A dynamic inequality with average condition and applications to predator-prey systems on time scales

Yang-Yang Yu; Lin-Lin Wang; Zhong-Xin Ma; Yang-Yang Yu; Lin-Lin Wang; Zhong-Xin Ma

doi:10.3934/math.2026246

AIMS Mathematics

2026, Volume 11, Issue 3: 5954-5965. doi: 10.3934/math.2026246

Previous Article Next Article

Research article

A dynamic inequality with average condition and applications to predator-prey systems on time scales

1.
School of Mathematics and Statistics, Xuzhou University of Technology, 02 Lishui Rd., 221018 Xuzhou, China
2.
School of Mathematics and Statistics Science, Ludong University, 186 Hongqi Middle Rd., 264025 Yantai, China
3.
School of Science, Zhejiang University of Science & Technology, 318 Liuhe Rd., 310023 Hangzhou, China

Received: 01 January 2026 Revised: 11 February 2026 Accepted: 04 March 2026 Published: 09 March 2026
MSC : 34N05, 37N25

In this paper, a dynamic inequality with average condition on time scales is established. Using this inequality, we study the uniform ultimate boundedness of solutions of two predator-prey systems on time scales. Our results extend and generalize some known results about these types of problems. We verify our main results by means of a numerical example.
- average condition,
- dynamic inequality,
- uniform ultimate boundedness,
- time scale,
- predator-prey system
Citation: Yang-Yang Yu, Lin-Lin Wang, Zhong-Xin Ma. A dynamic inequality with average condition and applications to predator-prey systems on time scales[J]. AIMS Mathematics, 2026, 11(3): 5954-5965. doi: 10.3934/math.2026246

Related Papers:

Abstract

In this paper, a dynamic inequality with average condition on time scales is established. Using this inequality, we study the uniform ultimate boundedness of solutions of two predator-prey systems on time scales. Our results extend and generalize some known results about these types of problems. We verify our main results by means of a numerical example.

