results and comprehend the parameter impact on system dynamics.

Article Details


  1. M. Teixeira Alves, F.M. Hilker, Hunting Cooperation and Allee Effects in Predators, J. Theor. Biol. 419 (2017), 13–22. https://doi.org/10.1016/j.jtbi.2017.02.002.
  2. J.D. Murray, Mathematical Biology, Springer – Verlag, New-York, 1989.
  3. A.D. Bazykin, A.I. Khibnik, B. Krauskopf, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998. https://doi.org/10.1142/2284.
  4. R.M. May, Stability and Complexity in Model Ecosystems, (2nd edition), Princeton University Press, Princeton, 2001.
  5. D.L. Mech, The Wolf: The Ecology and Behavior of an Endangered Species, The Natural History Press, Garden City, New York, 1970.
  6. D.P. Hector, Cooperative Hunting and its Relationship to Foraging Success and Prey Size in an Avian Predator, Ethology. 73 (1986), 247–257. https://doi.org/10.1111/j.1439-0310.1986.tb00915.x.
  7. P.E. Stander, Cooperative Hunting in Lions: The Role of the Individual, Behav. Ecol. Sociobiol. 29 (1992), 445–454. https://doi.org/10.1007/BF00170175.
  8. C. Boesch, Cooperative Hunting in Wild Chimpanzees, Animal Behav. 48 (1994), 653–667. https://doi.org/10.1006/anbe.1994.1285.
  9. S. Creel, N.M. Creel, Communal Hunting and Pack Size in African Wild Dogs, Lycaon Pictus, Animal Behav. 50 (1995), 1325–1339. https://doi.org/10.1016/0003-3472(95)80048-4.
  10. P.M. Kappeler, C.P. Van Schaik, Cooperation in Primates and Humans: Mechanisms and Evolution, Springer, Berlin, 2006.
  11. I. Bailey, J.P. Myatt, A.M. Wilson, Group Hunting Within the Carnivora: Physiological, Cognitive and Environmental Influences on Strategy and Cooperation, Behav. Ecol. Sociobiol. 67 (2012), 1–17. https://doi.org/10.1007/s00265-012-1423-3.
  12. A.A. Berryman, A.P. Gutierrez, R. Arditi, Credible, Parsimonious and Useful Predator‐Prey Models: A Reply to Abrams, Gleeson, and Sarnelle, Ecology. 76 (1995), 1980–1985. https://doi.org/10.2307/1940728.
  13. R.K. Naji, R.K. Upadhyay, V. Rai, Dynamical Consequences of Predator Interference in a Tri-Trophic Model Food Chain, Nonlinear Anal.: Real World Appl. 11 (2010), 809–818. https://doi.org/10.1016/j.nonrwa.2009.01.026.
  14. R.P. Gupta, P. Chandra, Bifurcation Analysis of Modified Leslie–Gower Predator–Prey Model With Michaelis–Menten Type Prey Harvesting, J. Math. Anal. Appl. 398 (2013), 278–295. https://doi.org/10.1016/j.jmaa.2012.08.057.
  15. H.A. Satar, R.K. Naji, Stability and Bifurcation of a Prey-Predator-Scavenger Model in the Existence of Toxicant and Harvesting, Int. J. Math. Math. Sci. 2019 (2019), 1573516. https://doi.org/10.1155/2019/1573516.
  16. X. Wang, L. Zanette, X. Zou, Modelling the Fear Effect in Predator–Prey Interactions, J. Math. Biol. 73 (2016), 1179-1204. https://doi.org/10.1007/s00285-016-0989-1.
  17. S. Pal, N. Pal, S. Samanta, J. Chattopadhyay, Fear Effect in Prey and Hunting Cooperation Among Predators in a Leslie-Gower Model, Math. Biosci. Eng. 16 (2019), 5146-5179. https://doi.org/10.3934/mbe.2019258.
  18. L. Přibylová, A. Peniašková, Foraging Facilitation Among Predators and Its Impact on the Stability of Predator–Prey Dynamics, Ecol. Complex. 29 (2017), 30-39. https://doi.org/10.1016/j.ecocom.2016.11.006.
  19. S. Pal, N. Pal, J. Chattopadhyay, Hunting Cooperation in a Discrete-Time Predator–Prey System, Int. J. Bifurcation Chaos. 28 (2018), 1850083. https://doi.org/10.1142/s0218127418500839.
  20. S. Pal, N. Pal, S. Samanta, J. Chattopadhyay, Effect of Hunting Cooperation and Fear in a Predator-Prey Model, Ecol. Complex. 39 (2019), 100770. https://doi.org/10.1016/j.ecocom.2019.100770.
