zbMATH — the first resource for mathematics

Numerical solutions of a fractional predator-prey system. (English) Zbl 1217.35205
Summary: We implement a relatively new analytical technique, the homotopy perturbation method, for solving nonlinear fractional partial differential equations arising in predator-prey biological population dynamics system. Numerical solutions are given, and some properties exhibit biologically reasonable dependence on the parameter values. And the fractional derivatives are described in the Caputo sense.

35R11 Fractional partial differential equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)
37N25 Dynamical systems in biology
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
26A33 Fractional derivatives and integrals
Full Text: DOI EuDML
[1] Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, New York, NY, USA; 1999:xxiv+340. · Zbl 0924.34008
[2] Metzler, R; Klafter, J, The random walks guide to anomalous diffusion: a fractional dynamics approach, Physics Reports A, 339, 1-77, (2000) · Zbl 0984.82032
[3] Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000:viii+463. · Zbl 0998.26002
[4] Golmankhaneh, AK; Golmankhaneh, AK; Baleanu, D, On nonlinear fractional Klein-Gordon equation, Signal Processing, 91, 446-451, (2011) · Zbl 1203.94031
[5] Rida, SZ; El-Sherbiny, HM; Arafa, AAM, On the solution of the fractional nonlinear Schrödinger equation, Physics Letters A, 372, 553-558, (2008) · Zbl 1217.81068
[6] Jiang, XY; Xu, MY, Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media, International Journal of Non-Linear Mechanics, 41, 156-165, (2006)
[7] Wang, S; Xu, M, Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus, Nonlinear Analysis: Real World Applications, 10, 1087-1096, (2009) · Zbl 1167.76311
[8] He, J-H, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257-262, (1999) · Zbl 0956.70017
[9] He, J-H, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Non-Linear Mechanics, 35, 37-43, (2000) · Zbl 1068.74618
[10] He, J-H, The homotopy perturbation method nonlinear oscillators with discontinuities, Applied Mathematics and Computation, 151, 287-292, (2004) · Zbl 1039.65052
[11] He, JH, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons & Fractals, 26, 695-700, (2005) · Zbl 1072.35502
[12] Li, X; Xu, M; Jiang, X, Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition, Applied Mathematics and Computation, 208, 434-439, (2009) · Zbl 1159.65106
[13] Wang, Q, Homotopy perturbation method for fractional KdV-Burgers equation, Chaos, Solitons & Fractals, 35, 843-850, (2008) · Zbl 1132.65118
[14] Momani, S; Odibat, Z, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A, 365, 345-350, (2007) · Zbl 1203.65212
[15] Odibat, Z; Momani, S, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos, Solitons & Fractals, 36, 167-174, (2008) · Zbl 1152.34311
[16] Shakeri, F; Dehghan, M, Numerical solution of a biological population model using He’s variational iteration method, Computers & Mathematics with Applications, 54, 1197-1209, (2007) · Zbl 1137.92033
[17] Rida, SZ; Arafa, AAM, Exact solutions of fractional-order biological population model, Communications in Theoretical Physics, 52, 992-996, (2009) · Zbl 1184.92038
[18] Tan, Y; Xu, H; Liao, S-J, Explicit series solution of travelling waves with a front of Fisher equation, Chaos, Solitons & Fractals, 31, 462-472, (2007) · Zbl 1143.35313
[19] Petrovskii, S; Shigesada, N, Some exact solutions of a generalized Fisher equation related to the problem of biological invasion, Mathematical Biosciences, 172, 73-94, (2001) · Zbl 0983.92031
[20] Dunbar, SR, Travelling wave solutions of diffusive Lotka-Volterra equations, Journal of Mathematical Biology, 17, 11-32, (1983) · Zbl 0509.92024
[21] Gourley, SA; Britton, NF, A predator-prey reaction-diffusion system with nonlocal effects, Journal of Mathematical Biology, 34, 297-333, (1996) · Zbl 0840.92018
[22] Petrovskii, S; Malchow, H; Li, B-L, An exact solution of a diffusive predator-prey system, Proceedings of The Royal Society of London A, 461, 1029-1053, (2005) · Zbl 1145.92341
[23] Kadem, A; Baleanu, D, Homotopy perturbation method for the coupled fractional Lotka-Volterra equations, No. 56, (2011) · Zbl 1231.65134
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.