Editor in Chief - Anry Nersessian (Institute of Mathematics NAS, Armenia)
Deputy Editor - Rafayel Barkhudaryan (Institute of Mathematics NAS, Armenia)
Managing Editor - Linda Khachatryan (Institute of Mathematics NAS, Armenia)
e-mail: ajm@instmath.sci.am
According to the latest Scimago Journal Rank, Armenian Journal of Mathematics has maintained its position in Q3 with an SJR score of 0.36.
Read more about Armenian Journal of Mathematics has been recognized as a Q3 journal in SJROver the last few decades, the numerical approximation of spatial segregation equations for reaction-diffusion systems with $m$ population densities has gained significant interest. These problems are governed by a minimization problem subject to a closed but non-convex set. In the present work, we deal with the numerical approximation of equations of stationary states for a certain class of the spatial segregation of reaction-diffusion systems with two population densities having disjoint support. We prove the convergence of the numerical algorithm for two competing populations with non-negative internal dynamics $f_i(x)\geq 0$, $i=1,2$. At the end of the paper, we present computational tests.
A. Arakelyan, A finite difference method for two-phase parabolic obstacle-like problem. Armen. J. Math., 7 (2015), no. 1, pp. 32-49.
A. Arakelyan, Convergence of the finite difference scheme for a general class of the spatial segregation of reaction--diffusion systems. Comput. Math. Appl., 75 (2018), no. 12, pp. 4232-4240.
A. Arakelyan and R. Barkhudaryan, A numerical approach for a general class of the spatial segregation of reaction-diffusion systems arising in population dynamics. Comput. Math. Appl., 72 (2016), no. 11, pp. 2823-2838.
A. Arakelyan, R. Barkhudaryan and M. Poghosyan, Numerical solution of the two-phase obstacle problem by finite difference method. Armen. J. Math., 7 (2015), no. 2, pp. 164-182.
A. Arakelyan and F. Bozorgnia, Uniqueness of limiting solution to a strongly competing system. Electron. J. Differ. Equ., 2017 (2017), no. 96, pp. 1-8.
A.G. Arakelyan, R.H. Barkhudaryan and M.P. Poghosyan, Finite difference scheme for two-phase obstacle problem. Dokl. Nats. Akad. Nauk Armen., 111 (2011), no. 3, pp. 224-231.
F. Bozorgnia, Numerical algorithm for spatial segregation of competitive systems. SIAM J. Sci. Comput., 31 (2009), no. 5, 3946-3958.
F. Bozorgnia, Numerical solutions of a two-phase membrane problem. Appl. Numer. Math., 61 (2011), no. 1, pp. 92-107.
F. Bozorgnia and A. Arakelyan, Numerical algorithms for a variational problem of the spatial segregation of reaction--diffusion systems. Appl. Math. Comput., 219 (2013), no. 17, pp. 8863-8875.
F. Bozorgnia and V. Kungurtsev, Optimal control of two-phase membrane problem. Appl. Math. Optim., 92 (2025), article number 8.
M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math., 195 (2005), no. 2, pp. 524-560.
M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems. Indiana Univ. Math. J., 54 (2005), no. 3, pp. 779-815.
M. Conti, S. Terracini and G. Verzini, Uniqueness and least energy property for solutions to strongly competing systems. Interfaces Free Bound., 8 (2006), no. 4, pp. 437-446.
E.C.M. Crooks, E.N. Dancer and D. Hilhorst, Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), no. 1, pp. 39-44.
E.C.M. Crooks, E.N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions. Topol. Methods Nonlinear Anal., 30 (2007), no. 1, pp. 1-36.
E.C.M. Crooks, E.N. Dancer, D. Hilhorst, M. Mimura and H. Ninomiya, Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions. Nonlinear Anal. Real World Appl., 5 (2004), no. 4, pp. 645-665.
E.N. Dancer and Y.H. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities. J. Differ. Equ., 114 (1994), no. 2, pp. 434-475.
E.N. Dancer, D. Hilhorst, M. Mimura and L.A. Peletier, Spatial segregation limit of a competition-diffusion system. European J. Appl. Math., 10 (1999), no. 2, pp. 97-115.
E.N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction. J. Differ. Equ., 182 (2002), no. 2, pp. 470-489.
A. Petrosyan, H. Shahgholian and N.N. Ural'ceva, Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics 136, American Mathematical Soc., 2012.
M. Squassina, On the long term spatial segregation for a competition-diffusion system. Asymptot. Anal., 57 (2008), no. 1-2, pp. 83-103.
M. Squassina and S. Zuccher, Numerical computations for the spatial segregation limit of some 2D competition-diffusion systems. Adv. Math. Sci. Appl., textbf{18} (2008), no. 1, pp. 83--104.
G.S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem. Comm. Partial Differ. Equ., 23 (1998), no. 3-4, pp. 439-455.
Armenian Journal of Mathematics (Armen.J.Math.) aims to publish original research papers and survey articles in all areas of mathematics, enhance the interests, talents, and achievements of all individuals in theoretical mathematics and its applications.
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