×

High-order semi-implicit linear multistep LG scheme for a three species competition-diffusion system in two-dimensional spatial domain arising in ecology. (English) Zbl 1450.65068

Summary: In this paper, we consider a robust numerical approximation of a three-species fully-coupled competition-diffusion system of Lotka-Volterra-type in a two-dimensional spatial domain. The model is characterized by the presence of a very small diffusion parameter. If the diffusivity coefficient is sufficiently small, a spatially segregated pattern with very thin internal layers occur. For such problems, it is a challenging task to develop an efficient numerical method that is also capable of capturing the various transient regimes and fine spatial structures of the solutions. In this paper, we develop a high-order semi-implicit multistep scheme based on the Lagrange temporal interpolation coupled with a conforming finite element method for the nonlinear competition-diffusion problem in two spatial dimensions. A major advantage of the proposed method is that it is essentially linear in terms of the current time-step values (no need for nonlinear iterative treatment), while its order of convergence is higher. Moreover, the couplings of current step values of the unknowns are one sided, which is a very desirable property in terms of algorithmic efficiency since each unknown is solved sequentially. This avoids solving for all unknowns simultaneously. We also discuss the stability and convergence of the proposed schemes. Furthermore, various simulations are carried out to demonstrate the performance of the proposed method in simulating different type of interaction patterns such as the onset of spiral-like coexistence pattern, complex spatio-temporal patterns and competitive exclusion of the species.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
35K57 Reaction-diffusion equations
35B36 Pattern formations in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D40 Ecology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anastassi, ZA; Simos, TE, A six-step p-stable trigonometrically-fitted method for the numerical integration of the radial Schrödinger equation, MATCH Commun MathComput Chem, 60, 3, 803-830 (2008) · Zbl 1199.65229
[2] Anastassi, ZA; Simos, TE, Numerical multistep methods for the efficient solution of quantum mechanics and related problems, Phys Rep, 482, 1-240 (2009)
[3] Anastassi, ZA, A new symmetric linear eight-step method with fifth trigonometric order for the efficient integration of the Schrödinger equation, Appl Math Lett, 24, 8, 1468-1472 (2011) · Zbl 1217.65144
[4] Boscarino, S.; LeFloch, PG; Russo, G., High-order asymptotic-preserving methods for fully non linear relaxation problems, SIAM J Sci Comput, 36, 2, A377-A395 (2014) · Zbl 1426.76455
[5] Boscarino, S.; Filbet, F.; Russo, G., High order semi-implicit schemes for time dependent partial differential equations, J Sci Comput, 68, 3, 975-1001 (2016) · Zbl 1353.65075
[6] Cia, X., Numerical simulation technique for nonlinear singularly perturbed predator-prey reaction diffusion system in biomathematics, Nat Comput ICNC, 5, 44-48 (2007)
[7] Cangiani, A.; Georgoulis, EH; Morozov, AY; Sutton, OJ, Revealing new dynamical patterns in a reaction-diffusion model with cyclic competition via a novel computational framework, Proc R Soc A, 474, 2213, 20170608 (2018) · Zbl 1402.35146
[8] Contento, L.; Mimura, M.; Tohma, M., Two-dimensional travelling waves arising from planar front interaction in a three-species competition-diffusion system, Jpn J Ind ApplMath, 32, 3, 707-747 (2015) · Zbl 1329.35173
[9] Ei, SI; Ikota, R.; Mimura, M., Segregating partition problem in competition-diffusion systems, Interfaces Free Boundaries, 1, 1, 57-80 (1999) · Zbl 0980.92037
[10] Filbet, F.; Jin, S., A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J Comput Phys, 229, 20, 7625-7648 (2010) · Zbl 1202.82066
[11] Giraldo, FX; Restelli, M.; Láuter, M., Semi-implicit formulations of the Navier-Stokes equations: application to nonhydrostatic atmosphereric modeling, SIAM J Sci Comput, 32, 6, 3394-3425 (2010) · Zbl 1237.76153
[12] Giraldo, FX; Kelly, JF; Costantinescu, EM, Implicit-explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA), SIAM J Sci Comput, 35, 5, B1162-B1194 (2013) · Zbl 1280.86008
[13] Higueras, I.; Mantas, JM; Roldán, T., Design and implementation of predictors for additive semi-implicit Runge-Kutta methods, SIAM J Sci Comput, 31, 3, 2131-2150 (2009) · Zbl 1200.65066
[14] Jia-qi, M.; Xiang-lin, H., Nonlinear predator-prey singularly perturbed Robin problems for reaction diffusion systems, J Zhejiang Univ Sci, 4, 5, 511-513 (2003) · Zbl 1067.35036
[15] Kan-On, Y.; Mimura, M., Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J Math Anal, 29, 6, 1519-1536 (1998) · Zbl 0920.35015
[16] Kan-On, Y.; Mimura, M., Predation-mediated coexistence and segregation structures, Stud Math Appl, 18, 129-155 (1986) · Zbl 0613.92023
[17] Kishimoto, K., Instability of non-constant equilibrium solutions of a system of competition-diffusion equations, J Math Biol, 13, 1, 105-114 (1981) · Zbl 0471.92015
[18] Kishimoto, K.; Weinberger, HF, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J DifferEqs, 58, 1, 15-21 (1985) · Zbl 0599.35080
[19] Mimura, M.; Tohma, M., Dynamic coexistence in a three-species competition-diffusion system, Ecol Complexity, 21, 215-232 (2015)
[20] Mimura, M.; Fife, PC, A 3-component system of competition -diffusion system, Hiroshima Math J, 16, 189-207 (1986) · Zbl 0615.92016
[21] Mo, J.; Tang, R., Nonlinear singularly perturbed predator-prey reaction diffusion systems, Appl Math, 19, 1, 57-66 (2004) · Zbl 1059.35005
[22] Owen, MR; Lewis, MA, How predation can slow, stop or reverse a prey invasion, Bull Math Biol, 63, 4, 655-684 (2001) · Zbl 1323.92181
[23] Schiesser, W. E.; Griffiths, G. W., A compendium of partial differential equation models: method of lines analysis with matlab (2009), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1172.65002
[24] Schiesser, W. E., Numerical method of lines: integration of partial differential equations (1991), Academic Press, Inc.: Academic Press, Inc. San Diego · Zbl 0763.65076
[25] Sewalt, L.; Doelman, A.; Meijer, HGE; Rottschäfer, V.; Zagaris, A., Tracking pattern evolution through extended center manifold reduction and singular perturbations, Phys D, 298, 48-67 (2015) · Zbl 1364.35040
[26] Smereka, P., Semi-implicit level set methods for curvature and surface diffusion motion, J Sci Comput, 19, 1-3, 439-456 (2003) · Zbl 1035.65098
[27] Zhang, H.; Sandu, A.; Blaise, S., High order implicit-explicit general linear methods with optimized stability regions, SIAM J Sci Comput, 38, 3, A1430-A1453 (2016) · Zbl 1337.65008
[28] Zhong, X., Additive semi-implicit Runge Kutta methods for computing high-speed nonequilibrium reactive flows, J Comput Phys, 128, 1, 19-31 (1996) · Zbl 0861.76057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.