The interaction of different time scales in predator-prey models has become a common research topic. In the present article, we concentrated on the dynamics of interactions at two time scales in a plankton-fish system. To investigate the effects of the two time scales on plankton-fish dynamics, we constructed a new parameter with a corrected type that differs from the traditional slow parameter. In addition, zooplankton's refuge from the predator and phytoplankton mortality due to competition are incorporated into the model. Positivity and boundedness of solutions were proved. We then discussed feasibility and stability conditions of the equilibrium. We used a variety of means to support the existence of chaos in the system. Hopf bifurcation conditions were also obtained. Chaos control in the plankton-fish model is one of the main motivations for this study. In the slow-variable parameter case, we explored the control mechanism of gestation delay on chaotic systems, which are calmed by different periodic solutions. Moreover, under seasonal mechanisms, external driving forces can stabilize the system from chaos to periodic oscillations. Meanwhile, the sliding mode control (SMC) approach quickly calms chaotic oscillations and stabilizes it to an internal equilibrium state. The necessary numerical simulation experiments support the theoretical results.
Citation: Guilin Tang, Ning Li. Chaotic behavior and controlling chaos in a fast-slow plankton-fish model[J]. AIMS Mathematics, 2024, 9(6): 14376-14404. doi: 10.3934/math.2024699
The interaction of different time scales in predator-prey models has become a common research topic. In the present article, we concentrated on the dynamics of interactions at two time scales in a plankton-fish system. To investigate the effects of the two time scales on plankton-fish dynamics, we constructed a new parameter with a corrected type that differs from the traditional slow parameter. In addition, zooplankton's refuge from the predator and phytoplankton mortality due to competition are incorporated into the model. Positivity and boundedness of solutions were proved. We then discussed feasibility and stability conditions of the equilibrium. We used a variety of means to support the existence of chaos in the system. Hopf bifurcation conditions were also obtained. Chaos control in the plankton-fish model is one of the main motivations for this study. In the slow-variable parameter case, we explored the control mechanism of gestation delay on chaotic systems, which are calmed by different periodic solutions. Moreover, under seasonal mechanisms, external driving forces can stabilize the system from chaos to periodic oscillations. Meanwhile, the sliding mode control (SMC) approach quickly calms chaotic oscillations and stabilizes it to an internal equilibrium state. The necessary numerical simulation experiments support the theoretical results.
| [1] |
R. N. Premakumari, C. Baishya, M. K. Kaabar, Dynamics of a fractional plankton-fish model under the influence of toxicity, refuge, and combine-harvesting efforts, J. Inequal. Appl., 2022 (2022), 137. http://dx.doi.org/10.1186/s13660-022-02876-z doi: 10.1186/s13660-022-02876-z
|
| [2] |
