Dynamical analysis of fractional-order Holling type-II food chain model

Cuimin Liu; Zhen Wang; Bo Meng; Cuimin Liu; Zhen Wang; Bo Meng

doi:10.3934/mbe.2021265

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 5: 5221-5235. doi: 10.3934/mbe.2021265

Previous Article Next Article

Research article

Dynamical analysis of fractional-order Holling type-II food chain model

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Received: 05 April 2021 Accepted: 27 May 2021 Published: 11 June 2021

This paper proposed a fractional-order Holling type-II food chain model. First, we verified the existence, uniqueness, nonnegativity and boundedness of the solution of the model, and some conditions for equilibrium existence and local stability were studied. Second, a controller was proposed, and the Lyapunov method was used to study the global stability of the positive equilibrium point. Finally, numerical simulations were performed to verify the theoretical results.
- prey-predator model,
- fractional-order system,
- global stability,
- Holling II functional response
Citation: Cuimin Liu, Zhen Wang, Bo Meng. Dynamical analysis of fractional-order Holling type-II food chain model[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 5221-5235. doi: 10.3934/mbe.2021265

Related Papers:

Abstract

This paper proposed a fractional-order Holling type-II food chain model. First, we verified the existence, uniqueness, nonnegativity and boundedness of the solution of the model, and some conditions for equilibrium existence and local stability were studied. Second, a controller was proposed, and the Lyapunov method was used to study the global stability of the positive equilibrium point. Finally, numerical simulations were performed to verify the theoretical results.

