DYNAMICS OF A PREY-PREDATOR MODEL WITH ALLEE EFFECTS AND HOLLING TYPE IV FUNCTIONAL RESPONSE: LOCAL STABILITY AND NUMERICAL EXPLORATION OF BIFURCATIONS

  • Resmawan Resmawan Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Brawijaya, Indonesia https://orcid.org/0000-0001-7921-2804
  • Agus Suryanto Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Brawijaya, Indonesia https://orcid.org/0000-0002-1335-5631
  • Isnani Darti Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Brawijaya, Indonesia https://orcid.org/0000-0002-4163-8030
  • Hasan S. Panigoro Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Gorontalo, Indonesia https://orcid.org/0000-0002-0118-2745
Keywords: Allee Effect, Forward bifurcation, Hopf Bifurcation, Prey group defence, Prey-predator model

Abstract

This study presents a prey-predator model incorporating the Allee effect and Holling Type IV Functional Response. The model identifies three equilibrium points: the zero-equilibrium, the predator extinction equilibrium, and the positive equilibrium. Under specific conditions, all these points exhibit local asymptotic stability. The Allee effect is an important factor in determining the stability of the equilibrium point. A weak Allee effect can destabilize the zero-equilibrium point, while a strong Allee effect ensures its local asymptotic stability, potentially leading to the extinction of both species. Additionally, forward and Hopf bifurcation under weak Allee conditions occur at the predator extinction equilibrium point. In contrast, a strong Allee effect may cause bistability between the zero-equilibrium and predator extinction equilibrium points. This evidence suggests that prey can survive without predators; however, a strong Allee effect might result in prey extinction if the population decreases significantly. The Holling Type IV functional response illustrates the impact of prey group defense, which diminishes predation pressure as prey density increases, thereby facilitating the development of limit cycles and establishing a positive equilibrium under specific parameter conditions. This mechanism is crucial for managing predator-prey cohabitation and influencing the system's bifurcation structure. The final section of the study includes numerical simulations to support the analytical findings. The interplay between the Allee effect and the Holling Type IV functional response yields complex dynamics, encompassing bistability, oscillation behavior, and sensitivity to initial conditions. Their collaborative interaction amplifies the system's nonlinearity, enabling the creation of various dynamic behaviors that are extremely sensitive to fluctuations in parameter values.

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References

D. Leclère et al., “BENDING THE CURVE OF TERRESTRIAL BIODIVERSITY NEEDS AN INTEGRATED STRATEGY,” Nature, vol. 585, no. 7826, pp. 551–556, Sep. 2020, doi: https://doi.org/10.1038/s41586-020-2705-y.

L. Berec, E. Angulo, and F. Courchamp, “MULTIPLE ALLEE EFFECTS AND POPULATION MANAGEMENT,” Trends Ecol Evol, vol. 22, no. 4, pp. 185–191, Apr. 2007, doi: https://doi.org/10.1016/j.tree.2006.12.002.

G. Beauchamp, SOCIAL PREDATION: HOW GROUP LIVING BENEFITS PREDATORS AND PREY. New York: Elsevier, 2014. doi: https://doi.org/10.1016/C2012-0-03532-6.

T. R. Malthus, An Essay on the Principle of Population. London: Reeves and Turner, 1872.

A. J. Lotka, Elements of Physical Biology. Baltimore: Williams and Wikkins, 1925.

V. Volterra, VARIAZIONI E FLUTTUAZIONI DEL NUMERO D’INDIVIDUI IN SPECIE ANIMALI CONVIVENTI. Societá anonima tipografica" Leonardo da Vinci", 1927.

P. H. Leslie and J. C. Gower, “THE PROPERTIES OF A STOCHASTIC MODEL FOR THE PREDATOR-PREY TYPE OF INTERACTION BETWEEN TWO SPECIES,” Biometrika, vol. 47, no. 3/4, p. 219, 1960, doi: https://doi.org/10.2307/2333294.

