Qualitative properties and optimal control strategy on a novel fractional three-species food chain model. (English) Zbl 07896488
Summary: In this study, the dynamics of a novel three-species food chain model featuring the Sokol-Howell functional response are explored. The fear of predators is incorporated into prey reproduction, and refuge is integrated into the middle predators within the framework of the Caputo fractional derivative. Theoretical aspects such as the existence and uniqueness of equilibria, their boundedness, and stability analysis are encompassed in the investigation. To examine the existence of chaos, Lyapunov exponents are computed. The optimal control measure concerning the growth of the prey population was considered, and the conditions that must be met for the optimal response to exist in the optimal control issue were determined using Pontryagin’s Maximum Principle. The theoretical outcomes were validated by using numerical simulation powered by the Adams-Bashforth-Moulton type predictor-corrector technique. Numerical justifications are provided for the influences of fear and refuge factors. When fear is absent, a numerical analysis is conducted on the global stability of the system for fractional order derivative.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34A08 | Fractional ordinary differential equations |
| 92D25 | Population dynamics (general) |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
| 49J15 | Existence theories for optimal control problems involving ordinary differential equations |
Keywords:
Caputo fractional derivative; three-species food chain; equilibria; optimal control; qualitative propertiesReferences:
| [1] | Malthus, T.R.: An essay on the principle of population. Reprinted from 1798 edition, Johnson, London, as. Malthus-population: the first essay. Ann Arbor Paper-backs, University of Michigan, Ann Arbor Michigan, USA (1959) |
| [2] | Verhulst, P-F, Notice sur la loi que la population suit dans son accroissement, Correspondence Mathematique et Physique, 10, 113-129, 1838 |
| [3] | Lotka, AJ, Elements of Physical Biology, 1925, Williams & Wilkins · JFM 51.0416.06 |
| [4] | Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, vol. 2. Societá anonima tipografica “Leonardo da Vinci” (1927) |
| [5] | Samardzija, N.; Greller, LD, Explosive route to chaos through a fractal torus in a generalized Lotka-Volterra model, Bull. Math. Biol., 50, 5, 465-491, 1988 · Zbl 0668.92010 · doi:10.1007/BF02458847 |
| [6] | Cui, Q.; Xu, C.; Ou, W.; Pang, Y.; Liu, Z.; Li, P.; Yao, L., Bifurcation behavior and hybrid controller design of a 2D Lotka-Volterra commensal symbiosis system accompanying delay, Mathematics, 11, 23, 2023 · doi:10.3390/math11234808 |
| [7] | Ou, W.; Xu, C.; Cui, Q.; Pang, Y.; Liu, Z.; Shen, J.; Baber, MZ; Farman, M.; Ahmad, S.; Ou, W.; Xu, C.; Cui, Q.; Pang, Y.; Liu, Z.; Shen, J.; Baber, MZ; Farman, M.; Ahmad, S., Hopf bifurcation exploration and control technique in a predator-prey system incorporating delay, AIMS Math., 9, 1, 1622-1651, 2024 · doi:10.3934/math.2024080 |
| [8] | Rihan, FA; Alsakaji, HJ, Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species, Discrete Contin. Dyn. Syst., 15, 2, 245-263, 2022 · Zbl 1486.92181 · doi:10.3934/dcdss.2020468 |