M. Gao, D. Jiang, J. Ding, Dynamical behavior of a nutrient-plankton model with Ornstein-Uhlenbeck process and nutrient recycling, Chaos Soliton. Fract., 174 (2023), 113763. http://dx.doi.org/10.1016/j.chaos.2023.113763 doi: 10.1016/j.chaos.2023.113763
|
| [3] |
K. K. Choudhary, B. Dubey, A non-autonomous approach to study the impact of environmental toxins on nutrient-plankton system, Appl. Math. Comput., 458 (2023), 128236. http://dx.doi.org/10.1016/j.amc.2023.128236 doi: 10.1016/j.amc.2023.128236
|
| [4] |
C. P. Lynam, M. Llope, C. Möllmann, P. Helaouët, G. A. B. Brown, N. C. Stenseth, Interaction between top-down and bottom-up control in marine food webs, P. Natl. Acad. Sci. USA, 114 (2017), 1952–1957. http://dx.doi.org/10.1073/pnas.1621037114 doi: 10.1073/pnas.1621037114
|
| [5] |
A. B. Medvinsky, I. A. Tikhonova, R. R. Aliev, B. L. Li, Z. S. Lin, H. Malchow, Patchy environment as a factor of complex plankton dynamics, Phys. Rev. E, 64 (2001), 021915. http://dx.doi.org/10.1103/PhysRevE.64.021915 doi: 10.1103/PhysRevE.64.021915
|
| [6] |
B. Mukhopadhyay, R. Bhattacharyya, Role of gestation delay in a plankton-fish model under stochastic fluctuations, Math. Biosci., 215 (2008), 26–34. http://dx.doi.org/10.1016/j.mbs.2008.05.007 doi: 10.1016/j.mbs.2008.05.007
|
| [7] |
D. Tikhonov, J. Enderlein, H. Malchow, A. B. Medvinsky, Chaos and fractals in fish school motion, Chaos Soliton. Fract., 12 (2001), 277–288. http://dx.doi.org/10.1016/s0960-0779(00)00049-7 doi: 10.1016/s0960-0779(00)00049-7
|
| [8] |
B. Dubey, S. K. Sasmal, Chaotic dynamics of a plankton-fish system with fear and its carry over effects in the presence of a discrete delay, Chaos Soliton. Fract., 160 (2022), 112245. http://dx.doi.org/10.1016/j.chaos.2022.112245 doi: 10.1016/j.chaos.2022.112245
|
| [9] |
R. P. Kaur, A. Sharma, A. K. Sharma, Impact of fear effect on plankton-fish system dynamics incorporating zooplankton refuge, Chaos Soliton. Fract., 143 (2021), 110563. http://dx.doi.org/10.1016/j.chaos.2020.110563 doi: 10.1016/j.chaos.2020.110563
|
| [10] |
S. N. Raw, B. Tiwari, P. Mishra, Analysis of a plankton-fish model with external toxicity and nonlinear harvesting, Ric. Mat., 69 (2020), 653–681. http://dx.doi.org/10.1007/s11587-019-00478-4 doi: 10.1007/s11587-019-00478-4
|
| [11] |
S. Sajan, S. K. Sasmal, B. Dubey, A phytoplankton-zooplankton-fish model with chaos control: In the presence of fear effect and an additional food, Chaos, 32 (2022), 013114. http://dx.doi.org/10.1063/5.0069474 doi: 10.1063/5.0069474
|
| [12] |
G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347–386. http://dx.doi.org/10.1007/s00285-009-0266-7 doi: 10.1007/s00285-009-0266-7
|
| [13] |
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, 73 (2023), 103888. http://dx.doi.org/10.1016/j.nonrwa.2023.103888 doi: 10.1016/j.nonrwa.2023.103888
|
| [14] |
P. R. Chowdhury, S. Petrovskii, M. Banerjee, Effect of slow-fast time scale on transient dynamics in a realistic prey-predator system, Mathematics, 10 (2022), 699. http://dx.doi.org/10.3390/math10050699 doi: 10.3390/math10050699
|
| [15] |
S. H. Piltz, F. Veerman, P. K. Maini, M. A. Porter, A predator-2 prey fast-slow dynamical system for rapid predator evolution, SIAM J. Appl. Dyn. Syst., 16 (2017), 54–90. http://dx.doi.org/10.1137/16M1068426 doi: 10.1137/16M1068426
