### Mathematical Biosciences and Engineering

2020, Issue 4: 2805-2819. doi: 10.3934/mbe.2020154
Research article Special Issues

# Stability analysis for a new fractional order N species network

• Received: 17 November 2019 Accepted: 10 March 2020 Published: 17 March 2020
• The present paper considers a fractional-order N species network, in which, the general functions are used for finding general theories. The existence, uniqueness, and non-negativity of the solutions for the considered model are proved. Moreover, the local and global asymptotic stability of the equilibrium point are studied by using eigenvalue method and Lyapunov direct method. Finally, some simple examples and numerical simulations are provided to demonstrate the theoretical results.

Citation: Yingkang Xie, Junwei Lu, Bo Meng, Zhen Wang. Stability analysis for a new fractional order N species network[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 2805-2819. doi: 10.3934/mbe.2020154

### Related Papers:

• The present paper considers a fractional-order N species network, in which, the general functions are used for finding general theories. The existence, uniqueness, and non-negativity of the solutions for the considered model are proved. Moreover, the local and global asymptotic stability of the equilibrium point are studied by using eigenvalue method and Lyapunov direct method. Finally, some simple examples and numerical simulations are provided to demonstrate the theoretical results.

 [1] F. Peter, Modelling with differential and difference equations, Cambridge University Press, Cambridge, 1997. [2] T. Zhang, T. Zhang, X. Meng, Stability analysis of a chemostat model with maintenance energy, Appl. Math. Lett., 68 (2017), 1-7. doi: 10.1016/j.aml.2016.12.007 [3] X. Wang, Z. Wang, H. Shen, Dynamical analysis of a discrete-time SIS epidemic model on complex networks, Appl. Math. Lett., 94 (2019), 292-299. doi: 10.1016/j.aml.2019.03.011 [4] X. Zhao, Dynamical systems in population biology, Springer, New York, 2000. [5] T. Zhang, X. Liu, X. Meng, T. Zhang, Spatio-temporal dynamics near the steady state of a planktonic system, Comput. Math. Appl., 75 (2018), 4490-4504. doi: 10.1016/j.camwa.2018.03.044 [6] X. Yu, S. Yuan, T. Zhang, Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin producing phytoplankton in an impulsive polluted environment, Appl. Math. Comput., 347 (2019), 249-264. [7] T. Zhang, Y. Xing, H. Zang, M. Han, Spatio-temporal patterns in a predator-prey model with hyperbolic mortality, Nonlinear Dyn., 78 (2014), 265-277. doi: 10.1007/s11071-014-1438-6 [8] C. L. Wolin, L. R. Lawlor, Models of facultative mutualism: density effects, Am. Nat., 124 (1984), 843-862. doi: 10.1086/284320 [9] S. Ahmad, A. C. Lazer, Average growth and total permanence in a competitive Lotka-Volterra System, Ann. Mat. Pura. Appl., 185 (2006), S47-S67. doi: 10.1007/s10231-004-0136-2 [10] T. K. Kar, H. Matsuda, Global dynamics and controllability of a harvested prey-predator system with Holling type Ⅲ functional response, Nonlinear Anal.-Hybrid Syst., 1 (2007), 59-67. doi: 10.1016/j.nahs.2006.03.002 [11] I. Al-Darabsah, X. Tang, Y. Yuan, A prey-predator model with migrations and delays, Discrete Contin. Dyn. Syst.-Ser. B, 21 (2017), 737-761. [12] B. I. Camara, M. Haque, H. Mokrani, Patterns formations in a diffusive ratio-dependent predatorprey model of interacting populations, Physica A, 461 (2016), 374-383. doi: 10.1016/j.physa.2016.05.054 [13] G. M. Abernethy, R. Mullan, D. H. Glass, M. Mccartney, A multiple phenotype predator-prey model with mutation, Physica A, 465 (2017), 762-774. doi: 10.1016/j.physa.2016.08.037 [14] P. S. Mandal, Noise-induced extinction for a ratio-dependent predator-prey model with strong Allee effect in prey, Physica A, 496 (2018), 40-52. doi: 10.1016/j.physa.2017.12.057 [15] T. W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401. doi: 10.1016/S0022-247X(02)00395-5 [16] M. Sen, M. Banerjeea, A. Morozov, Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecol. Complex, 11 (2012), 12-27. doi: 10.1016/j.ecocom.2012.01.002 [17] S. Chakraborty, S. Pal, N. Bairagi, Predator-prey interaction with harvesting: mathematical study with biological ramifications, Appl. Math. Model., 36 (2012), 4044-4059. doi: 10.1016/j.apm.2011.11.029 [18] X. Huang, Y. Fan, J. Jia, Z. Wang, Y. Li, Quasi-synchronization of fractional-order memristorbased neural networks with parameter mismatches, IET Contr. Theory Appl., 11 (2017), 2317-2327. doi: 10.1049/iet-cta.2017.0196 [19] J. Jia, X. Huang, Y. Li, J. Cao, A. Alsaedi, Global Stabilization of Fractional-Order MemristorBased Neural Networks with Time Delay, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 997-1009. doi: 10.1109/TNNLS.2019.2915353 [20] Y. Fan, X. Huang, Z. Wang, Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function, Nonlinear Dyn., 9 (2018), 1-17. [21] C. Huang, L. Cai, J. Cao, Linear control for synchronization of a fractional-order time-delayed chaotic financial system, Chaos Soliton. Fract., 113 (2018), 326-332. doi: 10.1016/j.chaos.2018.05.022 [22] X. Wang, Z. Wang, J. Xia, Stability and bifurcation control of a delayed fractional-order ecoepidemiological model with incommensurate orders, J. Frankl. Inst.