| [9] | Xu, C.; Pang, Y.; Liu, Z.; Shen, J.; Liao, M.; Li, P., Insights into COVID-19 stochastic modelling with effects of various transmission rates: simulations with real statistical data from UK, Australia, Spain, and India, Phys. Scr., 99, 2, 2024 |
| [10] | Xu, C., Zhao, Y., Lin, J., Pang, Y., Liu, Z., Farman, M., Ahmad, S.: Mathematical exploration on control of bifurcation for a plankton-oxygen dynamical model owning delay. J. Math. Chem. (2023) doi:10.1007/s10910-023-01543-y |
| [11] | Xu, C.; Cui, Q.; Liu, Z.; Pan, Y.; Cui, X.; Ou, W.; Rahman, M.; Farman, M.; Ahmad, S.; Zeb, A., Extended hybrid controller design of bifurcation in a delayed chemostat model, MATCH Commun. Math. Comput. Chem., 90, 609-648, 2023 · Zbl 1531.92061 · doi:10.46793/match.90-3.609X |
| [12] | Hua, F.; Sieving, KE; Fletcher, J.; Robert, J.; Wright, CA, Increased perception of predation risk to adults and offspring alters avian reproductive strategy and performance, Behav. Ecol., 25, 3, 509-519, 2014 · doi:10.1093/beheco/aru017 |
| [13] | Wang, J.; Cai, Y.; Fu, S.; Wang, W., The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge, Chaos Interdiscip. J. Nonlinear Sci., 29, 8, 2019 · Zbl 1420.92122 · doi:10.1063/1.5111121 |
| [14] | Zhang, H.; Cai, Y.; Fu, S.; Wang, W., Impact of the fear effect in a prey-predator model incorporating a prey refuge, Appl. Math. Comput., 356, 328-337, 2019 · Zbl 1428.92099 |
| [15] | Hossain, M.; Pal, N.; Samanta, S., Impact of fear on an eco-epidemiological model, Chaos Solitons Fractals, 134, 2020 · Zbl 1483.92133 · doi:10.1016/j.chaos.2020.109718 |
| [16] | Belew, B.; Melese, D., Modeling and analysis of predator-prey model with fear effect in prey and hunting cooperation among predators and harvesting, J. Appl. Math., 2022, 2776698, 2022 · Zbl 1517.92010 · doi:10.1155/2022/2776698 |
| [17] | Tian, Y.; Li, HM, The study of a predator-prey model with fear effect based on state-dependent harvesting strategy, Complexity, 2022, 9496599, 2022 · doi:10.1155/2022/9496599 |
| [18] | Haque, M.; Sarwardi, S., Dynamics of a harvested prey-predator model with prey refuge dependent on both species, Int. J. Bifurc. Chaos, 28, 1830040, 2017 · Zbl 1404.34060 · doi:10.1142/S0218127418300409 |
| [19] | Sih, A., Prey refuges and predator-prey stability, Theor. Popul. Biol., 31, 1, 1-12, 1987 · doi:10.1016/0040-5809(87)90019-0 |
| [20] | Ma, Z.; Wang, S.; Li, W.; Li, Z., The effect of prey refuge in a patchy predator-prey system, Math. Biosci., 243, 1, 126-130, 2013 · Zbl 1279.92089 · doi:10.1016/j.mbs.2013.02.011 |
| [21] | Kar, TK, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul., 10, 6, 681-691, 2005 · Zbl 1064.92045 · doi:10.1016/j.cnsns.2003.08.006 |
| [22] | Ko, W.; Ryu, K., Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, J. Differ. Equ., 231, 2, 534-550, 2006 · Zbl 1387.35588 · doi:10.1016/j.jde.2006.08.001 |
| [23] | Premakumari, RN; Baishya, C.; Kaabar, M., Dynamics of a fractional plankton-fish model under the influence of toxicity, refuge, and combine-harvesting efforts, J. Inequal. Appl., 2022, 1-26, 2022 · Zbl 1509.37125 · doi:10.1186/s13660-022-02876-z |
| [24] | Seo, G.; Deangelis, D., A predator-prey model with a Holling type I functional response including a predator mutual interference, J. Nonlinear Sci., 21, 811-833, 2011 · Zbl 1238.92049 · doi:10.1007/s00332-011-9101-6 |
