DYNAMIC ANALYSIS AND OPTIMAL CONTROL OF A FRACTIONAL ORDER SINGULAR LESLIE-GOWER PREY-PREDATOR MODEL

doi:10.1007/s10473-020-0520-z

Abstract

Abstract: In this article, we investigate a fractional-order singular Leslie-Gower prey-predator bioeconomic model, which describes the interaction between populations of prey and predator, and takes into account the economic interest. We firstly obtain the solvability condition and the stability of the model system, and discuss the singularity induced bifurcation phenomenon. Next, we introduce a state feedback controller to eliminate the singularity induced bifurcation phenomenon, and discuss the optimal control problems. Finally, numerical solutions and their simulations are considered in order to illustrate the theoretical results and reveal the more complex dynamical behavior.

Key words: fractional order system, differential-algebraic system, prey-predator bioeconomic model, singularity induced bifurcation, optimal control

CLC Number:

34A08

Linjie MA, Bin LIU. DYNAMIC ANALYSIS AND OPTIMAL CONTROL OF A FRACTIONAL ORDER SINGULAR LESLIE-GOWER PREY-PREDATOR MODEL[J].Acta mathematica scientia,Series B, 2020, 40(5): 1525-1552.

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References

[1] Huang C X, Huang Y C, Huang L H, et al. Dynamical behaviors of a food-chain model with stage structure and time delays. Adv Differ Equ, 2018, 2018(186):1-26
[2] Huang C X, Zhang H, Huang L H. Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term. Commun Pure Appl Anal, 2019, 18(6):3337-3349
[3] Clark C W. Mathematical models in the economics of renewable resources. SIAM Review, 197921(1):81-99
[4] Du Y, Peng R, Wang M. Effect of a protection zone in the diffusive Leslie predator-prey model. J Differ Equ, 2009, 246(10):3932-3956
[5] Ji C, Jiang D, Shi N. Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation. J Math Anal Appl, 2009, 359(2):482-498
[6] Ji C, Jiang D, Shi N. A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation. J Math Anal Appl, 2011, 377(1):435-440
[7] Gupta R P, Chandra P. Bifurcation analysis of modified Leslie-Gower predator-prey model with MichaelisMenten type prey harvesting. J Math Anal Appl, 2013, 398(1):278-295
[8] Gupta R P, Banerjee M, Chandra P. Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting. Differ Equ Dyn Syst, 2012, 20(3):339-366
[9] Chakraborty K, Das K, Yu H. Modeling and analysis of a modified Leslie-Gower type three species food chain model with an impulsive control strategy. Nonlinear Analysis:Hybrid Systems, 2015, 15:171-184
[10] Gordon H S. The economic theory of a common property resource:The fishery. J Political Economy, 1954, 62(2):124-142
[11] Kaczorek T. Reduced-order fractional descriptor observers for a class of fractional descriptor continuoustime nonlinear systems. Int J Appl Math Comput Sci, 2016, 26(2):277-283
[12] Liu C, Lu N, Zhang Q. Modeling and analysis in a prey-predator system with commercial harvesting and double time delays. Appl Math Comput, 2016, 281:77-101
[13] Zhang Q, Liu C, Zhang X. Complexity, Analysis and Control of Singular Biological Systems. London:Springer, 2012
[14] Zhao H, Zhang X, Huang X. Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion. Appl Math Comput, 2015, 266:462-480
[15] Zhang Y, Zhang Q, Yan X G. Complex dynamics in a singular Leslie-Gower predator-prey bioeconomic model with time delay and stochastic fluctuations. Physica A:Stat Mech Appl, 2014, 404(24):180-191
[16] Chakraborty K, Das S, Kar T K. Optimal control of effort of a stage structured prey-predator fishery model with harvesting. Nonlinear Analysis:Real World Applications, 2011, 12(6):3452-3467