M. L. Rosenzweig and R. H. MacArthur, “GRAPHICAL REPRESENTATION AND STABILITY CONDITIONS OF PREDATOR-PREY INTERACTIONS,” Am Nat, vol. 97, no. 895, pp. 209–223, 1963, doi: https://doi.org/10.1086/282272.

L. K. Beay, A. Suryanto, I. Darti, and Trisilowati, “HOPF BIFURCATION AND STABILITY ANALYSIS OF THE ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL WITH STAGE-STRUCTURE IN PREY,” Mathematical Biosciences and Engineering, vol. 17, no. 4, pp. 4080–4097, 2020, doi: https://doi.org/10.3934/mbe.2020226.

S. Zhang, S. Yuan, and T. Zhang, “A PREDATOR-PREY MODEL WITH DIFFERENT RESPONSE FUNCTIONS TO JUVENILE AND ADULT PREY IN DETERMINISTIC AND STOCHASTIC ENVIRONMENTS,” Appl Math Comput, vol. 413, p. 126598, 2022, doi: https://doi.org/10.1016/j.amc.2021.126598.

S. Sirisubtawee, N. Khansai, and A. Charoenloedmongkhon, “INVESTIGATION ON DYNAMICS OF AN IMPULSIVE PREDATOR–PREY SYSTEM WITH GENERALIZED HOLLING TYPE IV FUNCTIONAL RESPONSE AND ANTI-PREDATOR BEHAVIOR,” Adv Differ Equ, vol. 2021, no. 1, p. 160, Mar. 2021, doi: https://doi.org/10.1186/s13662-021-03324-w.

A. S. Purnomo, I. Darti, A. Suryanto, and W. M. Kusumawinahyu, “FEAR EFFECT ON A MODIFIED LESLIE-GOWER PREDATOR-PREY MODEL WITH DISEASE TRANSMISSION IN PREY POPULATION.,” Engineering Letters, vol. 31, no. 2, pp. 764–773, 2023.

Y. Long, L. Wang, and J. Li, “UNIFORM PERSISTENCE AND MULTISTABILITY IN A TWO-PREDATOR–ONE-PREY SYSTEM WITH INTER-SPECIFIC AND INTRA-SPECIFIC COMPETITION,” J Appl Math Comput, vol. 68, no. 2, pp. 767–794, 2022, doi https://doi.org/10.1007/s12190-021-01551-8.

H. S. Panigoro, E. Rahmi, and R. Resmawan, “BIFURCATION ANALYSIS OF A PREDATOR–PREY MODEL INVOLVING AGE STRUCTURE, INTRASPECIFIC COMPETITION, MICHAELIS–MENTEN TYPE HARVESTING, AND MEMORY EFFECT,” Front Appl Math Stat, vol. 8, 2023, doi: https://doi.org/10.3389/fams.2022.1077831.

E. Rahmi, I. Darti, A. Suryanto, Trisilowati, and H. S. Panigoro, “STABILITY ANALYSIS OF A FRACTIONAL-ORDER LESLIE-GOWER MODEL WITH ALLEE EFFECT IN PREDATOR,” J Phys Conf Ser, vol. 1821, no. 1, p. 12051, 2021, doi: https://doi.org/10.1088/1742-6596/1821/1/012051.

H. S. Panigoro and E. Rahmi, “COMPUTATIONAL DYNAMICS OF A LOTKA-VOLTERRA MODEL WITH ADDITIVE ALLEE EFFECT BASED ON ATANGANA-BALEANU FRACTIONAL DERIVATIVE,” Jambura Journal of Biomathematics (JJBM), vol. 2, no. 2, pp. 96–103, 2021, doi: https://doi.org/10.34312/jjbm.v2i2.11886.

H. S. Panigoro and E. Rahmi, “IMPACT OF FEAR AND STRONG ALLEE EFFECTS ON THE DYNAMICS OF A FRACTIONAL-ORDER ROSENZWEIG-MACARTHUR MODEL,” Nonlinear Dynamics and Applications, pp. 611–619, 2022, doi: https://doi.org/10.1007/978-3-030-99792-2_50.