References

[1]	Z. Wang, Y. Xie, J. Lu, Y. Li, Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition, App. Math. Comput., 347 (2019), 360-369.
[2]	A. J. Lotka, Elements of Physical Biology, Dover Publications, 1956.
[3]	C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Cambridge University Press, Cambridge, 2012.
[4]	B. Ritwick, D. Pritha, M. Debasis, Global dynamics of a Holling Type-III two prey one predator discrete model with optimal harvest strategy, Nonlinear Dynam., 99 (2020), 3285-3300. doi: 10.1007/s11071-020-05490-0
[5]	A. Hastings, T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903. doi: 10.2307/1940591
[6]	K. Das, S. Chatterjee, J. Chattopadhyay, Disease in prey population and body size of intermediate predator reduce the prevalence of chaos-conclusion drawn from Hastings-Powell model, Ecol. Complex, 6 (2009), 363-374. doi: 10.1016/j.ecocom.2009.03.003
[7]	C. Tian, L. Zhang, Hopf bifurcation analysis in a diffusive food-chain model with time delay, Comput. Math. Appl., 66 (2013), 2139-2153. doi: 10.1016/j.camwa.2013.09.002
[8]	Y. Chen, J. Yu, C. Sun, Stability and Hopf bifurcation analysis in a three-level food chain system with delay, Chaos Soliton Fract., 31 (2007), 683-694. doi: 10.1016/j.chaos.2005.10.020
[9]	A. E. Matouk, A. A. Elsadany, E. Ahmed, H. N. Agiza, Dynamical behavior of fractional-order Hastings-Powell food chain model and its discretization, Commun. Nonlinear Sci., 27 (2015), 153-167. doi: 10.1016/j.cnsns.2015.03.004
[10]	I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[11]	X. Wang, Z. Wang, J. Xia, Stability and bifurcation control of a delayed fractional-order eco-epidemiological model with incommensurate orders, J. Franklin. I., 356 (2019), 8278-8295. doi: 10.1016/j.jfranklin.2019.07.028
[12]	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.
[13]	K. M. Owolabi, B. Karaagac, D. Baleanu, Pattern formation in superdiffusion predator-prey-like problems with integer- and noninteger-order derivatives, Math. Method Appl. Sci., 44 (2020), 4018-4036.
[14]	C. Wu, Comments on "Stability analysis of Caputo fractional-order nonlinear systems revisited", Nonlinear Dynam., 104 (2021), 551-555. doi: 10.1007/s11071-021-06279-5
[15]	K. M. Owolabi, B. Karaagac, Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system, Chaos Soliton Fract., 141 (2020), 110302. doi: 10.1016/j.chaos.2020.110302
[16]	K. M. Owolabi, Dynamical behaviour of fractional-order predator-prey system of Holling-type, Discrete. Cont. Dyn-S., 13 (2018), 823-834.
[17]	N. Sene, Introduction to the fractional-order chaotic system under fractional operator in Caputo sense, Alex. Eng. J., 60 (2021), 3997-4014. doi: 10.1016/j.aej.2021.02.056
[18]	H. Delavari, D. Baleanu, J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dynam., 67 (2012), 2433-2439.
[19]	K. M. Owolabi, High-dimensional spatial patterns in fractional reaction-diffusion system arising in biology, Chaos Soliton Fract., 134 (2020), 109723. doi: 10.1016/j.chaos.2020.109723
[20]	K. M. Owolabi, D. Baleanu, Emergent patterns in diffusive Turing-like systems with fractional-order operator, Neural Comput. Appl., (2021), 1-18.
[21]	K. M. Owolabi, Numerical simulation of fractional-order reaction-diffusion equations with the Riesz and Caputo derivatives, Neural Comput. Appl., 32 (2019), 4093-4104.
[22]	K. M. Owolabi, study of symbiosis dynamics via the Caputo and Atangana-Baleanu fractional derivatives, Chaos Soliton Fract., 122 (2019), 89-101. doi: 10.1016/j.chaos.2019.03.014
[23]	E. Ahmed, A. M. A. El-Sayed, H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl., 325 (2007), 542-553. doi: 10.1016/j.jmaa.2006.01.087
[24]	Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810-1821. doi: 10.1016/j.camwa.2009.08.019
[25]	A. Jajarmi, M. Hajipour, E. Mohammadzadeh, D.Baleanu, A new approach for the nonlinear fractional optimal control problems with external persistent disturbances, J. Franklin I., 355 (2018), 3938-3967. doi: 10.1016/j.jfranklin.2018.03.012
[26]	N. Jia, L. Ding, Y. Liu, P. Hu, Global stability and optimal control of epidemic spreading on multiplex networks with nonlinear mutual interaction, Physica A, 502 (2018), 0378-4371.
[27]	S. Rosa, D. F. M. Torres, Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection, Chaos Soliton Fract., 117 (2018), 142-149. doi: 10.1016/j.chaos.2018.10.021
[28]	H. Kheiri, M. Jafari, Fractional optimal control of an HIV/AIDS epidemic model with random testing and contact tracing, J. Appl. Math. Comput., 60 (2019), 387-411. doi: 10.1007/s12190-018-01219-w
[29]	C. Xu, M. Liao, P. Li, Bifurcation control for a fractional-order competition model of Internet with delays, Nonlinear Dynam., 95 (2019), 3335-3356. doi: 10.1007/s11071-018-04758-w
[30]	S. Gakkhar, A. Singh, Control of chaos due to additional predator in the Hastings-Powell food chain model, J. Math. Anal. Appl., 385 (2012), 423-438. doi: 10.1016/j.jmaa.2011.06.047
[31]	M. Yavuz, N. Sene, Stability analysis and numerical computation of the fractional Predator-Prey model with the harvesting rate, Fractal Fract., 4 (2020).
[32]	H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2017), 1075-1081.
[33]	V. D. L. Cruz, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci., 24 (2015), 75-85. doi: 10.1016/j.cnsns.2014.12.013
[34]	F. Mansal, N. Sene, Analysis of fractional fishery model with reserve area in the context of time-fractional order derivative, Chaos Soliton Fract., 140 (2020), 110200. doi: 10.1016/j.chaos.2020.110200
[35]	E. Ahmed, E. S. Ama, E. S. Haa, On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems, Phys. Lett. A, 358 (2006), 1-4. doi: 10.1016/j.physleta.2006.04.087
[36]	A. E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified Autonomous Van der Pol-Duffing circuit, Commun. Nonlinear Sci., 16 (2011), 975-986.
[37]	D. Kai, N. J. Ford, A. D. Freed, A Predictor-Corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22. doi: 10.1023/A:1016592219341
[38]	D. Kai, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52. doi: 10.1023/B:NUMA.0000027736.85078.be
[39]	X. Wang, Z. Wang, J. Lu, B. Meng, Stability, bifurcation and chaos of a discrete-time pair approximation epidemic model on adaptive networks, Math. Comput. Simulat., 182 (2021), 182-194. doi: 10.1016/j.matcom.2020.10.019

Reader Comments

Your name:*

Email:*
© 2021 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)