[17] Glöckle W G, Nonnenmacher T F. A fractional calculus approach to self-similar protein dynamics. Biophysical Journal, 1995, 68(1):46-53
[18] Heymans N, Podlubny I. Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta, 2005, 45(5):765-771
[19] Hilfer R. Applications of Fractional Calculus in Physics. Singapore:World Scientific, 2000
[20] Podlubny I. Fractional Differential Equations. San Diego:Mathematics in Science and Engineering, 1999
[21] Li X W, Liu Z H, Li J, et al. Existence and controllability for nonlinear fractional control systems with damping in Hibert spaces. Acta Math Sci, 2019, 39B(1):229-242
[22] Hu C Z, Liu B, Xie S F. Monotone iterative solutions for nonlinear boundary value problems of fractional differential equation with deviating arguments. Appl Math Comput, 2013, 222(1):72-81
[23] Jian H, Liu B, Xie S F. Monotone iterative solutions for nonlinear fractional differential systemswith deviating arguments. Appl Math Comput, 2015, 262(1):1-14
[24] Wei Z, Li Q, Che J. Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J Math Anal Appl, 2010, 367(1):260-272
[25] Ghaziani R K, Alidousti J, Eshkaftaki A B. Stability and dynamics of a fractional order Leslie-Gower prey-predator model. Appl Math Modell, 2016, 40(3):2075-2086
[26] Moustafa M, Mohd M H, Ismail A I, et al. Dynamical analysis of a fractional-order Rosenzweig-MacArthur model incorporating a prey refuge. Chaos Solitons and Fractals, 2018, 109:1-13
[27] Huang C, Cao J, Xiao M, et al. Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders. Appl Math Comput, 2017, 293:293-310
[28] Dai L. Singular Control Systems. Berlin:Springer, 1989
[29] N'Doye I, Darouach M, Zasadzinski M, et al. Robust stabilization of uncertain descriptor fractional-order systems. Automatica, 2013, 49(6):1907-1913
[30] Nosrati K, Shafiee M. Fractional-order singular logistic map:stability, bifurcation and chaos analysis. Chaos, Solitons and Fractals, 2018, 115:224-238
[31] Nosrati K, Shafiee M. Kalman filtering for discrete-time linear fractional-order singular systems. IET Control Theory and Applications, 2018, 12(9):1254-1266
[32] Kaczorek T, Rogowski K. Fractional Linear Systems and Electrical Circuits. Switzerland:Springer, 2015
[33] Nosrati K, Shafiee M. Dynamic analysis of fractional-order singular Holling type-Ⅱ predator-prey system. Appl Math Comput, 2017, 313:159-179
[34] Kilbas A, Srivastava H, Trujillo J. Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier, 2006
[35] Kai D. The Analysis of Fractional Differential Equations. Berlin:Springer, 2010
[36] Zhang H, Cao J D, Jiang W. Reachability and controllability of fractional singular dynamical systems with control delay. J Appl Math, 2013, 2013(1):1-10
[37] Petráš I. Fractional-Order Nonlinear Systems. Beijing:Higher Education Press, 2011
[38] Zhang X F, Chen Y Q. Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order, α:The 0 < α < 1 case. ISA Transactions, 2018, 82:42-50
[39] Yang C, Zhang Q, Zhou L. Stability Analysis and Design for Nonlinear Singular Systems. Berlin:Springer, 2013
[40] Venkatasubramanian V, Schattler H, Zaborszky J. Local bifurcations and feasibility regions in differentialalgebraic systems. IEEE Trans Automatic Control, 1995, 40(12):1992-2013
[41] Beardmore R E. The singularity-induced bifurcation and its Kronecker normal form. SIAM J Matrix Anal Appl, 2001, 23(1):126-137
[42] Agrawal O P. A formulation and numerical scheme for fractional optimal control problems. J Vibr Contr, 2008, 14(9/10):1291-1299
[43] Atanackovic T M, Stankovic B. On a numerical scheme for solving differential equations of fractional order. Mechanics Research Communications, 2008, 35(7):429-438