Abstract：At present, most ecological competition networks are characterized by integer-order systems, and system behavior is only affected by the current moment. In this paper, a fractional time-delay predator-prey model with independent non-cross propagation is proposed. Time delay is selected as the bifurcation parameter. The stability and bifurcation problems of the model are studied through analyzing the distribution of the roots of the corresponding characteristic equation under the influence of different orders, and the stability conditions and Hopf bifurcation criteria induced by time delay are established. Finally, the accuracy of the theoretical results is verified by numerical simulations.
[1] VOLTERRA V. Fluctuations in the abundance of a species considered mathematically[J]. Nature, 1926, 118(2972): 558-560. [2] LOTKA A J. Elements of Physical Biology[M]. London: Dover, 1956. [3] Meats A. The relative importance to population increase of fluctuations in mortality, fecundity and the time variables of the reproductive schedule[J]. Oecologia, 1971, 6(3): 223-237. [4] MA L, GUO S J. Stability and bifurcation in a diffusive Lotka-Volterra system with delay[J]. Computers and Mathematics with Applications, 2016, 72(1): 147-177. [5] DAWES J H P, SOUZA M O. A derivation of Holling's type I, II and III functional responses in predator-prey systems [J]. Journal of Theoretical Biology, 2013, 327(1): 11-22. [6] SEO G, KOT M. A comparison of two predator-prey models with Holling's type I functional response[J]. Mathematical Biosciences, 2008, 212(2): 161-179. [7] WOLVERTON S. Harvest pressure and environmental carrying capacity: an ordinal-scale model of effects on ungulate prey[J]. American Antiquity, 2008, 73(2): 179-199. [8] LI C H. Dynamics of a network-based SIS epidemic model with nonmonotone incidence rate[J]. Physica A: Statistical Mechanics and its Applications, 2015, 427: 234-243. [9] LIU B, TENG Z D, CHEN L S. Analysis of a predator-prey model with Holling-II functional response concerning impulsive control strategy[J]. Computational and Applied Mathematics, 2006, 193(1): 347-362. [10] 张拥军, 王美娟, 徐金瑞. 捕食者具有传染病的时滞捕食系统模型分析[J]. 复杂系统与复杂性科学, 2009, 6(4): 83-90. ZHANG Y J, WANG M J, XU J R. Time-delay pradator-prey model with disease in the predator[J]. Complex Systems and Complexity Science, 2009, 6(4): 83-90. [11] 林蓓,祝光湖,傅新楚. 具有出生和死亡的复合种群疾病传播动力学[J]. 复杂系统与复杂性科学, 2013, 10(4): 56-61. LIN B, ZHU G H, FU X C. The dynamics of an mata-population model with birth and death[J]. Complex Systems and Complexity Science, 2013, 10(4): 56-61. [12] WANG L, XU R, FENG G. Modelling and analysis of an eco-epidemiological model with time delay and stage structure[J]. Journal of Applied Mathematics and Computing, 2015, 50(1): 175-197. [13] 李鞠鑫. 一类有接触性跨物种传染病的捕食-被捕食模型[D]. 大连:大连理工大学, 2017. LI J X. A predator-prey model of cross-species infectious diseases transmitted by contact[D]. Dalian: Dalian University of Technology, 2017. [14] 芦雪娟. 斑块扩散的生态与传染病动力模型研究[D]. 哈尔滨:哈尔滨工业大学, 2018. LU X J. Studies on patch diffusion ecological and epidemic dynamical models[D]. Harbin: Harbin Institute of Technology, 2018. [15] TOBIA D. Uniform persistence in a prey-predator model with a diseased predator[J]. Journal of Mathematical Biology, 2020, 80(4): 1077-1093. [16] CHATTOPADHYAY J, ARINO O. A predator-prey model with disease in the prey[J]. Nonlinear Analysis Theory Methods and Applications, 1999, 36(6): 747-766. [17] ZHANG J F, LI W T, YAN X P. Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system[J]. Applied Mathematics and Computation, 2008, 198(2): 865-876. [18] WANG X, WANG Z, SHEN H. Dynamical analysis of a discrete-time SIS epidemic model on complex networks[J]. Applied Mathematics Letters. 