Control and bifurcation analysis of a discrete predator-prey model with Holling type-Ⅳ and Leslie-Gower dynamics

Zhenxin Zhong; Wenlong Wang; Zhenxin Zhong; Wenlong Wang

doi:10.3934/era.2026350

Electronic Research Archive

2026, Volume 34, Issue 11: 8232-8277. doi: 10.3934/era.2026350

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Research article

Control and bifurcation analysis of a discrete predator-prey model with Holling type-Ⅳ and Leslie-Gower dynamics

Zhenxin Zhong ,
Wenlong Wang ^,

College of Science, Northeast Forestry University, Harbin 150040, China

Received: 21 July 2026 Revised: 20 August 2026 Accepted: 26 August 2026 Published: 22 September 2026

This paper investigates the dynamical behavior of a Euler-generated discrete predator-prey map incorporating a Holling type-Ⅳ functional response and Leslie-Gower predator dynamics. Since the Leslie-Gower term is singular at zero prey density, the biologically feasible state space is restricted to $ x > 0 $ and $ y\geq0 $. Within this domain, the existence and local stability of the boundary and positive equilibria are analyzed, and sufficient conditions for the possible configurations of positive equilibria are derived from the discriminant of the associated cubic equation. The positivity of the uncontrolled and controlled maps is also examined through analytical one-step conditions and direct numerical feasibility checks. By employing center manifold reduction and normal form theory, analytical conditions for Flip and Neimark-Sacker bifurcations at the positive equilibrium are obtained. Two stabilization approaches are then investigated. The state-feedback scheme yields an explicit triangular stability region in the gain-parameter plane, whereas the weighted hybrid scheme is equivalent to replacing the Euler step size $ h $ by an effective step size $ h_{\mathrm{eff}} = \omega h $. Numerical experiments, including convergence tests, bifurcation diagrams, maximum Lyapunov exponent (MLE) spectra, time series, and phase portraits, support the theoretical analysis and illustrate the transition among stable, periodic, quasi-periodic, and chaotic dynamics of the Euler-generated map.
- discrete predator-prey model,
- Holling type-Ⅳ,
- Leslie-Gower,
- bifurcation analysis,
- chaos control
Citation: Zhenxin Zhong, Wenlong Wang. Control and bifurcation analysis of a discrete predator-prey model with Holling type-Ⅳ and Leslie-Gower dynamics[J]. Electronic Research Archive, 2026, 34(11): 8232-8277. doi: 10.3934/era.2026350

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Abstract

This paper investigates the dynamical behavior of a Euler-generated discrete predator-prey map incorporating a Holling type-Ⅳ functional response and Leslie-Gower predator dynamics. Since the Leslie-Gower term is singular at zero prey density, the biologically feasible state space is restricted to $ x > 0 $ and $ y\geq0 $. Within this domain, the existence and local stability of the boundary and positive equilibria are analyzed, and sufficient conditions for the possible configurations of positive equilibria are derived from the discriminant of the associated cubic equation. The positivity of the uncontrolled and controlled maps is also examined through analytical one-step conditions and direct numerical feasibility checks. By employing center manifold reduction and normal form theory, analytical conditions for Flip and Neimark-Sacker bifurcations at the positive equilibrium are obtained. Two stabilization approaches are then investigated. The state-feedback scheme yields an explicit triangular stability region in the gain-parameter plane, whereas the weighted hybrid scheme is equivalent to replacing the Euler step size $ h $ by an effective step size $ h_{\mathrm{eff}} = \omega h $. Numerical experiments, including convergence tests, bifurcation diagrams, maximum Lyapunov exponent (MLE) spectra, time series, and phase portraits, support the theoretical analysis and illustrate the transition among stable, periodic, quasi-periodic, and chaotic dynamics of the Euler-generated map.