Y. Ye, H. Liu, Y. Wei, K. Zhang, M. Ma, and J. Ye, “DYNAMIC STUDY OF A PREDATOR-PREY MODEL WITH ALLEE EFFECT AND HOLLING TYPE-I FUNCTIONAL RESPONSE,” Adv Differ Equ, vol. 2019, no. 1, p. 369, Dec. 2019, doi: https://doi.org/10.1186/s13662-019-2311-1.

K. Baisad and S. Moonchai, “ANALYSIS OF STABILITY AND HOPF BIFURCATION IN A FRACTIONAL GAUSS-TYPE PREDATOR–PREY MODEL WITH ALLEE EFFECT AND HOLLING TYPE-III FUNCTIONAL RESPONSE,” Adv Differ Equ, vol. 2018, no. 1, p. 82, Dec. 2018, doi: https://doi.org/10.1186/s13662-018-1535-9.

E. Rahmi, I. Darti, A. Suryanto, and Trisilowati, “A MODIFIED LESLIE–GOWER MODEL INCORPORATING BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE, DOUBLE ALLEE EFFECT AND MEMORY EFFECT,” Fractal and Fractional, vol. 5, no. 3, p. 84, 2021, doi: https://doi.org/10.3390/fractalfract5030084.

H. S. Panigoro, E. Rahmi, A. Suryanto, and I. Darti, “A FRACTIONAL ORDER PREDATOR–PREY MODEL WITH STRONG ALLEE EFFECT AND MICHAELIS–MENTEN TYPE OF PREDATOR HARVESTING,” in The 8th Symposium on Biomathematics (SYMOMATH) 2021: Bridging Mathematics and Covid-19 Through Multidisciplinary Collaboration, AIP Conference Proceedings , 2022, p. 20018. doi: https://doi.org/10.1063/5.0082684.

E. Rahmi, I. Darti, A. Suryanto, and Trisilowati, “GLOBAL DYNAMICS OF A FRACTIONAL-ORDER LESLIE-GOWER MODEL WITH ALLEE EFFECT,” in The 8th Symposium on Biomathematics (SYMOMATH) 2021: Bridging Mathematics and Covid-19 Through Multidisciplinary Collaboration, AIP Conference Proceedings, 2022, p. 20014. doi: https://doi.org/10.1063/5.0082943.

N. Anggriani, H. S. Panigoro, E. Rahmi, O. J. Peter, and S. A. Jose, “A PREDATOR–PREY MODEL WITH ADDITIVE ALLEE EFFECT AND INTRASPECIFIC COMPETITION ON PREDATOR INVOLVING ATANGANA–BALEANU–CAPUTO DERIVATIVE,” Results Phys, vol. 49, p. 106489, 2023, doi: https://doi.org/10.1016/j.rinp.2023.106489.

E. Rahmi, I. Darti, A. Suryanto, and T. Trisilowati, “A FRACTIONAL-ORDER ECO-EPIDEMIOLOGICAL LESLIE–GOWER MODEL WITH DOUBLE ALLEE EFFECT AND DISEASE IN PREDATOR,” International Journal of Differential Equations, vol. 2023, pp. 1–24, 2023, doi: https://doi.org/10.1155/2023/5030729.

A. T. R. Sidik, H. S. Panigoro, R. Resmawan, and E. Rahmi, “THE EXISTENCE OF NEIMARK-SACKER BIFURCATION ON A DISCRETE-TIME SIS-EPIDEMIC MODEL INCORPORATING LOGISTIC GROWTH AND ALLEE EFFECT,” Jambura Journal of Biomathematics (JJBM), vol. 3, no. 2, pp. 58–62, 2022, doi: https://doi.org/10.34312/jjbm.v3i2.17515.