|
| [16] |
M. H. Cortez, S. P. Ellner, Understanding rapid evolution in predator-prey interactions using the theory of fast-slow dynamical systems, Am. Nat., 176 (2010), E109–E127. http://dx.doi.org/10.1086/656485 doi: 10.1086/656485
|
| [17] |
J. Shen, Z. Zhou, Fast-slow dynamics in logistic models with slowly varying parameters, Commun. Nonlinear Sci., 18 (2013), 2213–2221. http://dx.doi.org/10.1016/j.cnsns.2012.12.036 doi: 10.1016/j.cnsns.2012.12.036
|
| [18] |
T. Grozdanovski, J. J. Shepherd, A. Stacey, Multi-scaling analysis of a logistic model with slowly varying coefficients, Appl. Math. Lett., 22 (2009), 1091–1095. http://dx.doi.org/10.1016/j.aml.2008.10.002 doi: 10.1016/j.aml.2008.10.002
|
| [19] |
F. M. Alharbi, A slow single-species model with non-symmetric variation of the coefficients, Fractal Fract., 6 (2022), 72. http://dx.doi.org/10.3390/fractalfract6020072 doi: 10.3390/fractalfract6020072
|
| [20] |
V. A. Jansen, Regulation of predator-prey systems through spatial interactions: A possible solution to the paradox of enrichment, Oikos, 74 (1995), 384–390. http://dx.doi.org/10.2307/3545983 doi: 10.2307/3545983
|
| [21] |
M. Scheffer, Should we expect strange attractors behind plankton dynamics-and if so, should we bother? J. Plankton Res., 13 (1991), 1291-1305. https://doi.org/10.1093/plankt/13.6.1291 doi: 10.1093/plankt/13.6.1291
|
| [22] |
H. Tian, Z. Wang, P. Zhang, M. Chen, Y. Wang, Dynamic analysis and robust control of a chaotic system with hidden attractor, Complexity, 2021 (2021), 1–11. http://dx.doi.org/10.1155/2021/8865522 doi: 10.1155/2021/8865522
|
| [23] |
D. Yan, X. Wu, Applicability of the 0–1 test for chaos in magnetized Kerr-Newman spacetimes, Eur. Phys. J. C, 83 (2023), 1–17. http://dx.doi.org/10.1140/epjc/s10052-023-11978-x doi: 10.1140/epjc/s10052-023-11978-x
|
| [24] |
G. A. Gottwald, I. Melbourne, On the implementation of the 0–1 test for chaos, SIAM J. Appl. Dyn. Syst., 8 (2009), 129–145. http://dx.doi.org/10.1137/080718851 doi: 10.1137/080718851
|
| [25] |
K. H. Sun, X. Liu, C. X. Zhu, The 0–1 test algorithm for chaos and its applications, Chinese Phys. B, 19 (2010), 110510. http://dx.doi.org/10.1088/1674-1056/19/11/110510 doi: 10.1088/1674-1056/19/11/110510
|
| [26] |
A. Hastings, C. L. Hom, S. Ellner, P. Turchin, H. C. J. Godfray, Chaos in ecology: Is mother nature a strange attractor? Annu. Rev. Ecol. Evol. S., 24 (1993), 1-33. http://dx.doi.org/10.1146/annurev.es.24.110193.000245 doi: 10.1146/annurev.es.24.110193.000245
|
| [27] |
A. Morozov, S. Petrovskii, B. L. Li, Bifurcations and chaos in a predator-prey system with the Allee effect, P. Roy. Soc. B, 271 (2004), 1407–1414. http://dx.doi.org/10.1098/rspb.2004.2733 doi: 10.1098/rspb.2004.2733
|
| [28] |
A. R. Herrera, Chaos in predator-prey systems with/without impulsive effect, Nonlinear Anal.-Real, 13 (2012), 977–986. http://dx.doi.org/10.1016/j.nonrwa.2011.09.004 doi: 10.1016/j.nonrwa.2011.09.004
|
| [29] |
S. Gakkhar, R. K. Naji, Existence of chaos in two-prey, one-predator system, Chaos Soliton. Fract., 17 (2003), 639–649. http://dx.doi.org/10.1016/S0960-0779(02)00473-3 doi: 10.1016/S0960-0779(02)00473-3