  21. S. Pal, N. Pal, S. Samanta, J. Chattopadhyay, Fear Effect in Prey and Hunting Cooperation Among Predators in a Leslie-Gower Model, Math. Biosci. Eng. 16 (2019), 5146–5179. https://doi.org/10.3934/mbe.2019258.
  22. J. Zhang, W. Zhang, Dynamics of a Predator–Prey Model with Hunting Cooperation and Allee Effects in Predators, Int. J. Bifurcation Chaos. 30 (2020) 2050199. https://doi.org/10.1142/s0218127420501990.
  23. J. Liu, B. Liu, P. Lv, T. Zhang, An Eco-Epidemiological Model With Fear Effect and Hunting Cooperation, Chaos Solitons Fractals. 142 (2021) 110494. https://doi.org/10.1016/j.chaos.2020.110494.
  24. N.H. Fakhry, R.K. Naji, The Dynamic of an Eco-Epidemiological Model Involving Fear and Hunting Cooperation, Commun. Math. Biol. Neurosci. 2023 (2023), 63. https://doi.org/10.28919/cmbn/7998.
  25. K.Q. Al-Jubouri, R.K. Naji, Delay in Eco-Epidemiological Prey-Predator Model With Predation Fear and Hunting Cooperation, Commun. Math. Biol. Neurosci. 2023 (2023), 89. https://doi.org/10.28919/cmbn/8081.
  26. X. Wang, X. Zou, Modeling the Fear Effect in Predator-Prey Interactions With Adaptive Avoidance of Predators, Bull. Math. Biol. 79 (2017), 1325–1359. https://doi.org/10.1007/s11538-017-0287-0.
  27. P. Panday, N. Pal, S. Samanta, J. Chattopadhyay, Stability and Bifurcation Analysis of a Three-Species Food Chain Model with Fear, Int. J. Bifurcation Chaos. 28 (2018), 1850009. https://doi.org/10.1142/s0218127418500098.
  28. S.K. Sasmal, Population Dynamics With Multiple Allee Effects Induced by Fear Factors – A Mathematical Study on Prey-Predator Interactions, Appl. Math. Model. 64 (2018), 1–14. https://doi.org/10.1016/j.apm.2018.07.021.
  29. S. Pal, S. Majhi, S. Mandal, N. Pal, Role of Fear in a Predator–Prey Model with Beddington–DeAngelis Functional Response, Z. Naturforsch., A. 74 (2019), 581-595. https://doi.org/10.1515/zna-2018-0449.
  30. A. Sha, S. Samanta, M. Martcheva, J. Chattopadhyay, Backward Bifurcation, Oscillations and Chaos in an Eco-Epidemiological Model With Fear Effect, J. Biol. Dyn. 13 (2019), 301–327. https://doi.org/10.1080/17513758.2019.1593525.
  31. X. Wang, Y. Tan, Y. Cai, W. Wang, Impact of the Fear Effect on the Stability and Bifurcation of a Leslie–Gower Predator–Prey Model, Int. J. Bifurcation Chaos. 30 (2020), 2050210. https://doi.org/10.1142/s0218127420502107.
  32. N.H. Fakhry, and R.K. Naji, The Dynamics of a Square Root Prey-Predator Model with Fear, Iraqi J. Sci. 61 (2020), 139–146. https://doi.org/10.24996/ijs.2020.61.1.15.
  33. K. Sarkar, S. Khajanchi, Impact of Fear Effect on the Growth of Prey in a Predator-Prey Interaction Model, Ecol. Complex. 42 (2020), 100826. https://doi.org/10.1016/j.ecocom.2020.100826.
  34. S.K. Sasmal, Y. Takeuchi, Dynamics of a Predator-Prey System with Fear and Group Defense, J. Math. Anal. Appl. 481 (2020), 123471. https://doi.org/10.1016/j.jmaa.2019.123471.
  35. F.H. Maghool, R.K. Naji, The Dynamics of a Tritrophic Leslie-Gower Food-Web System with the Effect of Fear, J. Appl. Math. 2021 (2021), 2112814. https://doi.org/10.1155/2021/2112814.
  36. S. Vinoth, R. Sivasamy, K. Sathiyanathan, B. Unyong, G. Rajchakit, R. Vadivel, N. Gunasekaran, The Dynamics of a Leslie Type Predator–Prey Model with Fear and Allee Effect, Adv. Differ. Equ. 2021 (2021), 338. https://doi.org/10.1186/s13662-021-03490-x.
  37. N. Santra, S. Mondal, G. Samanta, Complex Dynamics of a Predator–Prey Interaction with Fear Effect in Deterministic and Fluctuating Environments, Mathematics. 10 (2022), 3795. https://doi.org/10.3390/math10203795.
  38. F.H. Maghool, R.K. Naji. Chaos in the three-species Sokol-Howell food chain system with fear, Commun. Math. Biol. Neurosci. 2022 (2022), 14. https://doi.org/10.28919/cmbn/7056.
  39. S.M.A. Al-Momen, R.K. Naji, The Dynamics of Modified Leslie-Gower Predator-Prey Model Under the Influence of Nonlinear Harvesting and Fear Effect, Iraqi J. Sci. 63 (2022), 259-282. https://doi.org/10.24996/ijs.2022.63.1.27.
  40. A.R.M. Jamil, R.K. Naji, Modeling and Analysis of the Influence of Fear on the Harvested Modified Leslie–Gower Model Involving Nonlinear Prey Refuge, Mathematics. 10 (2022) 2857. https://doi.org/10.3390/math10162857.
  41. A.R.M. Jamil, R.K. Naji, Modeling and Analyzing the Influence of Fear on the Harvested Modified Leslie-Gower Model, Baghdad Sci. J. 20 (2023), 1701-1712. https://doi.org/10.21123/bsj.2023.7432.
  42. S. Creel, D. Christianson, S. Liley, J.A. Winnie Jr., Predation Risk Affects Reproductive Physiology and Demography of Elk, Science. 315 (2007), 960-960. https://doi.org/10.1126/science.1135918.
  43. S. Creel, P. Schuette, D. Christianson, Effects of Predation Risk on Group Size, Vigilance, and Foraging Behavior in an African Ungulate Community, Behav. Ecol. 25 (2014), 773–784. https://doi.org/10.1093/beheco/aru050.
  44. R.M. Anderson, R.M. May Population biology of infectious diseases: Part I, Nature, 280 (1979), 361–367. https://doi.org/10.1038/280361a0.
  45. D. Mukherjee, Hopf Bifurcation in an Eco-Epidemic Model, Appl. Math. Comput. 217 (2010), 2118–2124. https://doi.org/10.1016/j.amc.2010.07.010.
  46. K. Agnihotri. and N. Juneja, An Eco-Epidemic Model with Disease in Both Prey and Predator, Int. J. Adv. Electric. Electron. Eng. 4 (2015), 50-54.
  47. N. Juneja, K. Agnihotri, Conservation of a Predator Species in SIS Prey-Predator System Using Optimal Taxation Policy, Chaos Solitons Fractals. 116 (2018), 86–94. https://doi.org/10.1016/j.chaos.2018.09.024.
  48. X.Y. Meng, N.N. Qin, H.F. Huo, Dynamics Analysis of a Predator–Prey System with Harvesting Prey and Disease in Prey Species, J. Biol. Dyn. 12 (2018), 342–374. https://doi.org/10.1080/17513758.2018.1454515.
  49. R.K. Naji, A.N. Mustafa, The Dynamics of an Eco-Epidemiological Model with Nonlinear Incidence Rate, J. Appl. Math. 2012 (2012), 852631. https://doi.org/10.1155/2012/852631.
  50. A.S. Abdulghafour, R.K. Naji, A Study of a Diseased Prey-Predator Model with Refuge in Prey and Harvesting from Predator, J. Appl. Math. 2018 (2018), 2952791. https://doi.org/10.1155/2018/2952791.
  51. S. Hariprasad, M.A.S. Srinivas, N. Phani Kumar, K. Praveenkumar, The Dynamics of an Eco-Epidemiological Model of three species with Holling type-II and Type-IV functional responses, J. Phys.: Conf. Ser. 1850 (2021), 012003. https://doi.org/10.1088/1742-6596/1850/1/012003.
  52. H.A. Satar, H.A. Ibrahim, D.K. Bahlool, On the Dynamics of an Eco-Epidemiological System Incorporating a Vertically Transmitted Infectious Disease, Iraqi J. Sci. 62 (2021), 1642–1658. https://doi.org/10.24996/ijs.2021.62.5.27.
  53. N. SK, S. Pal, P. Majumdar, B. Mondal, Dynamics of an Eco-Epidemiological System: Predators Get Infected in Two Paths, J. Comput. Sci. 69 (2023), 102023. https://doi.org/10.1016/j.jocs.2023.102023.
  54. F. Chen, On a Nonlinear Nonautonomous Predator–Prey Model with Diffusion and Distributed Delay, J. Comput. Appl. Math. 180 (2005), 33–49. https://doi.org/10.1016/j.cam.2004.10.001.
  55. H.I. Freedman, P. Waltman, Persistence in Models of Three Interacting Predator-Prey Populations, Math. Biosci. 68 (1984), 213–231. https://doi.org/10.1016/0025-5564(84)90032-4.
  56. L. Perko, Differential Equations and Dynamical Systems, 3rd edition, Springer, New York, 2001.