带有经济效益的时滞分数阶微分-代数捕食-被捕食系统的Hopf分岔

摘要/Abstract

摘要：

该文首先提出了一类带有经济效益的时滞分数阶微分-代数捕食-被捕食系统. 利用稳定性理论, 得到了在零经济收益条件下, 系统的正平衡点是局部渐近稳定的; 在正经济收益条件下, 时滞产生Hopf分岔的充分条件. 最后借助于数值模拟验证了理论的正确性, 并进一步讨论了分数阶阶数、经济收益和时滞对系统稳定性的影响.

关键词: 分数阶模型, Hopf分岔, 时滞

Abstract:

A fractional differential-algebraic biological economic system with harvest and time delay is firstly proposed and investigated in this paper. By using the Hopf bifurcation theory, some sufficient conditions for the existence of Hopf bifurcation induced by delay are obtained. The results show that in the case of zero economic profit, the biological equilibrium point of the system is asymptotically stable. Under positive economic profit condition, the system produces limit cycles at the positive equilibrium point as the delay increases through a certain threshold. It is found that fractional exponent, economic interest and delay can affect the other dynamic behavior of the system through some numerical simulations.

Key words: Fractional system, Hopf bifurcation, Time delay

中图分类号:

O175.1

张道祥,李奔,陈丹丹,林雅婷,王鑫梅. 带有经济效益的时滞分数阶微分-代数捕食-被捕食系统的Hopf分岔[J]. 数学物理学报, 2022, 42(2): 570-582.

Daoxiang Zhang,Ben Li,Dandan Chen,Yating Lin,Xinmei Wang. Hopf Bifurcation for a Fractional Differential-Algebraic Predator-Prey System with Time Delay and Economic Profit[J]. Acta mathematica scientia,Series A, 2022, 42(2): 570-582.

参考文献

1	Legendre L . The significance of microalgal blooms for fisheries and for the export of particulate organic carbon in oceans. Journal of Plankton Research, 1990, 12 (4): 681- 699
2	Liu X X , Huang Q D . The dynamics of a harvested predator-prey system with Holling type IV functional response. BioSystems, 2018, 169: 26- 39
3	Nosrati K , Shafiee M . Dynamic analysis of fractional-order singular Holling type-Ⅱ predator-prey system. Applied Mathematics and Computation, 2017, 313: 159- 179
4	Scheffer M . Fish and nutrients interplay determines algal biomass: a minimal model. Oikos, 1991, 62: 271- 282
5	Rihan F A , Anwar M N . Qualitative analysis of delayed SIR epidemic model with a saturated incidence rate. International Journal of Differential Equations, 2012,
6	Zhang D X , Sun G X , Zhao L X , Yan P . Pattern formation and selection in a diffusive predator-prey system with ratio-dependent functional response. Acta Ecologica Sinica, 2017, 38 (5): 290- 297
7	Hillary R , Bees M . Plankton lattices and the role of chaos in plankton patchiness. Physical Review E, 2004, 69 (3): 1- 11
8	Han R J , Dai B X . Hopf bifurcation in a reaction-diffusive two-spcies model with nonlocal delay effect and general functional response. Chaos Solitons and Fractals, 2017, 96: 90- 109
9	Rihan F A . Numerical modelling in biosciences using delay differential equations. Joural of Computational and Applied Mathematics, 2000, 125 (1/2): 183- 199
10	Zhang D X , Ding W W , Zhu M . Existence of positive periodic solutions of competitor-competitor-mutualist Lotka-Volterra systems with infinite delays. Journal of Systems Science and Complexity, 2015, 28 (2): 316- 326
11	Rihan F A . Sensitivity analysis of dynamic systems with time lags. Joural of Computational and Applied Mathematics, 2003, 151 (2): 445- 462
12	Liao T C, Yu H G, Zhao M. Dynamics of delayed phytoplankton-zooplankton system with Crowley-Martin functional response. Advances in Difference Equations, 2017, 2017, Article number: 5
13	Crowley P H , Martin E K . Functional response and interference within and between year classes of a dragonfly population. Journal of the North American Benthological Society, 1989, 8 (3): 211- 221
14	Zhao H Y , Zhang X B , Huang X X . Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion. Applied Mathematics and Computation, 2015, 266: 462- 480
15	Battaglia J L , Cois O , Puigsegur L , Oustaloup A . Solving an inverse heat conduction problem using a non-integer identified model. International Journal of Heat and Mass Transf, 2001, 44 (14): 2671- 2680
16	Mainardi F. Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics//Carpenteri A, Mainardi F. Fractals and Fractional Calculus in Continuum Mechanics. Wien: Springer, 1997: 291-348
17	Hilfer R . Applications of Fractional Calculus in Physics. Singapore: World Scientific, 2000
18	Tlacuahuac A F , Biegler L T . Optimization of fractional order dynamic chemical processing systems. Industrial and Engineering Chemistry Research, 2014, 53 (13): 5110- 5127
19	Ahmed E , El-Sayed A M A , El-Saka H A . Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. Journal of Mathematical Analysis and Applications, 2007, 325 (1): 542- 553
20	Huang C D , Li H , Cao J D . A novel strategy of bifurcation control for a delayed fractional predator-prey model. Applied Mathematics and Computation, 2019, 347: 808- 838
21	Abdelouahab M S , Hamri N E , Wang J . Hopf bifurcation and chaos in fractional-order modified hybrid optical system. Nonlinear Dynamics, 2011, 69 (1/2): 275- 284
22	Gutierrez-Vega J C . Fractionalization of optical beams: I. Planar analysis. Optics Letters, 2007, 32 (11): 1521- 1523
23	Li H , Zhang L , Hu C . Dynamic analysis of a fractional-order single-species model with diffusion. Nonlinear Analysis: Modelling and Control, 2017, 22 (3): 303- 316
24	Gordon H . The economic theory of a common property resource: The fishery. Journal of Political Economy, 1954, 62: 124- 142
25	Zhang X , Zhang Q , Zhang Y . Bifurcations of a class of singular biological economic models. Chaos Solitons and Fractals, 2009, 40 (3): 1309- 1318
26	Zhang X , Zhang Q Q . Bifurcation analysis and control of a differential-algebraic predator-prey model with Allee effect and time delay. Journal of Applied Mathematics, 2014,
27	Zhang G , Zhu L , Chen B . Hopf bifurcation and stability for a differential-algebraic biological economic system. Applied Mathematics and Computation, 2010, 217 (1): 330- 338
28	El-Saka H A A , Lee S , Jang B . Dynamic analysis of fractional-order predator-prey biological economic system with Holling type Ⅱ functional response. Nonlinear Dynamics, 2019, 96: 407- 416
29	Nosrati K , Shafiee M . Dynamic analysis of fractional-order singular Holling type-Ⅱ predator-prey system. Applied Mathematics and Computation, 2017, 313: 159- 179

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