|
| [30] |
H. Kharbanda, S. Kumar, Chaos detection and optimal control in a cannibalistic prey-predator system with harvesting, Int. J. Bifurcat. Chaos, 30 (2020), 2050171. http://dx.doi.org/10.1142/S0218127420501710 doi: 10.1142/S0218127420501710
|
| [31] |
R. P. Kaur, A. Sharma, A. K. Sharma, G. P. Sahu, Chaos control of chaotic plankton dynamics in the presence of additional food, seasonality, and time delay, Chaos Soliton. Fract., 153 (2021), 111521. http://dx.doi.org/10.1016/j.chaos.2021.111521 doi: 10.1016/j.chaos.2021.111521
|
| [32] |
J. M. Nazzal, A. N. Natsheh, Chaos control using sliding-mode theory, Chaos Soliton. Fract., 33 (2007), 695–702. http://dx.doi.org/10.1016/j.chaos.2006.01.071 doi: 10.1016/j.chaos.2006.01.071
|
| [33] |
C. Huang, H. Li, J. Cao, A novel strategy of bifurcation control for a delayed fractional predator-prey model, Appl. Math. Comput., 347 (2019), 808–838. http://dx.doi.org/10.1016/j.amc.2018.11.031 doi: 10.1016/j.amc.2018.11.031
|
| [34] |
J. Laoye, U. Vincent, S. Kareem, Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller, Chaos Soliton. Fract., 39 (2009), 356–362. http://dx.doi.org/10.1016/j.chaos.2007.04.020 doi: 10.1016/j.chaos.2007.04.020
|
| [35] |
M. Lampart, A. Lampartová, Chaos control and anti-control of the heterogeneous Cournot oligopoly model, Mathematics, 8 (2020), 1670. http://dx.doi.org/10.3390/math8101670 doi: 10.3390/math8101670
|
| [36] |
U. E. Kocamaz, B. Cevher, Y. Uyaroğlu, Control and synchronization of chaos with sliding mode control based on cubic reaching rule, Chaos Soliton. Fract., 105 (2017), 92–98. http://dx.doi.org/10.1016/j.chaos.2017.10.008 doi: 10.1016/j.chaos.2017.10.008
|
| [37] |
S. Akhtar, R. Ahmed, M. Batool, N. A. Shah, J. D. Chung, Stability, bifurcation and chaos control of a discretized Leslie prey-predator model, Chaos Soliton. Fract., 152 (2021), 111345. http://dx.doi.org/10.1016/j.chaos.2021.111345 doi: 10.1016/j.chaos.2021.111345
|
| [38] |
A. Singh, S. Gakkhar, Controlling chaos in a food chain model, Math. Comput. Simulat., 115 (2015), 24–36. http://dx.doi.org/10.1016/j.matcom.2015.04.001 doi: 10.1016/j.matcom.2015.04.001
|
| [39] |
F. H. I. P. Pinto, A. M. Ferreira, M. A. Savi, Chaos control in a nonlinear pendulum using a semi-continuous method, Chaos Soliton. Fract., 22 (2004), 653–668. http://dx.doi.org/10.1016/j.chaos.2004.02.047 doi: 10.1016/j.chaos.2004.02.047
|
| [40] |
A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer, B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn., 14 (2008), 147–175. http://dx.doi.org/10.1080/13873950701742754 doi: 10.1080/13873950701742754
|
| [41] |
S. K. Behera, R. A. Ranjan, S. Sarangi, A. K. Samantaray, R. Bhattacharyya, Nonlinear dynamics and chaos control of circular dielectric energy generator, Commun. Nonlinear Sci., 128 (2024), 107608. http://dx.doi.org/10.1016/j.cnsns.2023.107608 doi: 10.1016/j.cnsns.2023.107608
|
| [42] |
M. N. Huda, Q. Q. A'yun, S. Wigantono, H. Sandariria, I. Raming, A. Asmaidi, Effects of harvesting and planktivorous fish on bioeconomic phytoplankton-zooplankton models with ratio-dependent response functions and time delays, Chaos Soliton. Fract., 173 (2023), 113736. http://dx.doi.org/10.1090/S0894-0347-1992-1124979-1 doi: 10.1090/S0894-0347-1992-1124979-1