-Eng. Appl. Math., 356 (2019), 8278-8295. doi: 10.1016/j.jfranklin.2019.07.028 [23] A. Alkhazzan, P. Jiang, D. Baleanu, H. Khan, A. Khan, Stability and existence results for a class of nonlinear fractional differential equations with singularity, Math. Meth. Appl. Sci., 41 (2018), 9321-9334. doi: 10.1002/mma.5263 [24] X. Wang, Z. Wang, X. Huang, Y. Li, Dynamic analysis of a delayed fractional-order SIR model with saturated incidence and treatment functions, Int. J. Bifurcat. Chaos, 28 (2018), 1850180. doi: 10.1142/S0218127418501808 [25] A. Yusuf, A. I. Aliyu, D. Baleanu, Conservation laws, soliton-like and stability analysis for the time fractional dispersive long-wave equation, Adv. Differ. Equ., 1 (2018), 319. [26] Y. Xie, J. Lu, Z. Wang, Stability analysis of a fractional-order diffused prey-predator model with prey refuges, Physica A, 526 (2019), 120773. doi: 10.1016/j.physa.2019.04.009 [27] Z. Wang, Y. Xie, J. Lu, Y. Li, Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition, Appl. Math. Comput., 347 (2019), 360-369. [28] G. C. Wu, D. Baleanu, L. L. Huang, Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse, Appl. Math. Lett., 82 (2018), 71-78. doi: 10.1016/j.aml.2018.02.004 [29] C. N. Angstmann, A. M. Erickson, B. I. Henry, A. V. McGann, J. M. Murray, J. A. Nichols, Fractional order compartment models, SIAM J. Appl. Math., 77 (2017), 430-446. doi: 10.1137/16M1069249 [30] E. Ahmed, A. S. Elgazzar, On fractional order differential equations model for nonlocal epidemics, Physica A, 379 (2012), 607-614. [31] 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 [32] C. Guo, S. Fang, Stability and approximate analytic solutions of the fractional Lotka-Volterra equations for three competitors, Adv. Differ. Equ., 1 (2016), 219. [33] A. A. Elsadany, A. E. Matouk, Dynamical behaviors of fractional-order Lotka-Volterra predatorprey model and its discretization, J. Appl. Math. Comput., 49 (2015), 269-283. doi: 10.1007/s12190-014-0838-6 [34] J. Tian, Y. Yu, H. Wang, Stability and bifurcation of two Kinds of three-dimensional fractional Lotka-Volterra systems, Math. Probl. Eng., 2014 (2014). [35] H. L. Li, L. Zhang, C. Hu, Y. L. Jiang, Z. Teng, Dynamical analysis of a fractional-order predatorprey model incorporating a prey refuge, J. Math. Anal. Appl., 54 (2016), 1-15. [36] D. Hu, H. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Anal.-Real. World Appl., 33 (2017), 58-82. doi: 10.1016/j.nonrwa.2016.05.010 [37] C. Huang, Z. Li, D. Ding, J. Cao, Bifurcation analysis in a delayed fractional neural network involving self-connection, Neurocomputing, 314 (2018), 186-197. doi: 10.1016/j.neucom.2018.06.016 [38] C. Huang, X. Song, B. Fang, M. Xiao, J. Cao, Modeling, analysis and bifurcation control of a delayed fractional-order predator-prey model, Int. J. Bifurcat. Chaos, 28 (2018), 1850117. doi: 10.1142/S0218127418501171 [39] Y. Li, Y. Q. 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 [40] G. P. Samanta, A. Maiti, M. Das, Stability analysis of a prey-predator fractional order model incorporating prey refuge, Ecol. Genet. Genomics, 7 (2018), 33-46. [41] R. Chinnathambi, F. A. Rihan, Stability of fractional-order prey-predator system with time-delay and Monod-Haldane functional response, Nonlinear Dyn., 92 (2018), 1-12. doi: 10.1007/s11071-018-4140-2 [42] N. Supajaidee, S. Moonchai, Stability analysis of a fractional-order two-species facultative mutualism model with harvesting, Adv. Differ. Equ., 1 (2017), 372. [43] J. Alidousti, Stability and bifurcation analysis for a fractional prey-predator scavenger model, Appl. Math. Model., 81 (2020), 342-355. doi: 10.1016/j.apm.2019.11.025 [44] C. Huang, H. Li, T. Li, S. Chen, Stability and bifurcation control in a fractional predator-prey model via extended delay feedback, Int. J. Bifurcat. Chaos, 29 (2019), 1950150. doi: 10.1142/S0218127419501505 [45] X. Wang, Z. Wang, X. Shen, Stability and Hopf bifurcation of a fractional-order food chain model with disease and two delays, J. Comput. Nonlinear Dyn., 15 (2020). [46] I. Podlubny, Fractional differential equations, Academic press, New York, 1999. [47] G. Ji, Q. Ge, J. Xu, Dynamic behaviors of a fractional order two-species cooperative systems with harvesting, Chaos Soliton. Fract., 92 (2016), 51-55. doi: 10.1016/j.chaos.2016.09.014 [48] C. Vargas-De-Len, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci. Numer. Simul., 24 (2015), 75-85. doi: 10.1016/j.cnsns.2014.12.013 [49] H. Delavari, D. Baleanu, J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dyn., 67 (2011), 2433-2439. [50] S. Ahmad, A. C. Lazer, On the nonautonomous N-competing species problems, Appl. Anal., 57 (2007), 309-323.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

Article outline

Figures(3)

• On This Site