Stable Numerical Solutions Preserving Qualitative Properties of Nonlocal Biological Dynamic Problems

[EN] This paper deals with solving numerically partial integrodifferential equations appearing in biological dynamics models when nonlocal interaction phenomenon is considered. An explicit finite difference scheme is proposed to get a numerical solution preserving qualitative properties of the solution. Gauss quadrature rules are used for the computation of the integral part of the equation taking advantage of its accuracy and low computational cost. Numerical analysis including consistency, stability, and positivity is included as well as numerical examples illustrating the efficiency of the proposed method.This work has been partially supported by the Ministerio de Economía y Competitividad Spanish grant MTM2017-89664-P.Piqueras-García, MÁ.; Company Rossi, R.; Jódar Sánchez, LA. (2019). Stable Numerical Solutions Preserving Qualitative Properties of Nonlocal Biological Dynamic Problems. Abstract and Applied Analysis. 2019:1-7. https://doi.org/10.1155/2019/5787329S172019Ahlin, A. C. (1962). On Error Bounds for Gaussian Cubature. SIAM Review, 4(1), 25-39. doi:10.1137/1004004Aronson, D. ., & Weinberger, H. . (1978). Multidimensional nonlinear diffusion arising in population genetics. Advances in Mathematics, 30(1), 33-76. doi:10.1016/0001-8708(78)90130-5Berestycki, H., Nadin, G., Perthame, B., & Ryzhik, L. (2009). The non-local Fisher–KPP equation: travelling waves and steady states. Nonlinearity, 22(12), 2813-2844. doi:10.1088/0951-7715/22/12/002Edelman, G. M., & Gally, J. A. (2001). Degeneracy and complexity in biological systems. Proceedings of the National Academy of Sciences, 98(24), 13763-13768. doi:10.1073/pnas.231499798Fakhar-Izadi, F., & Dehghan, M. (2012). An efficient pseudo-spectral Legendre-Galerkin method for solving a nonlinear partial integro-differential equation arising in population dynamics. Mathematical Methods in the Applied Sciences, 36(12), 1485-1511. doi:10.1002/mma.2698FISHER, R. A. (1937). THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES. Annals of Eugenics, 7(4), 355-369. doi:10.1111/j.1469-1809.1937.tb02153.xFurter, J., & Grinfeld, M. (1989). Local vs. non-local interactions in population dynamics. Journal of Mathematical Biology, 27(1), 65-80. doi:10.1007/bf00276081Genieys, S., Volpert, V., & Auger, P. (2006). Pattern and Waves for a Model in Population Dynamics with Nonlocal Consumption of Resources. Mathematical Modelling of Natural Phenomena, 1(1), 63-80. doi:10.1051/mmnp:2006004Genieys, S., Bessonov, N., & Volpert, V. (2009). Mathematical model of evolutionary branching. Mathematical and Computer Modelling, 49(11-12), 2109-2115. doi:10.1016/j.mcm.2008.07.018Hamel, F., & Ryzhik, L. (2014). On the nonlocal Fisher–KPP equation: steady states, spreading speed and global bounds. Nonlinearity, 27(11), 2735-2753. doi:10.1088/0951-7715/27/11/2735Shivanian, E. (2013). Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics. Engineering Analysis with Boundary Elements, 37(12), 1693-1702. doi:10.1016/j.enganabound.2013.10.002Tian, C., Ling, Z., & Zhang, L. (2017). Nonlocal interaction driven pattern formation in a prey–predator model. Applied Mathematics and Computation, 308, 73-83. doi:10.1016/j.amc.2017.03.017Apreutesei, N., Bessonov, N., Volpert, V., … Vougalter, V. (2010). Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 13(3), 537-557. doi:10.3934/dcdsb.2010.13.537Weinberger, H. F. (2002). On spreading speeds and traveling waves for growth and migration models in a periodic habitat. Journal of Mathematical Biology, 45(6), 511-548. doi:10.1007/s00285-002-0169-3Weinberger, H. F., Lewis, M. A., & Li, B. (2007). Anomalous spreading speeds of cooperative recursion systems. Journal of Mathematical Biology, 55(2), 207-222. doi:10.1007/s00285-007-0078-