mmnsa Mathematical Modelling and Numerical Simulation with Applications 2791-8564 Mehmet YAVUZ 10.53391/mmnsa.1408997 Numerical Analysis Biological Mathematics Sayısal Analiz Biyolojik Matematik A novel Touchard polynomial-based spectral matrix collocation method for solving the Lotka-Volterra competition system with diffusion https://orcid.org/0000-0002-6116-4928 Izadi Mohammad Shahid Bahonar University of Kerman, https://orcid.org/0000-0003-2009-3905 El-mesady Ahmed Menoufia university https://orcid.org/0000-0002-0557-8536 Adel Waleed Mansoura university 03 31 2024 4 1 37 65 12 23 2023 03 09 2024 Copyright © 2021, Mathematical Modelling and Numerical Simulation with Applications 2021 Mathematical Modelling and Numerical Simulation with Applications

This paper presents the computational solutions of a time-dependent nonlinear system of partial differential equations (PDEs) known as the Lotka-Volterra competition system with diffusion. We propose a combined semi-discretized spectral matrix collocation algorithm to solve this system of PDEs. The first part of the algorithm deals with the time-marching procedure, which is performed using the well-known Taylor series formula. The resulting linear systems of ordinary differential equations (ODEs) are then solved using the spectral matrix collocation technique based on the novel Touchard family of polynomials. We discuss and establish the error analysis and convergence of the proposed method. Additionally, we examine the stability analysis and the equilibrium points of the model to determine the stability condition for the system. We perform numerical simulations using diverse model parameters and with different Dirichlet and Neumann boundary conditions to demonstrate the utility and applicability of our combined Taylor-Touchard spectral collocation algorithm.

 Collocation points Convergent analysis Stability Touchard polynomials Taylor series formula [1] Kumar, S., Kumar, A. and Odibat, Z.M. A nonlinear fractional model to describe the population dynamics of two interacting species. Mathematical Methods in the Applied Sciences, 40(11), 4134–4148, (2017). [2] Lotka, A.J. Contribution to the theory of periodic reactions. The Journal of Physical Chemistry, 14(3), 271-274, (2022). [3] Owolabi, K.M. Computational dynamics of predator-prey model with the power-law kernel. Results in Physics, 21, 103810, (2021). [4] Owolabi, K.M., Pindza, E. and Atangana, A. Analysis and pattern formation scenarios in the superdiffusive system of predation described with Caputo operator. Chaos, Solitons & Fractals, 152, 111468, (2021). [5] Pan, M.X., Wang, S.Y., Wu, X.L., Zhang, M.W. and Schiavo, A.L. Study on the growth driving model of the enterprise innovation community based on the Lotka–Volterra model: a case study of the Chinese Automobile Manufacturing Enterprise Community. Mathematical Problems in Engineering, 2022, 8743167, (2023). [6] Han, J. The Impact of epidemic infectious diseases on the ecological environment of three species based on the Lotka–Volterra model. World Scientific Research Journal, 7(1), 340-345, (2021). [7] Ni, W., Shi, J. and Wang, M. Global stability and pattern formation in a nonlocal diffusive Lotka–Volterra competition model. Journal of Differential Equations, 264(11), 6891-6932, (2018). [8] Lin, G. and Ruan, S. Traveling wave solutions for delayed reaction–diffusion systems and applications to diffusive Lotka–Volterra competition models with distributed delays. Journal of Dynamics and Differential Equations, 26, 583-605, (2014). [9] Wijeratne, A.W., Yi, F. and Wei, J. Bifurcation analysis in the diffusive Lotka–Volterra system: an application to market economy. Chaos, Solitons & Fractals, 40(2), 902-911, (2009). [10] Cherniha, R. Construction and application of exact solutions of the diffusive Lotka–Volterra system: a review and new results. Communications in Nonlinear Science and Numerical Simulation, 113, 106579, (2022). [11] Zhang, S., Zhu, X. and Liu, X. A diffusive Lotka–Volterra model with Robin boundary condition and sign-changing growth rates in time-periodic environment. Nonlinear Analysis: Real World Applications, 72, 103856, (2023). [12] Ma, L., Gao, J., Li, D. and Lian, W. Dynamics of a delayed Lotka–Volterra competition model with directed dispersal. Nonlinear Analysis: Real World Applications, 71, 103830, (2023). [13] Barker, W. Existence of traveling waves of Lotka Volterra type models with delayed diffusion term and partial quasimonotonicity. ArXiv Preprint, ArXiv:2303.11145, (2023). [14] Guo, S. Global dynamics of a Lotka-Volterra competition-diffusion system with nonlinear boundary conditions. Journal of Differential Equations, 352, 308-353, (2023). [15] Kudryashov, N.A. and Zakharchenko, A.S. Analytical properties and exact solutions of the Lotka–Volterra competition system. Applied Mathematics and Computation, 254, 219-228, (2015). [16] Islam, M., Islam, B. and