A NEW TECHNIQUE TO SOLVE PREDATOR–PREY MODELS BY USING SHEHU TRANSFORMATION-AKBARI-GANJI’S METHOD WITH PADÉ APPROXIMANT
Keywords:
predator-prey systems, Shehu transform, Akbari -Ganji’s method, Padé approximant, convergence, stability
Abstract
In this paper, a new technique has been proposed and applied to find analytical approximate solutions for predator-prey systems. The new technique depends on combining the algorithms of Shehu transform and Akbari-Ganji’s method (AGM) with Padé approximant. Three examples are given to test the effectiveness, accuracy, and performance of the suggested method. In comparison to the existing methods used to identify the analytical approximate solutions of the current problems, the results of using the new technique demonstrate that it has excellent efficiency and accuracy. Also, the tables and errors graphs show the applicability and necessity of this technique.
References
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33. Boyd J., Padé Approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain. Computers in Physics, 11, (3), 299- 303, (1997).
34. Wazwaz A.M., Analytical approximations and Padé approximants for Volterra’s population model. Applied Mathematics and Mathematical Computation, 100, 31-35, (1999).
35. Wazwaz A.M., The Modified decomposition method and Pade’s approximants for solving Thomas-Fermi equation. Applied Mathematics and Mathematical Computations, 105, 11- 19, (1999).
36. Momani S., Rami Q., Numerical approximation and Padé approximation for a fractional population growth model, 31, 1907-1914, (2007).
37. Ghotbi, A. R., Barari, A., and Ganji, D., Solving ratio-dependent predator-prey system with constant effort harvesting using homotopy perturbation method. Mathematical Problems in Engineering, (2008).
38. Bougoffa L., Solvability of the predator and prey system with variable coefficients and com- parison of the results with modified decomposition, Applied Mathematics and Computation,182, (1),383–387, (2006).
39. Makinde O. D., Solving ratio-dependent predator-prey system with constant effort harvesting using Adomian decomposition method, Applied Mathematics and Computation, 186, (1), 17–22, (2007).
40. Al-Jaberi, A. K., Abdul-Wahab, M. S., and Buti R. H., A new approximate method for solving linear and non-linear differential equation systems. In AIP Conference Proceedings, 2398, (1), 060082,14, (2022).
2. Volterra V., Fluctuations in the abundance of a species considered mathematically, Nature 118, 558–560 (1926).
3. Craciun G., Nazarov F. and Pantea C., Persistence and permanence of mass-action and power- law dynamical systems. SIAM J. Appl. Math. 73, 305–329 (2013).
4. Xie L. and Wang Y. , On a fully parabolic chemotaxis system with Lotka–Volterra competitive kinetics.J. Math.Anal. Appl. 471, 584–598 (2019).
5. Su K., Wang C., Zhang S. and Liu S., Lotka Volterra equation based modeling of aerobic granulation process in sequencing batch reactors. Int. Biodeterior. Biodegrad. 115, 49–54 (2016).
6. Cropp R.A. and Norbury J., Population interactions in ecology:arule-based approach to modeling ecosystems in a mass-conserving framework. SIAM Rev. 57, 437–446 (2015).
7. Li C. and Zhu H., Canard cycles for predator–prey systems with Holling types of functional response. J. Differ. Equ. 254, 879–910 (2013).
8. Zhu C. and Yin G., On competitive Lotka–Volterra model in random environments. J.Math. Anal.Appl.357,154–170 (2009).
9. Badri V.,Yazdanpanah M.J. and Tavazoei M.S., Global stabilization of Lotka–Volterra systems with interval uncertainty. IEEE Trans. Autom. Control 64, 1209–1213 (2018).
10. Murray J. D., Mathematical Biology. I: An Introduction, Springer. (2002).
11. Biazar J. and Montazeri R., A computational method for solution of the prey and predator problem, Applied Mathematics and Computation, 163, (2), 841–847, (2005).
12. Bougoffa L., Solvability of the predator and prey system with variable coefficients and comparison of the results with modified decomposition, Applied Mathematics and Computation, 182, (1), 383–387,( 2006)
13. Chowdhury M. S. H., Hashim I., and Mawa S., Solution of prey predator problem by numeric analytic technique, Communications in Nonlinear Science and Numerical Simulation, 14, (4),1008–1012, (2009).
14. Makinde O. D., Solving ratio-dependent predator-prey system with constant effort harvesting using Adomian decomposition method, Applied Mathematics and Computation, 186, (1), 17–22, ( 2007).
15. Yusufoğlu, Elçin, and Barış E., He's variational iteration method applied to the solution of the prey and predator problem with variable coefficients. Physics Letters A 372.21 ,3829-3835, (2008).
16. Dobromir T. D. and Hristo V. K., Nonstandard finite-difference methods for predator-prey models with general functional response. Mathematics and Computers in Simulation, 78, (1),–11, ( 2008).
17. Erturk V. S. and Momani S., Solutions to the problem of prey and predator and the epidemic model via differential transform method, Kybernetes, 37, (8), 1180– 1188, (2008).