References

[1]	Food and Agriculture Organization of the United Nations, Scientific Review of the Impact of Climate Change on Plant Pests, FAO on behalf of the IPPC Secretariat, Rome, Italy, 2021.
[2]	S. Savary, L. Willocquet, S. J. Pethybridge, P. Esker, N. McRoberts, A. Nelson, The global burden of pathogens and pests on major food crops, Nat. Ecol. Evol., 3 (2019), 430–439. https://doi.org/10.1038/s41559-018-0793-y doi: 10.1038/s41559-018-0793-y
[3]	T. Van Leeuwen, J. Vontas, A. Tsagkarakou, W. Dermauw, L. Tirry, Acaricide resistance mechanisms in the two-spotted spider mite Tetranychus urticae and other important Acari: A review, Insect Biochem. Mol. Biol., 40 (2010), 563–572. https://doi.org/10.1016/j.ibmb.2010.05.008 doi: 10.1016/j.ibmb.2010.05.008
[4]	C. Vásquez, Y. C. Colmenárez, N. Greco, M. Ramos, Current status of phytoseiid mites as biological control agents in Latin America and experiences from Argentina using Neoseiulus californicus, Neotrop. Entomol., 52 (2023), 240–250. https://doi.org/10.1007/s13744-023-01026-4 doi: 10.1007/s13744-023-01026-4
[5]	Y. Yang, K. Zhang, Z. Q. Zhang, Predatory mites Amblydromalus limonicus and Amblyseius herbicolus (Acari: Phytoseiidae) as potential biocontrol agents of Eotetranychus sexmaculatus (Acari: Tetranychidae) in avocado: Examining predation on different prey life stages, J. Econ. Entomol., 118 (2025), 1335–1343. https://doi.org/10.1093/jee/toaf036 doi: 10.1093/jee/toaf036
[6]	J. C. van Lenteren, K. Bolckmans, J. Köhl, W. J. Ravensberg, A. Urbaneja, Biological control using invertebrates and microorganisms: Plenty of new opportunities, BioControl, 63 (2018), 39–59. https://doi.org/10.1007/s10526-017-9801-4 doi: 10.1007/s10526-017-9801-4
[7]	W. Qin, Z. Dong, L. Huang, Impulsive effects and complexity dynamics in the anti-predator model with IPM strategies, Mathematics, 12 (2024), 1043. https://doi.org/10.3390/math12071043 doi: 10.3390/math12071043
[8]	C. Li, D. Cheng, X. Feng, M. Liu, Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response, AIMS Math., 9 (2024), 14454–14472. https://doi.org/10.3934/math.2024702 doi: 10.3934/math.2024702
[9]	X. Dai, J. Jiao, Q. Quan, A. Zhou, Dynamics of a predator-prey system with sublethal effects of pesticides on pests and natural enemies, Int. J. Biomath., 17 (2024), 2350007. https://doi.org/10.1142/S1793524523500079 doi: 10.1142/S1793524523500079
[10]	C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385–398. https://doi.org/10.4039/Ent91385-7 doi: 10.4039/Ent91385-7
[11]	C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 5–60. https://doi.org/10.4039/entm9745fv doi: 10.4039/entm9745fv
[12]	J. H. P. Dawes, M. O. Souza, A derivation of Holling's type Ⅰ, Ⅱ and Ⅲ functional responses in predator-prey systems, J. Theor. Biol., 327 (2013), 11–22. https://doi.org/10.1016/j.jtbi.2013.02.017 doi: 10.1016/j.jtbi.2013.02.017
[13]	C. Arancibia-Ibarra, J. D. Flores, G. J. Pettet, P. van Heijster, A Holling-Tanner predator-prey model with strong Allee effect, Int. J. Bifurcation Chaos, 29 (2019), 1930032. https://doi.org/10.1142/S0218127419300325 doi: 10.1142/S0218127419300325
[14]	M. A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population model, Chaos Solitons Fractals, 14 (2002), 1275–1293. https://doi.org/10.1016/S0960-0779(02)00079-6 doi: 10.1016/S0960-0779(02)00079-6