| [25] | Achar, SJ; Baishya, C.; Veeresha, P.; Akinyemi, L., Dynamics of fractional model of biological pest control in tea plants with Beddington-DeAngelis functional response, Fractal Fract., 6, 1, 2022 · doi:10.3390/fractalfract6010001 |
| [26] | Upadhyay, R.; Raw, S., Complex dynamics of a three species food-chain model with Holling type IV functional response, Nonlinear Anal. Model. Control, 16, 353-374, 2011 · Zbl 1272.49085 |
| [27] | Sokol, W.; Howell, J., Kinetics of phenol oxidation by washed cells, Biotechnol. Bioeng., 23, 9, 2039-2050, 1981 · doi:10.1002/bit.260230909 |
| [28] | Ali, S.; Arifin, N.; Naji, R.; Ismail, F.; Bachok, N., Controlling chaotic dynamics of a continuous ecological model, Int. J. Pure Appl. Math., 109, 177-191, 2016 |
| [29] | Alsakaji, H.; Kundu, S.; Rihan, F., Delay differential model of one-predator two-prey system with Monod-Haldane and Holling type II functional responses, Appl. Math. Comput., 397, 2021 · Zbl 1508.92195 |
| [30] | Rihan, FA; Rajivganthi, C., Dynamics of fractional-order delay differential model of prey-predator system with Holling-type III and infection among predators, Chaos Solitons Fractals, 141, 2020 · Zbl 1496.92095 · doi:10.1016/j.chaos.2020.110365 |
| [31] | Priyadarshi, A.; Gakkhar, S., Dynamics of Leslie-Gower type generalist predator in a tri-trophic food web system, Commun. Nonlinear Sci. Numer. Simul., 18, 11, 3202-3218, 2013 · Zbl 1328.92065 · doi:10.1016/j.cnsns.2013.03.001 |
| [32] | Li, P.; Gao, R.; Xu, C.; Shen, J.; Ahmad, S.; Li, Y., Exploring the impact of delay on Hopf bifurcation of a type of BAM neural network models concerning three nonidentical delays, Neural Process. Lett., 55, 11595, 2023 · doi:10.1007/s11063-023-11392-0 |
| [33] | Zafar, Z.; Yusuf, A.; Musa, S.; Qureshi, S.; Alshomrani, A.; Baleanu, D., Impact of public health awareness programs on covid-19 dynamics: a fractional modeling approach, Fractals, 31, 10, 2340005, 2023 · doi:10.1142/S0218348X23400054 |
| [34] | Tassaddiq, A.; Qureshi, S.; Soomro, A.; Arqub, OA; Senol, M., Comparative analysis of classical and Caputo models for COVID-19 spread: vaccination and stability assessment, Fixed Point Theory Algorithms Sci. Eng., 1, 2024, 2024 · Zbl 07833567 |
| [35] | Chinnamuniyandi, M.; Chandran, S.; Xu, C., Fractional order uncertain BAM neural networks with mixed time delays: an existence and quasi-uniform stability analysis, J. Intell. Fuzzy Syst. Appl. Eng. Technol., 46, 2, 4291-4313, 2024 |
| [36] | Tariq, M.; Ahmad, H.; Shaikh, A.; Ntouyas, S.; Hincal, E.; Qureshi, S., Fractional Hermite-Hadamard-type inequalities for differentiable Preinvex mappings and applications to modified Bessel and q-digamma functions, Math. Comput. Appl., 28, 108, 2023 |
| [37] | Baishya, C.; Achar, SJ; Veeresha, P.; Prakasha, DG, Dynamics of a fractional epidemiological model with disease infection in both the populations, Chaos (Woodbury, N.Y.), 31, 4, 2021 · Zbl 1460.92184 · doi:10.1063/5.0028905 |
| [38] | Akinyemi, L.; Iyiola, O., Exact and approximate solutions of time-fractional models arising from physics via Shehu transform, Math. Methods Appl. Sci., 43, 7442, 2020 · Zbl 1454.34011 · doi:10.1002/mma.6484 |
| [39] | Shah, SMA; Tahir, H.; Khan, A.; Khan, WA; Arshad, A., Stochastic model on the transmission of worms in wireless sensor network, J. Math. Tech. Model., 1, 1, 75-88, 2024 |
| [40] | Baishya, C.; Premakumari, RN; Samei, M., Chaos control of fractional order nonlinear Bloch equation by utilizing sliding mode controller, Chaos Solitons Fractals, 174, 2023 · doi:10.1016/j.chaos.2023.113773 |