18. Ghotbi, A. R., Barari A., and Ganji D. D., Solving ratio-dependent predator-prey system with constant effort harvesting using homotopy perturbation method. Mathematical Problems in Engineering 2008, 2-5, (2008).
19. Vijayalakshmi, T., and Rathinam S., Application of homotopy perturbation and variational iteration methods for nonlinear imprecise prey–predator model with stability analysis. The Journal of Supercomputing 78. (2) ,2477-2502, (2022).
20. Jawad S., Modelling, dynamics and analysis of multi-species systems with prey refuge. Msc.thesis, Brunel University, London, (2018).
21. Ali N., Haque M., Venturino E., and Chakravarty S., Dynamics of a three species ratio-dependent food chain model with intra-specific competition within the top predator, Comput. Biol. Med., 85, 63–74, (2017).
22. Jawad S., Sultan D., and Winter M., The dynamics of a modified Holling-Tanner prey-predator model with wind effect, Int. J.Nonlinear Anal. Appl., 12, 2203–2210, (2021).
23. Yasin, M. W., Ahmed, N., Iqbal, M. S., Raza, A., Rafiq, M., Eldin, E. M. T., and Khan, I.,Spatio-temporal numerical modeling of stochastic predator-prey model. Scientific Reports,13.(1): 1990, (2023).
24. Sun, Yihuan, and Shanshan C., Stability and bifurcation in a reaction–diffusion–advection predator–prey model. Calculus of Variations and Partial Differential Equations 62. (2), 61,1, (2023).
25. Arif, Abodayeh K., Shoaib M. and Ejaz A., On the stability of the diffusive and non-diffusive predator-prey system with consuming resources and disease in prey species. Mathematical Biosciences and Engineering 20.(3), 5066-5093, (2023).
26. Abdulameer, Y. A., and Al-Saif A. S. J. A., A well-founded analytical technique to solve 2D viscous flow between slowly expanding or contracting walls with weak permeability. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 97. (2), 39-56 (2022).
27. Abdoon M.A., and Hasan F.L. Advantages of the differential equations for solving problems in mathematical physics with symbolic computation. Mathematical modelling of engineering problems 9, (1), 268-276, (2022).
28. Hasan Z. A. and Al-Saif A.S. J. A hybrid method with RDTM for solving the biological population model. Mathematical Theory and Modeling. 12.(2), 1, (2022).
29. Al‐Griffi, T. A. J., and Al‐Saif A. S. J. Yang transform–homotopy perturbation method for solving a non‐Newtonian viscoelastic fluid flow on the turbine disk. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 102. e202100116, (2022).
30. Mirgolbabaee H., Ledari S. T. and Ganji D. D., Semi-analytical investigation on micropolar fluid flow and heat transfer in a permeable channel using AGM, Journal of the Association of Arab Universities for Basic and Applied Sciences, 24:213-222, (2017).
31. Wu B. and Qian Y., Padé approximation based on orthogonal polynomial, Advances in Computer Science Research. 58, 249- 252, (2016).
32. Maitama S. and Zhao W., New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations. arXiv preprint arXiv1904.11370, (2019).
33. Boyd J., Padé Approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain. Computers in Physics, 11, (3), 299- 303, (1997).
34. Wazwaz A.M., Analytical approximations and Padé approximants for Volterra’s population model. Applied Mathematics and Mathematical Computation, 100, 31-35, (1999).
35. Wazwaz A.M., The Modified decomposition method and Pade’s approximants for solving Thomas-Fermi equation. Applied Mathematics and Mathematical Computations, 105, 11- 19, (1999).
36. Momani S., Rami Q., Numerical approximation and Padé approximation for a fractional population growth model, 31, 1907-1914, (2007).
37. Ghotbi, A. R., Barari, A., and Ganji, D., Solving ratio-dependent predator-prey system with constant effort harvesting using homotopy perturbation method. Mathematical Problems in Engineering, (2008).
38. Bougoffa L., Solvability of the predator and prey system with variable coefficients and com- parison of the results with modified decomposition, Applied Mathematics and Computation,182, (1),383–387, (2006).
39. Makinde O. D., Solving ratio-dependent predator-prey system with constant effort harvesting using Adomian decomposition method, Applied Mathematics and Computation, 186, (1), 17–22, (2007).
40. Al-Jaberi, A. K., Abdul-Wahab, M. S., and Buti R. H., A new approximate method for solving linear and non-linear differential equation systems. In AIP Conference Proceedings, 2398, (1), 060082,14, (2022).
Published
2023-08-31
How to Cite
Rania O. Al–Sadi, & Abdul-Sattar J. Al-Saif. (2023). A NEW TECHNIQUE TO SOLVE PREDATOR–PREY MODELS BY USING SHEHU TRANSFORMATION-AKBARI-GANJI’S METHOD WITH PADÉ APPROXIMANT. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 4(8), 102-119. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/508
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