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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Fractional Differential Equations System for Commercial Fishing under Predator-Prey Interaction

Journal of Applied Nonlinear Dynamics 2(4) (2013) 409--417 | DOI:10.5890/JAND.2013.11.007

G.H. Erjaee; M.H. Ostadzad; K. Okuguchi; E. Rahimi

$^{1}$ Mathematics Department, Shiraz University, Shiraz, Iran

$^{2}$ Department of Economics, Tokyo Metropolitan University, Hachioji-shi, Tokyo, Japan

$^{3}$ Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran

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In this article we introduce and analyze a mathematical model of commercial fishing as a dynamical system presented by fractional differential equations (FDE). In the model two different species of fish are interacting as predator and prey. We discuss the long-run properties of this system, including stability of equilibrium points, non-existence of limit cycle and periodic solution for the movement of fish stocks for both species. In deriving our mathematical model, instead of customary use of classical positive integer order of differential equations, we use fractional order derivatives. As we will discuss, non-local properties of FDE and its advantages in modeling problems with non-smooth domains will yield more accurate results in comparison with ordinary differential equation counterpart.


  1. [1]  Ruseski, G. (1998), International fish wars: the strategic roles for fleet licensing and effort subsidies, Journal of Environmental Economics and Management, 36, 70–88.
  2. [2]  Szidarovszky, F., Okuguchi, K. and Kope, M. (2005), International fishery with several countries, Pure Mathematics and Applications, 16, 493.
  3. [3]  Erjaee, G. H. and Okuguchi, K. (2006), Bifurcation and stability in imperfectly competitive international commercial fishing, Keio Economic Studies, 43, 61.
  4. [4]  Das, T., Mukherjee, R. N. and Chaudhuri, K. S. (2009), Harvesting of a prey-predator fishery in the presence of toxicity, Applied Mathematical Modelling, 33, 2282.
  5. [5]  Kar T.K. and Chakraborty, K. (2010), Effort dynamics in a prey-predator model with harvesting, International Journal of Information and System Sciences, 6, 318.
  6. [6]  Okuguchi, K. (1984), Commercial fishing with predator-prey interaction, Keio Economic Studies 21, 37.
  7. [7]  Borredon, L., Henry, B., & Wearne, S. (1999). Differentiating the non-differentiable fractional calculus, Parabola, 35(2) 1.
  8. [8]  Baleanu, D., Defterli, O. and Agrawal, O.P. (2009), A central difference numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 15(4), 583.
  9. [9]  Oustaloup, A. (1981), Fractional order sinusoidal oscillators: optimization and their use in highly linear F. M. modulation, IEEE Transactions on Circuits and Systems, 28(10), 1007.
  10. [10]  Machado, J.T., Kiryakova, V. and Mainardi, F. (2011), Recent history of fractional calculus, Communi cations in Nonlinear Science and Numerical Simulation, 16(3), 1140.
  11. [11]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, New York.
  12. [12]  Oldham, K.B. and Spanier, J. (1974), The Fractional Calculus, Academic Press, New York.
  13. [13]  Matignon, D. (1996), Stability results of fractional differential equations with applications to control processing, in: IMACS, IEEE-SMC, Lille, France, 963.
  14. [14]  Clark, C.W. (1976), Mathematical Bio-economics, John Wiley, New York.
  15. [15]  Tavazoei, M.S. and Haeri, M. (2009), A proof for non-existence of periodic solutions in time invariant fractional order systems, Automatica, 45, 1886.
  16. [16]  Gandolfo, G. (1997), Economic Dynamics, study edition, Springer-Verlag, Berlin.