| [41] | Ain, QT, Nonlinear stochastic cholera epidemic model under the influence of noise, J. Math. Tech. Model., 1, 1, 52-74, 2024 |
| [42] | Veeresha, P.; Baleanu, D., A unifying computational framework for fractional Gross-Pitaevskii equations, Phys. Scr., 96, 2021 · doi:10.1088/1402-4896/ac28c9 |
| [43] | Premakumari, RN; Baishya, C.; Veeresha, P.; Akinyemi, L., A fractional atmospheric circulation system under the influence of a sliding mode controller, Symmetry, 14, 2618, 2022 · doi:10.3390/sym14122618 |
| [44] | Din, A.; Li, Y.; Yusuf, A., Delayed hepatitis B epidemic model with stochastic analysis, Chaos Solitons Fractals, 146, 2021 · Zbl 1498.92206 · doi:10.1016/j.chaos.2021.110839 |
| [45] | Din, A., Bifurcation analysis of a delayed stochastic HBV epidemic model: cell-to-cell transmission, Chaos Solitons Fractals, 181, 2021 · doi:10.1016/j.chaos.2024.114714 |
| [46] | Khan, FM; Khan, ZU; Abdullah, Numerical analysis of fractional order drinking mathematical model, J. Math. Tech. Model., 1, 1, 11-24, 2024 |
| [47] | Khan, WA; Zarin, R.; Zeb, A.; Khan, Y.; Khan, A., Navigating food allergy dynamics via a novel fractional mathematical model for antacid-induced allergies, J. Math. Tech. Model., 1, 1, 25-51, 2024 |
| [48] | Alidousti, J.; Mostafavi, M., Dynamical behavior of a fractional three-species food chain model, Nonlinear Dyn., 95, 1841, 2019 · Zbl 1432.37115 · doi:10.1007/s11071-018-4663-6 |
| [49] | Devi, A.; Kumar, A.; Baleanu, D.; Khan, A., On stability analysis and existence of positive solutions for a general non-linear fractional differential equations, Adv. Differ. Equ., 2020, 300, 2020 · Zbl 1485.34033 · doi:10.1186/s13662-020-02729-3 |
| [50] | Devi, A.; Kumar, A.; Abdeljawad, T.; Khan, A., Stability analysis of solutions and existence theory of fractional Lagevin equation, Alex. Eng. J., 60, 4, 3641-3647, 2021 · doi:10.1016/j.aej.2021.02.011 |
| [51] | Xu, C.; Liu, Z.; Li, P.; Yan, J.; Yao, L., Bifurcation mechanism for fractional-order three-triangle multi-delayed neural networks, Neural Process. Lett., 55, 6125, 2022 · doi:10.1007/s11063-022-11130-y |
| [52] | Mahmoud, E.; Trikha, P.; Jahanzaib, L.; Almaghrabi, O., Dynamical analysis and chaos control of the fractional chaotic ecological model, Chaos Solitons Fractals, 141, 2020 · Zbl 1496.37095 · doi:10.1016/j.chaos.2020.110348 |
| [53] | Shah, A.; Khan, RA; Khan, A.; Khan, H.; Gomez-Aguilar, JF, Investigation of a system of nonlinear fractional order hybrid differential equations under usual boundary conditions for existence of solution, Math. Methods Appl. Sci., 44, 2, 1628-1638, 2021 · Zbl 1471.34022 · doi:10.1002/mma.6865 |
| [54] | Cui, Z.; Zhou, Y.; Li, R., Complex dynamics analysis and Chaos control of a fractional-order three-population food chain model, Fractal Fract., 7, 7, 2023 · doi:10.3390/fractalfract7070548 |
| [55] | Aslam, M.; Murtaza, R.; Abdeljawad, T.; Rahman, G.; Khan, A.; Khan, H.; Gulzar, H., A fractional order HIV/AIDS epidemic model with Mittag-Leffler kernel, Adv. Differ. Equ., 2021, 107, 2021 · Zbl 1494.92123 · doi:10.1186/s13662-021-03264-5 |
| [56] | Padder, A.; Almutairi, L.; Qureshi, S.; Soomro, A.; Afroz, A.; Hınçal, E.; Tassaddiq, A., Dynamical analysis of generalized tumor model with Caputo fractional-order derivative, Fractal Fract., 7, 258, 2023 · doi:10.3390/fractalfract7030258 |