2019, 94: 292-299. [19] HUANG C X, ZHANG H, CAO J D, et al. Stability and hopf bifurcation of a delayed prey-predator model with disease in the predator[J]. International Journal of Bifurcation and Chaos, 2019, 29(7): 1950091. [20] MCCLUSKEY C C. Complete global stability for an SIR epidemic model with delay-distributed or discrete[J]. Nonlinear Analysis Real World Applications, 2010, 11(1): 55-59. [21] BERETTA E, TAKEUCHI Y. Global stability of an SIR epidemic model with time delays[J]. Journal of Mathematical Biology, 1995, 33(3): 250-260. [22] WANG X H, WANG Z, XIA J W. Stability and bifurcation control of a delayed fractional-order eco-epidemiological model with incommensurate orders[J]. Journal of the Franklin Institute, 2019, 356(15): 8278-8295. [23] 朱伟, 陈坤, 王谦, 等. 基于自适应滑模控制的分数阶蔡氏电路系统动力学分析与控制[J]. 复杂系统与复杂性科学, 2018, 15(2): 92-98. ZHU W, CHEN K, WANG Q, et al. Dynamical analysis and control of fractional-order Chua's circuit system based on adaptive slding model control[J]. Complex Systems and Complexity Science, 2018, 15(2): 92-98. [24] 赵小国, 阎晓妹, 孟欣. 基于RBF神经网络的分数阶混沌系统的同步[J]. 复杂系统与复杂性科学, 2010, 7(1): 40-46. ZHAO X G, YAN X M, MENG X. Synchronization of fractional order chaotic systems via RBF neural network[J]. Complex Systems and Complexity Science, 2010, 7(1): 40-46. [25] KUMAR S, KUMAR R, AGARWAL R P, et al. A study of fractional lotka-volterra population model using haar wavelet and adams-bashforth-moulton methods[J]. Mathematical Methods in the Applied Sciences, 2020, 43(8): 5564-5578. [26] 高金峰, 张成芬. 基于非线性观测器实现一类分数阶混沌系统的同步[J]. 复杂系统与复杂性科学, 2007, 4(2):50-55. GAO J F, ZHANG C F. Synchronization of a class of fractional-order chaotic systems by employing nonlinear observers[J]. Complex Systems and Complexity Science, 2007, 4(2):50-55. [27] ALIDOUSTI J, GHAHFAROKHI M M. Dynamical behavior of a fractional three-species food chain model[J]. Nonlinear Dynamics, 2019, 7(3): 776-784. [28] LI Q, LIANG J L. Dissipativity of the stochastic Markovian switching CVNNs with randomly occurring uncertainties and general uncertain transition rates[J]. International Journal of Systems Science, 2020, 51(6): 1102-1118. [29] ALMEIDA R, CRUZ A M C B D, MARTINS N, et al. An epidemiological MSEIR model described by the caputo fractional derivative[J]. International Journal of Dynamics and Control, 2019(3): 776-784. [30] MAITI A P, JANA C, MAITI D K. A delayed eco-epidemiological model with nonlinear incidence rate and crowley-martin functional response for infected prey and predator[J]. Nonlinear Dynamics, 2019, 98(2): 1137-1167. [31] MOUSTAFA M, MOHD M H, ISMAIL A I, et al. Dynamical analysis of a fractional order eco-epidemiological model with nonlinear incidence rate and prey refuge[J]. Journal of Applied Mathematics and Computing, 2021, 65(1-2): 623-650. [32] CHINNATHAMBI R, RIHAN F A. Stability of fractional-order prey-predator system with time-delay and monod-haldane functional response[J]. Nonlinear Dynamics, 2018, 92(4): 1637-1648. [33] WANG Z, XIE Y K, LU J W, et al. Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition[J]. Applied Mathematics and Computation, 2019, 34(7): 360-369. [34] PODLUBNY I. Fractional Differential Equations[M]. New York: Academic Press, 1999: 1-23. [35] 高旭彬. 带有传染病的捕食-被捕食模型[D]. 大连: 大连理工大学,2015. GAO X B. The predator-prey models with epidemics[D]. Dalian: Dalian University of Technology, 2015. [36] TAO B B, XIAO M, ZHENG W X, et al. Dynamics analysis and design for a bidirectional super-ring-shaped neural network with n neurons and multiple delays[J]. IEEE Transactions on Neural Networks and Learning Systems, 2021, 32(7): 2978-2992.