Islam, N. Exact solution of the prey-predator model with diffusion using an expansion method. Applied Sciences, 15, 85-93, (2013). [17] Wang, J., Liu, Q. and Luo, Y. The numerical analysis of the long time asymptotic behavior for Lotka-Volterra competition model with diffusion. Numerical Functional Analysis and Optimization, 40(6), 685-705, (2019). [18] Sabawi, Y.A., Pirdawood, M.A. and Sadeeq, M.I. A compact fourth-order implicit-explicit Runge-Kutta type method for solving diffusive Lotka–Volterra system. In Proceedings, Journal of Physics: Conference Series (Vol. 1999, No. 1, p. 012103). IOP Publishing, (2021, April). [19] Izadi, M. Numerical approximation of Hunter-Saxton equation by an efficient accurate approach on long time domains. UPB Scientific Bulletin Series A Applied Mathematics and Physics, 83(1), 291-300, (2021). [20] Izadi, M. and Yuzbasi, S. A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-diffusion problems. Mathematical Communications, 27(1), 47-62, (2022). [21] Izadi, M. and Roul, P. Spectral semi-discretization algorithm for a class of nonlinear parabolic PDEs with applications. Applied Mathematics and Computation, 429, 127226, (2022). [22] Izadi, M. and Zeidan, D. A convergent hybrid numerical scheme for a class of nonlinear diffusion equations. Computational and Applied Mathematics, 41, 318, (2022). [23] Günerhan, H., Dutta, H., Dokuyucu, M.A. and Adel, W. Analysis of a fractional HIV model with Caputo and constant proportional Caputo operators. Chaos, Solitons & Fractals, 139, 110053, (2020). [24] El-Sayed, A.A., Baleanu, D. and Agarwal, P. A novel Jacobi operational matrix for numerical solution of multi-term variable-order fractional differential equations. Journal of Taibah University for Science, 14(1), 963-974, (2020). [25] Srivastava, H.M. and Izadi, M. Generalized shifted airfoil polynomials of the second kind to solve a class of singular electrohydrodynamic fluid model of fractional order. Fractal and Fractional, 7(1), 94, (2023). [26] Sabermahani, S., Ordokhani, Y. and Hassani, H. General Lagrange scaling functions: application in general model of variable order fractional partial differential equations. Computational and Applied Mathematics, 40, 269, (2021). [27] Abbasi, Z., Izadi, M. and Hosseini, M.M. A highly accurate matrix method for solving a class of strongly nonlinear BVP arising in modeling of human shape corneal. Mathematical Methods in the Applied Sciences, 46(2), 1511-1527, (2023). [28] Razavi, M., Hosseini, M.M. and Salemi, A. Error analysis and Kronecker implementation of Chebyshev spectral collocation method for solving linear PDEs. Computational Methods for Differential Equations, 10(4), 914–927, (2022). [29] Srivastava, H.M., Adel, W., Izadi, M. and El-Sayed, A.A. Solving some physics problems involving fractional-order differential equations with the Morgan-Voyce polynomials. Fractal and Fractional, 7(4), 301, (2023). [30] Izadi, M., Yüzbası, S. and Adel, W. Accurate and efficient matrix techniques for solving the fractional Lotka–Volterra population model. Physica A: Statistical Mechanics and its Applications, 600, 127558, (2022). [31] Mihoubi, M. and Maamra, M.S. Touchard polynomials, partial Bell polynomials and polynomials of binomial type. Journal of Integer Sequences, 14(3), (2011). [32] Boyadzhiev, K.N. Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals. Abstract and Applied Analysis, 2009, 168672, (2009). [33] Sabermahani, S. and Ordokhani, Y. A computational method to solve fractional-order Fokker-Planck equations based on Touchard polynomials. Computational Mathematics and Computer Modeling with Applications (CMCMA), 1(2), 65-73, (2022). [34] Aldurayhim, A., Elsonbaty, A. and Elsadany, A.A. Dynamics of diffusive modified Previte Hoffman food web model. Mathematical Biosciences and Engineering, 17(4), 4225-4256, (2020). [35] Ahmed, N., Elsonbaty, A., Raza, A., Rafiq, M. and Adel, W. Numerical simulation and stability analysis of a novel reaction–diffusion COVID-19 model. Nonlinear Dynamics, 106, 1293-1310, (2021). [36] Touchard, J. Sur les cycles des substitutions. Acta Mathematica, 70, 243-297, (1939). [37] Bell, E.T. Exponential polynomials. Annals of Mathematics, 35(2), 258-277, (1934). [38] Mansour, T. and Schork, M. The generalized Touchard polynomials revisited. Applied Mathematics and Computation, 219(19), 9978-9991, (2013). [39] Kim, T., Herscovici, O., Mansour, T. and Rim, S.H. Differential equations for p, q-Touchard polynomials. Open Mathematics, 14(1), 908-912, (2016). [40] Comtet, L. The art of finite and infinite expansions. In Advanced Combinatorics (pp. xi-343). D. Reidel Publishing Co. Dordrecht, (1974). [41] Harper, L.H. Stirling behavior is asymptotically normal. The Annals of Mathematical Statistics, 38(2), 410-414, (1967). [42] Isaacson, E. and Keller, H.B. Analysis of Numerical Methods. Courier Corporation: North Chelmsford, United States, (1994).