N. Anggriani, H. S. Panigoro, E. Rahmi, O. J. Peter, and S. A. Jose, “A PREDATOR–PREY MODEL WITH ADDITIVE ALLEE EFFECT AND INTRASPECIFIC COMPETITION ON PREDATOR INVOLVING ATANGANA–BALEANU–CAPUTO DERIVATIVE,” Results Phys, vol. 49, p. 106489, 2023, doi: https://doi.org/10.1016/j.rinp.2023.106489.

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.doi: https://doi.org/10.34312/jjom.v2i2.4566

D. Bai and X. Zhang, “DYNAMICS OF A PREDATOR–PREY MODEL WITH THE ADDITIVE PREDATION IN PREY,” Mathematics, vol. 10, no. 4, 2022, doi: https://doi.org/10.3390/math10040655.

C. C. García and J. V. Cuenca, “ADDITIVE ALLEE EFFECT ON PREY IN THE DYNAMICS OF A GAUSE PREDATOR–PREY MODEL WITH CONSTANT OR PROPORTIONAL REFUGE ON PREY AT LOW OR HIGH DENSITIES,” Commun Nonlinear Sci Numer Simul, vol. 126, p. 107427, 2023, doi: https://doi.org/10.1016/j.cnsns.2023.107427.

B. Xie and Z. Zhang, “IMPACT OF ALLEE AND FEAR EFFECTS IN A FRACTIONAL ORDER PREY–PREDATOR SYSTEM INCORPORATING PREY REFUGE,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 33, no. 1, p. 13131, Feb. 2023, doi: https://doi.org/10.1063/5.0130809.

Y. Ye and Y. Zhao, “BIFURCATION ANALYSIS OF A DELAY-INDUCED PREDATOR–PREY MODEL WITH ALLEE EFFECT AND PREY GROUP DEFENSE,” International Journal of Bifurcation and Chaos, vol. 31, no. 10, p. 2150158, 2021, doi: https://doi.org/10.1142/S0218127421501583.

C. Zhang, R. Wu, and M. Chen, “HOPF BIFURCATION IN A DELAYED PREDATOR-PREY SYSTEM WITH GENERAL GROUP DEFENCE FOR PREY,” Journal of Applied Analysis & Computation, vol. 11, no. 2, pp. 810–840, 2021, doi: https://doi.org/10.11948/20200011.

X. Jiao, X. Li, and Y. Yang, “DYNAMICS AND BIFURCATIONS OF A FILIPPOV LESLIE-GOWER PREDATOR-PREY MODEL WITH GROUP DEFENSE AND TIME DELAY,” Chaos Solitons Fractals, vol. 162, p. 112436, 2022, doi: https://doi.org/10.1016/j.chaos.2022.112436.

R. R. Patra, S. Kundu, and S. Maitra, “EFFECT OF DELAY AND CONTROL ON A PREDATOR-PREY ECOSYSTEM WITH GENERALIST PREDATOR AND GROUP DEFENCE IN THE PREY SPECIES,” The European Physical Journal Plus, vol. 137, no. 1, p. 28, 2022, doi: https://doi.org/10.1140/epjp/s13360-021-02225-x.

S. Wiggins, INTRODUCTION TO APPLIED NONLINEAR DYNAMICAL SYSTEMS AND CHAOS, vol. 2. in Texts in Applied Mathematics, vol. 2. New York: Springer-Verlag, 2003. doi: https://doi.org/10.1007/b97481.

Published
2025-09-01
How to Cite
[1]
R. Resmawan, A. Suryanto, I. Darti, and H. S. Panigoro, “DYNAMICS OF A PREY-PREDATOR MODEL WITH ALLEE EFFECTS AND HOLLING TYPE IV FUNCTIONAL RESPONSE: LOCAL STABILITY AND NUMERICAL EXPLORATION OF BIFURCATIONS”, BAREKENG: J. Math. & App., vol. 19, no. 4, pp. 2891-2906, Sep. 2025.