[15]	R. M. May, G. F. Oster, Bifurcations and dynamic complexity in simple ecological models, Am. Naturalist, 110 (1976), 573–599. https://doi.org/10.1086/283092 doi: 10.1086/283092
[16]	M. A. Aziz-Alaoui, M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069–1075. https://doi.org/10.1016/S0893-9659(03)90096-6 doi: 10.1016/S0893-9659(03)90096-6
[17]	A. Oaten, W. W. Murdoch, Functional response and stability in predator-prey systems, Am. Nat., 109 (1975), 289–298. https://doi.org/10.1086/282998 doi: 10.1086/282998
[18]	T. W. Hwang, Uniqueness of the limit cycle for Gause-type predator-prey systems, J. Math. Anal. Appl., 238 (1999), 179–195. https://doi.org/10.1006/jmaa.1999.6520 doi: 10.1006/jmaa.1999.6520
[19]	D. Xiao, S. Ruan, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445–1472. https://doi.org/10.1137/S0036139999361896 doi: 10.1137/S0036139999361896
[20]	P. Georgescu, Y. H. Hsieh, Global dynamics of a predator-prey model with stage structure for the predator, SIAM J. Appl. Math., 67 (2007), 1379–1395. https://doi.org/10.1137/060670377 doi: 10.1137/060670377
[21]	S. B. Hsu, T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763–783. https://doi.org/10.1137/S0036139993253201 doi: 10.1137/S0036139993253201
[22]	M. L. Rosenzweig, R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209–223. https://doi.org/10.1086/282272 doi: 10.1086/282272
[23]	H. I. Freedman, R. M. Mathsen, Persistence in predator-prey systems with ratio-dependent predator influence, Bull. Math. Biol., 55 (1993), 817–827. https://doi.org/10.1016/S0092-8240(05)80190-9 doi: 10.1016/S0092-8240(05)80190-9
[24]	W. Ko, K. Ryu, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, J. Differ. Equations, 231 (2006), 534–550. https://doi.org/10.1016/j.jde.2006.08.001 doi: 10.1016/j.jde.2006.08.001
[25]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331–340. https://doi.org/10.2307/3866 doi: 10.2307/3866
[26]	G. T. Skalski, J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the Holling type Ⅱ model, Ecology, 82 (2001), 3083–3092. https://doi.org/10.2307/2679836 doi: 10.2307/2679836
[27]	J. P. Tripathi, S. Bugalia, V. Tiwari, Y. Kang, A predator-prey model with Crowley-Martin functional response: A nonautonomous study, Nat. Resour. Modeling, 33 (2020), e12287. https://doi.org/10.1111/nrm.12287 doi: 10.1111/nrm.12287
[28]	R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459–467. https://doi.org/10.1038/261459a0 doi: 10.1038/261459a0
[29]	A. Hastings, T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896–903. https://doi.org/10.2307/1940591 doi: 10.2307/1940591
[30]	T. Y. Li, J. A. Yorke, Period three implies chaos, Am. Math. Mon., 82 (1975), 985–992. https://doi.org/10.2307/2318254
[31]	X. Liu, D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals, 32 (2007), 80–94. https://doi.org/10.1016/j.chaos.2005.10.081 doi: 10.1016/j.chaos.2005.10.081
[32]	Q. Din, Complexity and chaos control in a discrete-time prey–predator model, Commun. Nonlinear Sci. Numer. Simul., 49 (2017), 113–134. https://doi.org/10.1016/j.cnsns.2017.01.025 doi: 10.1016/j.cnsns.2017.01.025
[33]	D. Auerbach, P. Cvitanović, J. P. Eckmann, G. Gunaratne, I. Procaccia, Exploring chaotic motion through periodic orbits, Phys. Rev. Lett., 58 (1987), 2387–2389. https://doi.org/10.1103/PhysRevLett.58.2387 doi: 10.1103/PhysRevLett.58.2387
[34]	E. Ott, C. Grebogi, J. A. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 1196–1199.
