Mode Locking, Farey Sequence, and Bifurcation in a Discrete Predator-Prey Model with Holling Type IV Response
This paper presents and examines a discrete-time predator–prey model of the Leslie type, integrating a Holling type IV functional response for analysis. The mathematical analysis succinctly identifies fixed points and evaluates their local stability within the model. The study employs the normal for...
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2025-05-01
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| author | Yun Liu Xijuan Liu |
| author_facet | Yun Liu Xijuan Liu |
| author_sort | Yun Liu |
| collection | DOAJ |
| description | This paper presents and examines a discrete-time predator–prey model of the Leslie type, integrating a Holling type IV functional response for analysis. The mathematical analysis succinctly identifies fixed points and evaluates their local stability within the model. The study employs the normal form approach and bifurcation theory to explore codimension-one and two bifurcation behaviors for this model. The primary conclusions are substantiated by a combination of rigorous theoretical analysis and meticulous computational simulations. Additionally, utilizing fractal basin boundaries, periodicity variations, and Lyapunov exponent distributions within two-parameter spaces, we observe a mode-locking structure akin to Arnold tongues. These periods are arranged in a Farey tree sequence and embedded within quasi-periodic/chaotic regions. These findings enhance comprehension of bifurcation cascade emergence and structural patterns in diverse biological systems with discrete dynamics. |
| format | Article |
| id | doaj-art-9d33a9a33e0049858450c55bda8ae47e |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-9d33a9a33e0049858450c55bda8ae47e2025-08-20T03:27:13ZengMDPI AGAxioms2075-16802025-05-0114641410.3390/axioms14060414Mode Locking, Farey Sequence, and Bifurcation in a Discrete Predator-Prey Model with Holling Type IV ResponseYun Liu0Xijuan Liu1College of Information Engineering, Tarim University, Alar 843300, ChinaCollege of Information Engineering, Tarim University, Alar 843300, ChinaThis paper presents and examines a discrete-time predator–prey model of the Leslie type, integrating a Holling type IV functional response for analysis. The mathematical analysis succinctly identifies fixed points and evaluates their local stability within the model. The study employs the normal form approach and bifurcation theory to explore codimension-one and two bifurcation behaviors for this model. The primary conclusions are substantiated by a combination of rigorous theoretical analysis and meticulous computational simulations. Additionally, utilizing fractal basin boundaries, periodicity variations, and Lyapunov exponent distributions within two-parameter spaces, we observe a mode-locking structure akin to Arnold tongues. These periods are arranged in a Farey tree sequence and embedded within quasi-periodic/chaotic regions. These findings enhance comprehension of bifurcation cascade emergence and structural patterns in diverse biological systems with discrete dynamics.https://www.mdpi.com/2075-1680/14/6/414predator–prey systemstability analysisbifurcationArnold tonguefractal basin |
| spellingShingle | Yun Liu Xijuan Liu Mode Locking, Farey Sequence, and Bifurcation in a Discrete Predator-Prey Model with Holling Type IV Response Axioms predator–prey system stability analysis bifurcation Arnold tongue fractal basin |
| title | Mode Locking, Farey Sequence, and Bifurcation in a Discrete Predator-Prey Model with Holling Type IV Response |
| title_full | Mode Locking, Farey Sequence, and Bifurcation in a Discrete Predator-Prey Model with Holling Type IV Response |
| title_fullStr | Mode Locking, Farey Sequence, and Bifurcation in a Discrete Predator-Prey Model with Holling Type IV Response |
| title_full_unstemmed | Mode Locking, Farey Sequence, and Bifurcation in a Discrete Predator-Prey Model with Holling Type IV Response |
| title_short | Mode Locking, Farey Sequence, and Bifurcation in a Discrete Predator-Prey Model with Holling Type IV Response |
| title_sort | mode locking farey sequence and bifurcation in a discrete predator prey model with holling type iv response |
| topic | predator–prey system stability analysis bifurcation Arnold tongue fractal basin |
| url | https://www.mdpi.com/2075-1680/14/6/414 |
| work_keys_str_mv | AT yunliu modelockingfareysequenceandbifurcationinadiscretepredatorpreymodelwithhollingtypeivresponse AT xijuanliu modelockingfareysequenceandbifurcationinadiscretepredatorpreymodelwithhollingtypeivresponse |