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/10.1155/2019/5787329 Ahlin, A. C. (1962). On Error Bounds for Gaussian Cubature. SIAM Review, 4(1), 25-39. doi:10.1137/1004004 Aronson, D. ., & Weinberger, H. . (1978). Multidimensional nonlinear diffusion arising in population genetics. Advances in Mathematics, 30(1), 33-76. doi:10.1016/0001-8708(78)90130-5 Berestycki, 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:10.1088/0951-7715/22/12/002 Edelman, 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.231499798 Fakhar-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.2698 FISHER, 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.x Furter, J., & Grinfeld, M. (1989). Local vs. non-local interactions in population dynamics. Journal of Mathematical Biology, 27(1), 65-80. doi:10.1007/bf00276081 Genieys, 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:2006004 Genieys, S., Bessonov, N., & Volpert, V. (2009).