[35]	Z. Wei, W. Tan, A. A. Elsadany, I. Moroz, Complexity and chaos control in a Cournot duopoly model based on bounded rationality and relative profit maximization, Nonlinear Dyn., 111 (2023), 17561–17589. https://doi.org/10.1007/s11071-023-08782-3 doi: 10.1007/s11071-023-08782-3
[36]	C. Wang, X. Li, Stability and Neimark-Sacker bifurcation of a semi-discrete population model, J. Appl. Anal. Comput., 4 (2014), 419–435. https://doi.org/10.11948/2014024 doi: 10.11948/2014024
[37]	S. Yu, Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Discrete Dyn. Nat. Soc., 2012 (2012), 208167. https://doi.org/10.1155/2012/208167 doi: 10.1155/2012/208167
[38]	S. G. Mortoja, P. Panja, S. K. Mondal, Dynamics of a predator-prey model with stage-structure on both species and anti-predator behavior, Inf. Med. Unlocked, 10 (2018), 50–57. https://doi.org/10.1016/j.imu.2017.12.004 doi: 10.1016/j.imu.2017.12.004
[39]	Y. Li, M. Rafaqat, T. J. Zia, I. Ahmed, C. Y. Jung, Flip and Neimark-Sacker bifurcations of a discrete time predator-pre model, IEEE Access, 7 (2019), 123430–123435. https://doi.org/10.1109/ACCESS.2019.2937956 doi: 10.1109/ACCESS.2019.2937956
[40]	D. Xiao, S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differ. Equations, 176 (2001), 494–510. https://doi.org/10.1006/jdeq.2000.3982 doi: 10.1006/jdeq.2000.3982
[41]	P. S. Mandal, M. Banerjee, Stochastic persistence and stability analysis of a modified Holling–Tanner model, Math. Methods Appl. Sci., 36 (2013), 1263–1280. https://doi.org/10.1002/mma.2680 doi: 10.1002/mma.2680
[42]	L. M. Pecora, T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821–824. https://doi.org/10.1103/PhysRevLett.64.821 doi: 10.1103/PhysRevLett.64.821
[43]	X. Li, Y. Liu, Transcritical bifurcation and Flip bifurcation of a new discrete ratio-dependent predator-prey system, Qual. Theory Dyn. Syst., 21 (2022), 122. https://doi.org/10.1007/s12346-022-00646-2 doi: 10.1007/s12346-022-00646-2
[44]	R. Ma, Y. Bai, F. Wang, Dynamical behavior analysis of a two-dimensional discrete predator-prey model with prey refuge and fear factor, J. Appl. Anal. Comput., 10 (2020), 1683–1697. https://doi.org/10.11948/20190426 doi: 10.11948/20190426
[45]	X. Jia, K. Huang, C. Li, Bifurcation analysis of a modified Leslie-Gower predator-prey system, Int. J. Bifurcation Chaos, 33 (2023), 2350024. https://doi.org/10.1142/S0218127423500244 doi: 10.1142/S0218127423500244
[46]	B. Roy, S. K. Roy, D. B. Gurung, Holling–Tanner model with Beddington–DeAngelis functional response and time delay introducing harvesting, Math. Comput. Simul., 142 (2017), 1–14. https://doi.org/10.1016/j.matcom.2017.03.010 doi: 10.1016/j.matcom.2017.03.010
[47]	E. Sáez, E. González-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867–1878. https://doi.org/10.1137/S0036139997318457 doi: 10.1137/S0036139997318457
[48]	M. Bucolo, A. Buscarino, L. Fortuna, S. Gagliano, Multidimensional discrete chaotic maps, Front. Phys., 10 (2022), 862376. https://doi.org/10.3389/fphy.2022.862376 doi: 10.3389/fphy.2022.862376
[49]	Z. Wei, F. Wang, H. Li, W. Zhang, Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo, Discrete Cont. Dyn. Syst. B, 27 (2022), 5029–5045. https://doi.org/10.3934/dcdsb.2021263 doi: 10.3934/dcdsb.2021263
[50]	S. Elaydi, An Introduction to Difference Equations, 3rd edition, Springer, New York, 2005.
[51]	Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edition, Springer, New York, 2004.

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