| [57] | May, R., Allen, P.: Stability and complexity in model ecosystems. IEEE Trans. Syst. Man Cybern. SMC-6(12), 887-887 (1976) |
| [58] | Xu, C.; Liao, M.; Li, P.; Yao, L.; Qin, Q.; Shang, Y., Chaos control for a fractional-order Jerk system via time delay feedback controller and mixed controller, Fractal Fract., 5, 4, 2021 · doi:10.3390/fractalfract5040257 |
| [59] | Johansyah, D.; Sambas, A.; Qureshi, S.; Zheng, S.; Alsbagh, T.; Vaidyanathan, S.; Ibrahim, S., Investigation of the hyperchaos and control in the fractional order financial system with profit margin, Partial Differ. Equ. Appl. Math., 9, 2024 · doi:10.1016/j.padiff.2023.100612 |
| [60] | Ding, Y.; Wang, Z.; Ye, H., Optimal control of a fractional-order HIV-immune system with memory, IEEE Trans. Control Syst. Technol., 20, 5, 763-769, 2012 · doi:10.1109/TCST.2011.2153203 |
| [61] | Ullah, S.; Khan, MA, Modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study, Chaos Solitons Fractals, 139, 2020 · doi:10.1016/j.chaos.2020.110075 |
| [62] | Shen, Z-H; Chu, Y-M; Khan, MA; Muhammad, S.; Al-Hartomy, OA; Higazy, M., Mathematical modeling and optimal control of the COVID-19 dynamics, Results Phys., 31, 2021 · doi:10.1016/j.rinp.2021.105028 |
| [63] | Din, A., The stochastic bifurcation analysis and stochastic delayed optimal control for epidemic model with general incidence function, Chaos, 31, 2021 · Zbl 07868475 · doi:10.1063/5.0063050 |
| [64] | Awadalla, M.; Alahmadi, J.; Cheneke, KR; Qureshi, S., Fractional optimal control model and bifurcation analysis of human syncytial respiratory virus transmission dynamics, Fractal Fract., 8, 1, 2024 · doi:10.3390/fractalfract8010044 |
| [65] | Ali, S.; Khan, A.; Shah, K.; Alqudah, MA; Abdeljawad, T.; Siraj-ul-IslamOn, computational analysis of highly nonlinear model addressing real world applications, Results Phys., 36, 2022 · doi:10.1016/j.rinp.2022.105431 |
| [66] | Qureshi, S.; Ramos, H.; Soomro, A.; Akinfenwa, OA; Akanbi, MA, Numerical integration of stiff problems using a new time-efficient hybrid block solver based on collocation and interpolation techniques, Math. Comput. Simul., 220, 237-252, 2024 · Zbl 1540.65184 · doi:10.1016/j.matcom.2024.01.001 |
| [67] | Qureshi, S.; Soomro, A.; Naseem, A.; Gdawiec, K.; Argyros, IK; Alshaery, AA; Secer, A., From Halley to Secant: redefining root finding with memory-based methods including convergence and stability, Math. Methods Appl. Sci., 47, 7, 5509-5531, 2024 · Zbl 07869339 · doi:10.1002/mma.9876 |
| [68] | Podlubny, I., Fractional Differential Equations, 1999, Academic Press · Zbl 0924.34008 |
| [69] | Li, H-L; Zhang, L.; Hu, C.; Jiang, Y-L; Teng, Z., Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 54, 1, 435-449, 2017 · Zbl 1377.34062 · doi:10.1007/s12190-016-1017-8 |
| [70] | Jodar, L.; Villanueva, RJ; Arenas, AJ; Gonalez, GC, Nonstandard numerical methods for a mathematical model for influenza disease, Math. Comput. Simul., 79, 3, 622-633, 2008 · Zbl 1151.92018 · doi:10.1016/j.matcom.2008.04.008 |
| [71] | Diethelm, K.; Ford, NJ, Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 2, 229-248, 2002 · Zbl 1014.34003 · doi:10.1006/jmaa.2000.7194 |
| [72] | Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5, 08, 1998 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