|
| [43] |
N. K. Thakur, A. Ojha, Complex dynamics of delay-induced plankton-fish interaction exhibiting defense, SN Appl. Sci., 2 (2020), 1–25. http://dx.doi.org/10.1007/s42452-020-2860-7 doi: 10.1007/s42452-020-2860-7
|
| [44] |
S. Rinaldi, S. Muratori, Y. Kuznetsov, Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities, B. Math. Biol., 55 (1993), 15–35. http://dx.doi.org/10.1016/s0092-8240(05)80060-6 doi: 10.1016/s0092-8240(05)80060-6
|
| [45] |
S. Gakkhar, R. K. Naji, Chaos in seasonally perturbed ratio-dependent preypredator system, Chaos Soliton. Fract., 15 (2003), 107–118. http://dx.doi.org/10.1016/s0960-0779(02)00114-5 doi: 10.1016/s0960-0779(02)00114-5
|
| [46] |
M. Gao, H. Shi, Z. Li, Chaos in a seasonally and periodically forced phytoplankton-zooplankton system, Nonlinear Anal.-Real, 10 (2009), 1643–1650. http://dx.doi.org/10.1016/j.nonrwa.2008.02.005 doi: 10.1016/j.nonrwa.2008.02.005
|
| [47] |
J. Shen, C. H. Hsu, T. H. Yang, Fast-slow dynamics for intraguild predation models with evolutionary effects, J. Dyn. Differ. Equ., 32 (2020), 895–920. http://dx.doi.org/10.1007/s10884-019-09744-3 doi: 10.1007/s10884-019-09744-3
|
| [48] | K. A. N. Al Amri, Q. J. Khan, Combining impact of velocity, fear and refuge for the predator-prey dynamics, J. Biol. Dynam., 17 (2023). 2181989. http://dx.doi.org/10.1080/17513758.2023.2181989 |
| [49] |
S. Pandey, U. Ghosh, D. Das, S. Chakraborty, A. Sarkar, Rich dynamics of a delay-induced stage-structure prey-predator model with cooperative behaviour in both species and the impact of prey refuge, Math. Comput. Simulat., 216 (2024), 49–76. http://dx.doi.org/10.1016/j.matcom.2023.09.002 doi: 10.1016/j.matcom.2023.09.002
|
| [50] |
W. Li, L. Huang, J. Wang, Global asymptotical stability and sliding bifurcation analysis of a general Filippov-type predator-prey model with a refuge, Appl. Math. Comput., 405 (2021), 126263. http://dx.doi.org/10.1016/j.amc.2021.126263 doi: 10.1016/j.amc.2021.126263
|
| [51] |
S. Vaidyanathan, K. Benkouider, A. Sambas, P. Darwin, Bifurcation analysis, circuit design and sliding mode control of a new multistable chaotic population model with one prey and two predators, Arch. Control Sci., 33 (2023), 127–153. http://dx.doi.org/10.24425/acs.2023.145117 doi: 10.24425/acs.2023.145117
|
| [52] |
G. C. Sabin, D. Summers, Chaos in a periodically forced predator-prey ecosystem model, Math. Biosci., 113 (1993), 91–113. http://dx.doi.org/10.1016/0025-5564(93)90010-8 doi: 10.1016/0025-5564(93)90010-8
|
| [53] |
T. L. Rogers, B. J. Johnson, S. B. Munch, Chaos is not rare in natural ecosystems, Nat. Ecol. Evol., 6 (2022), 1105–1111. http://dx.doi.org/10.1038/s41559-022-01787-y doi: 10.1038/s41559-022-01787-y
|
| [54] |
A. M. Edwards, M. A. Bees, Generic dynamics of a simple plankton population model with a non-integer exponent of closure, Chaos Soliton. Fract., 12 (2001), 289–300. http://dx.doi.org/10.1016/s0960-0779(00)00065-5 doi: 10.1016/s0960-0779(00)00065-5
|
Figures(13) / Tables(3)
Guilin Tang, Ning Li. Chaotic behavior and controlling chaos in a fast-slow plankton-fish model[J]. AIMS Mathematics, 2024, 9(6): 14376-14404. doi: 10.3934/math.2024699
DownLoad: