Dynamics in two competing predators-one prey system with two types of Holling and fear effect

Adin Lazuardy Firdiansyah, Nurhidayati Nurhidayati


In this article, it is formulated a predator-prey model of two predators consuming a single limited prey resource. On the other hand, two predators have to compete with each other for survival. The predation function for two predators is assumed to be different where one predator uses Holling type I while the other uses Holling type II. It is also assumed that the fear effect is considered in this model as indirect influence evoked by both predators. Non-negativity and boundedness is written to show the biological justification of the model. Here, it is found that the model has five equilibrium points existed under certain condition. We also perform the local stability analysis on the equilibrium points with three equilibrium points are stable under certain conditions and two equilibrium points are unstable. Hopf bifurcation is obtained by choosing the consumption rate of the second predator as the bifurcation parameter. In the last part, several numerical solutions are given to support the analysis results.


Holling Functional Response; Hopf Bifurcation; Local Stability; Predator-Prey Model

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Y. Cai, C. Zhao, W. Wang, and J. Wang, “Dynamics of a Leslie-Gower predator-prey model with additive Allee effect,” Applied Mathematical Modelling, vol. 39, no. 7, pp. 2092–2106, 2015.

J. Wang, Y. Cai, S. Fu, and W. Wang, “The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29, no. 8, 2019.

U. Salamah, A. Suryanto, and W. M. Kusumawinahyu, “Leslie-Gower predator-prey model with stage-structure, Beddington-DeAngelis functional response, and anti-predator behavior,” AIP Conference Proceedings, vol. 2084, 2019.

H. S. Panigoro, E. Rahmi, N. Achmad, and S. L. Mahmud, “The influence of additive Allee effect and periodic harvesting to the dynamics of Leslie-Gower predator-prey model,” Jambura Journal of Mathematics, vol. 2, no. 2, pp. 87–96, 2020.

A. L. Firdiansyah, “Effect of prey refuge and harvesting on dynamics of eco-epidemiological model with Holling type III,” Jambura Journal of Mathematics, vol. 3, no. 1, pp. 16–25, 2021.

M. L. Rosenzweig and R. H. MacArthur, “Graphical representation and stability conditions of predator-prey interactions,” The American Naturalist, vol. 97, no. 895, pp. 209–223, 1963.

L. K. Beay, A. Suryanto, I. Darti, and Trisilowati, “Stability of a stage-structure Rosenzweig-MacArthur model incorporating Holling type-II functional response,” IOP Conference Series: Materials Science and Engineering, vol. 546, no. 5, pp. 0–7, 2019.

L. K. Beay and M. Saija, “A stage-structure Rosenzweig-MacArthur model with effect of prey refuge,” Jambura Journal of Biomathematics, vol. 1, no. 1, pp. 1–7, 2020.

I. Djakaria, M. B. Gaib, and Resmawan, “Analysis of the Rosenzweig-MacArthur predator-prey model with anti-predator behavior,” CAUCHY: Jurnal Matematika Murni dan Aplikasi, vol. 6, no. 4, pp. 260–269, 2021.

D. Savitri, A. Suryanto, W. M. Kusumawinahyu, and Abadi, “A dynamics behaviour of two predators and one prey interaction with competition between predators,” IOP Conference Series: Materials Science and Engineering, vol. 546, no. 5, 2019.

J. Alebraheem and Y. Abu-Hasan, “Persistence of predators in a two predators-one prey model with non-periodic solution,” Applied Mathematical Sciences, vol. 6, no. 17-20, pp. 943–956, 2012.

S. Sarwardi, P. K. Mandal, and S. Ray, “Dynamical behaviour of a two-predator model with prey refuge,” Journal of Biological Physics, vol. 39, no. 4, pp. 701–722, 2013.

B. Mukhopadhyay and R. Bhattacharyya, “Effects of harvesting and predator interference in a model of two-predators competing for a single prey,” Applied Mathematical Modelling, vol. 40, no. 4, pp. 3264–3274, 2016.

R. D. Holt, “Predation, apparent competition, and the structure of prey communities,” Theoretical Population Biology, vol. 12, no. 2, pp. 197–229, 1977.

R. R. Vance, “Predation and resource partitioning in one predator – two prey model communities,” The American Naturalist, vol. 112, no. 987, pp. 797–813, 1978.

B. P. Kotler and R. D. Holt, “Predation and competition: The interaction of two types of species interactions,” Oikos, vol. 54, no. 2, pp. 256–260, 1989.

J. Gurevitch, J. A. Morrison, and L. V. Hedges, “The interaction between competition and predation: A meta-analysis of field experiments,” The American Naturalist, vol. 155, no. 4, pp. 435–453, 2000.

S. L. Lima, “Nonlethal effects in the ecology of predator-prey interactions,” BioScience, vol. 48, no. 1, pp. 25–34, 1998.

S. Creel, D. Christianson, S. Liley, and J. A. Winnie, “Predation risk affects reproductive physiology and demography of elk,” Science, vol. 315, no. 5814, p. 960, 2007.

S. Creel and D. Christianson, “Relationships between direct predation and risk effects,” Trends in Ecology & Evolution, vol. 23, no. 4, pp. 194–201, 2008.

H. Zhang, Y. Cai, S. Fu, and W. Wang, “Impact of the fear effect in a prey-predator model incorporating a prey refuge,” Applied Mathematics and Computation, vol. 356, pp. 328–337, 2019.

D. Mukherjee, “Impact of predator fear on two competing prey species,” Jambura Journal of Biomathematics, vol. 2, no. 1, pp. 1–12, 2021.

D. Barman, J. Roy, and S. Alam, “Dynamical Behaviour of an Infected Predator-Prey Model with Fear Effect,” Iranian Journal of Science and Technology, Transaction A, vol. 45, no. 1, pp. 309–325, 2021.

DOI: https://doi.org/10.34312/jjbm.v2i2.11264

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