References

[1]	R. Arditi, L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311–326. https://doi.org/10.1016/S0022-5193(89)80211-5 doi: 10.1016/S0022-5193(89)80211-5
[2]	M. Bohner, E. Girejko, A. B. Malinowska, T. Truong, The uncertain malthusian model on time scales, Proc. Amer. Math. Soc., 152 (2025), 2657–2668. https://doi.org/10.1090/proc/16701 doi: 10.1090/proc/16701
[3]	M. Bohner, A. Peterson, Dynamic equations on time scales: An introduction with applications, Birkhäuser Boston: Springer, 2001. https://doi.org/10.1007/978-1-4612-0201-1
[4]	J. Cui, Y. Takeuchi, Permanence, extinction and periodic solution of predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 317 (2006), 464–474. https://doi.org/10.1016/j.jmaa.2005.10.011 doi: 10.1016/j.jmaa.2005.10.011
[5]	S. Dey, S. Ghorai, M. Banerjee, Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth, J. Math. Biol., 87 (2023), 21. https://doi.org/10.1007/s00285-023-01957-x doi: 10.1007/s00285-023-01957-x
[6]	N. H. Du, N. N. Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Appl. Math. Lett., 64 (2017), 223–230. https://doi.org/10.1016/j.aml.2016.09.012 doi: 10.1016/j.aml.2016.09.012
[7]	Y. H. Fan, W. T. Li, Harmless delays in a discrete ratio-dependent periodic predator-prey system, Discrete Dyn. Nat. Soc., 2006 (2006), 012176. https://doi.org/10.1155/DDNS/2006/12176 doi: 10.1155/DDNS/2006/12176
[8]	Y. H. Fan, L. L. Wang, Average conditions for the permanence of a bounded discrete predator-prey system, Discrete Dyn. Nat. Soc., 2013 (2013), 508686. https://doi.org/10.1155/2013/508686 doi: 10.1155/2013/508686
[9]	Y. H. Fan, W. T. Li, Permanence in delayed ratio-dependent predator-prey models with monotonic functional responses, Nonlinear Anal. Real World Appl., 8 (2007), 424–434. https://doi.org/10.1016/J.NONRWA.2005.12.003 doi: 10.1016/J.NONRWA.2005.12.003
[10]	M. X. He, F. D. Chen, Z. Li, Permanence and global attractivity of an impulsive delay logistic model, Appl. Math. Lett., 62 (2016), 92–100. https://doi.org/10.1016/j.aml.2016.07.009 doi: 10.1016/j.aml.2016.07.009
[11]	S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrums mannigfaltigkeiten, Ph.D. thesis, Universität of Würzburg, 1988.
[12]	M. D. Johnston, C. Pantea, P. Donnell, A computational approach to persistence, permanence, and endotacticity of biochemical reaction systems, J. Math. Biol., 72 (2016), 467–498. https://doi.org/10.1007/s00285-015-0892-1 doi: 10.1007/s00285-015-0892-1
[13]	V. Lakshmikantham, S. Sivasundaram, B. Kaymakçalan, Dynamic systems on measure chains, New York: Springer, 1996. https://doi.org/10.1007/978-1-4757-2449-3
[14]	H. Y. Li, Z. K. She, Dynamics of a non-autonomous density-dependent predator-prey model with Beddington-DeAngelis type, Int. J. Biomath., 9 (2016), 1650050. https://doi.org/10.1142/S1793524516500509 doi: 10.1142/S1793524516500509
[15]	R. H. Liu, C. Zhang, Y. X. Guo, X. F. Zhang, Intermittent event-triggered control for exponential synchronization of delayed neural networks on time scales, Commun. Nonlinear Sci. Numer. Simul., 137 (2024), 108158. https://doi.org/10.1016/j.cnsns.2024.108158 doi: 10.1016/j.cnsns.2024.108158
[16]	X. K. Ma, Y. Su, X. F. Zou, Joint impact of maturation delay and fear effect on the population dynamics of a predator-Prey system, SIAM J. Appl. Math., 84 (2024), 1557–1579. https://doi.org/10.1137/23M1596569 doi: 10.1137/23M1596569
[17]	M. Q. Ouyang, X. Y. Li, Permanence and asymptotical behavior of stochastic prey-predator system with Markovian switching, Appl. Math. Comput., 266 (2015), 539–559. https://doi.org/10.1016/j.amc.2015.05.083 doi: 10.1016/j.amc.2015.05.083
[18]	A. Parwaliya, A. Singh, A. Kumar, Hopf bifurcation in a delayed prey-predator model with prey refuge involving fear effect, Int. J. Biomath., 17 (2024), 2350042. https://doi.org/10.1142/S1793524523500420 doi: 10.1142/S1793524523500420
[19]	M. J. S. Sahir, F. Chaudhry, D. Faryal, Affinity of several classical and dynamic inequalities with comprehensive approach on time scales, Math. Method Appl. Sci., 48 (2025), 13632–13638. https://doi.org/10.1002/mma.11131 doi: 10.1002/mma.11131
[20]	D. Sahoo, G. Samanta, Oscillatory and transient dynamics of a slow-fast predator-prey system with fear and its carry-over effect, Nonlinear Anal. Real World Appl., 73 (2023), 103888. https://doi.org/10.1016/j.nonrwa.2023.103888 doi: 10.1016/j.nonrwa.2023.103888
[21]	Z. Wang, M. M. Jin, Permanence and extinction of the delay tumor-immune system with the stochastic environment, Int. J. Biomath., 2024. https://doi.org/10.1142/S1793524524501341 doi: 10.1142/S1793524524501341
[22]	G. C. Wu, T. T. Song, S. Q. Wang, Caputo-Hadamard fractional differential equations on time scales: Numerical scheme, asymptotic stability, and chaos, Chaos, 32 (2022), 093143. https://doi.org/10.1063/5.0098375 doi: 10.1063/5.0098375
[23]	Y. Y. Zhao, C. J. Xu, Y. Y. Xu, J. T. Lin, Y. C. Pang, Z. X. Liu, et al., Mathematical exploration on control of bifurcation for a 3D predator-prey model with delay, AIMS Mathematics, 9 (2024), 29883–29915. https://doi.org/10.3934/math.20241445 doi: 10.3934/math.20241445

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)