# Fractional modeling for prey and predator problem by using optimal homotopy asymptotic method

## Authors

• Jafar Biazar guilan university
• Saghi Safaei guilan university

2020-09-30

## Keywords:

Optimal Homotopy Asymptotic Method, Prey and Predator Problem, Convergence Analysis, Caputo Derivative.

## Abstract

In this paper, a fractional-ordered prey and predator population model is introduced and applied to obtain an approximate solution with help of optimal homotopy asymptotic method (OHAM). Some plots for populations of the prey and the predator versus time are presented to show the efficiency and the accuracy of the method and confirm that the method is straightforward as well. The fractional derivatives are described in the Caputo sense.

Â

## References

[1] El-Borai M, El-Sayed W & M-Jawad A, Adomian Decomposition method for solving fractional differential equations, international Research Journal of Engineering and Technology, (2015).

[2] Mehmet Giyas Sakar, Fevzi Erdogan, â€œThe homotopy analysis method for solving the time-fractional Fornbergâ€“Whitham equation and comparison with Adomianâ€™s decomposition methodâ€ Applied Mathematical Modelling, vol 37, 8876-8885, (2013). https://doi.org/10.1016/j.apm.2013.03.074.

[3] J. Biazar, R. Montazeri, A computational method for solution of the prey and predator problem, Applied Mathematics and Computation 163 (2) 841â€“847, (2005). https://doi.org/10.1016/j.amc.2004.05.001.

[4] Mehmet Giyas Sakar, Hilmi ErgoÌˆren, â€œAlternative variational iteration method for solving the time-fractional Fornbergâ€“Whitham equationâ€ Applied Mathematical Modelling, Vol. 39, 3972-3979, (2015). https://doi.org/10.1016/j.apm.2014.11.048.

[5] Khan Y, Faraz N, Yildirim A & Wu Q, Fractional Variational Iteration Method for fractional initial-boundary value problems arising in the application of nonlinear science, Computers & Mathematics with Applications, 2011; 2273-2278. https://doi.org/10.1016/j.camwa.2011.07.014.

[6] Guo-Cheng Wu, A Fractional Variational Iteration Method for solving fractional nonlinear differential equations, 2011: 2186-2190. https://doi.org/10.1016/j.camwa.2010.09.010.

[7] Ibis B, Bayram M & Agargum A, Applications of fractional Differential Transform Method to fractional differential- Algebraic equations, European Journal of Pure and Applied Mathematics, 2011: 129-141.

[8] Zaid Odibat, â€œOn the optimal selection of the linear operator and the initial approximation in the application of the homotopy analysis method to nonlinear fractional differential equationsâ€ Applied Numerical Mathematics, Vol. 137, 203-212, (2019). https://doi.org/10.1016/j.apnum.2018.11.003.

[9] Ganjiani M, solution nonlinear fractional differential equations using Homotopy Analysis Method, Applied Mathematical Modelling, 2010: 1634-1641. https://doi.org/10.1016/j.apm.2009.09.011.

[10] S. Das & D. K Gupta, Homotopy Analysis Method for solving fractional hyperbolic partial differential equations, International Journal of Computer Mathematics, 2010: 578-588. https://doi.org/10.1080/00207161003631901.

[11] Kurulay M, solving the fractional nonlinear Kelin-Gordon equation by means of the Homotopy Analysis Method, Advances in Difference equations, 2012: 2012:187. https://doi.org/10.1186/1687-1847-2012-187.

[12] Hosseinnia S, Ranjbar A & Momani S, using enhanced Homotopy Perturbation Method in fractional differential equations via deforming the linear part, Computers & Mathematics with Applications, 2008: 3138-3149. https://doi.org/10.1016/j.camwa.2008.07.002.

[13] M. Jleli, S. Kumar, R, Kumar & B. Samet, â€œ Analytical approach for time fractional wave equations in the sense of Yang-Abdel-Aty-Cattani via the homotopy perturbation transform methodâ€ Alexandria Engineering Journal, in press, (2019). https://doi.org/10.1016/j.aej.2019.12.022.

[14] Marinca V & Herisanu N, Application of Optimal Homotopy Asymptotic Method for solving non-linear equations arising in heat transfer, International Communications in Heat and Mass Transfer, 2008; 35:710â€“715. https://doi.org/10.1016/j.icheatmasstransfer.2008.02.010.

[15] H. Khalil, R. A. Khan, M. M. Rashidi, Brenstien polynomials and applications to fractional differential equations, computational methods for differential equations, 2015: 14-35.

[16] S. Kumar, D. Kumar, S. Abbasbandy and M. M. Rashidi, Analytical solution of fractional Navier-Stokes equation by using modified Laplace decomposition method, Ain shams engineering Journal, 2014: 569-574. https://doi.org/10.1016/j.asej.2013.11.004.

[17] M. A. Abdelkawy, Antonio M. Lopes & Mohhamad M. Babtin, â€œShifted fractional Jacobi collocation method for solving fractional functional differential equations of variable orderâ€ Chaos, Solitons & Fractals, Vol. 134, 109721, May 2020. https://doi.org/10.1016/j.chaos.2020.109721.

[18] K. M. Saad, Eman. H.F. AL. Shareef, A.K. Alomari, Dumitru Baleanu, â€œOn exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burgerâ€™s equations using homotopy analysis transform methodâ€, Chinese Journal of Physics, Vol 63, 149-162, (2020). https://doi.org/10.1016/j.cjph.2019.11.004.

[19] J. Biazar, M. Ilie, A. khoshkenar, A new approach to the solution of the prey and predator problem and comparison of the results with the Adomian method, Applied Mathematics and Computation 171 (2005) 486â€“491. https://doi.org/10.1016/j.amc.2005.01.040.

[20] S. Das, P.K. Gupta, Rajeev, A fractional predator-prey model and its solution, Int. J. Nonlin. Sci. Numer. Simul. 10 (2009) 873-876. https://doi.org/10.1515/IJNSNS.2009.10.7.873.

[21] K. Logeswari, C. Ravichandran, A new exploration on existen ceoffractional neutral integro-differential equations in the concept of Atanganaâ€“Baleanu, PhysicaA, (2019).

[22] N. Valliammal, C. Ravichandran, Ju H. Park, on the controllability of fractional neutral integro-differential delay equations with nonlocal conditions, Mathematical Methods in the Applied Sciences, (2016), https://doi.org/10.1002/mma.4369.

[23] C. Ravichandran, N. Valliammal, Juan J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, Journal of the Franklin Institute, (2018), https://doi.org/10.1016/j.jfranklin.2018.12.001.

[24] N. Valliammal, C. Ravichandran, Z. Hammouch, H. M. Baskonu, A New Investigation on Fractional-Ordered Neutral Differential Systems with State-Dependent Delay, International Journal of Nonlinear Sciences and Numerical Simulation, (2019), https://doi.org/10.1515/ijnsns-2018-0362.

[25] R. Subashini, , K. Jothimani, K. S. Nisar, C. Ravichandran, New results on nonlocal functional integrodifferential equations via Hilfer fractional derivative, Alexandria Engineering Journal, (2020), https://doi.org/10.1016/j.aej.2020.01.055.

[26] S. Kumar , A. Kumar , S. Momani, M. Aldhaifallah, K. S. Nisar, Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems, Advance in Difference Equations, (2019), https://doi.org/10.1186/s13662-019-2334-7.