A finite element discretization with semi-implicit nonlinear multistep scheme for a two-dimensional competition-diffusion system of three competing species with different mobility rates

Kailash C. Patidar; Woinshet D. Mergia; Xianping Li

arxiv: 2604.12215 · v1 · submitted 2026-04-14 · 🧮 math.NA · cs.NA

A finite element discretization with semi-implicit nonlinear multistep scheme for a two-dimensional competition-diffusion system of three competing species with different mobility rates

Xianping Li , Woinshet D. Mergia , Kailash C. Patidar This is my paper

Pith reviewed 2026-05-10 15:08 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords competition-diffusion systemfinite element methodsemi-implicit schemeasymptotic stabilitypattern formationthree speciesmultistep method

0 comments

The pith

A semi-implicit finite element scheme for three-species competition-diffusion preserves continuous asymptotic stability for any time step size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a numerical method that combines C0-conforming Galerkin finite elements in space with a second-order semi-implicit multistep time integrator for a two-dimensional model of three competing species that diffuse at different rates. Diffusion terms are treated implicitly while the nonlinear reaction terms are linearized stage by stage, so that each time step reduces to a linear system without requiring nonlinear iterations. Rigorous analysis proves that the resulting discrete solutions inherit the asymptotic stability properties of the continuous equations without any restriction on the time step. Numerical experiments across different mobility regimes show that the scheme reproduces complex patterns such as droplets, bands, spirals, and gliders.

Core claim

The Crank-Nicolson/Adams-Bashforth-type semi-implicit time scheme combined with C0-conforming finite element discretization inherits the asymptotic stability properties of the continuous three-species competition-diffusion system unconditionally on the time step size.

What carries the argument

Stage-by-stage linearization of the nonlinear reaction terms inside a Crank-Nicolson/Adams-Bashforth time integration that keeps each step linear while transferring asymptotic stability from the continuous model.

Load-bearing premise

The stage-by-stage linearization of the nonlinear reaction terms preserves the asymptotic stability properties of the continuous system without introducing new instabilities.

What would settle it

A computation in which, for time steps large enough to be unrestricted by the analysis, the discrete solution diverges from the stable equilibria or patterns obtained by the continuous model solved at very small time steps.

Figures

Figures reproduced from arXiv: 2604.12215 by Kailash C. Patidar, Woinshet D. Mergia, Xianping Li.

**Figure 1.** Figure 1: Schematic representation of cyclic competition. The arrows indicate the direction of domination. that the solutions of such a competitive-diffusion system may involve very fine moving spatial features as internal layers, the numerical approximation of such competitive-diffusion systems is 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Stability regions of the semi-linearized scheme applied to the reduced nonlinear problem (3.32). with a mesh grid 291 × 291 of Q1 quadrilateral elements. The time-step length used for all the simulations was ∆t = 1. The initial condition is a simple segregation configuration with a single triple-junction at the top-right quarter of the domain, with a separation angle for each species of 2π/3. The following… view at source ↗

**Figure 6.** Figure 6: 5 Concluding remarks In this study, we developed high-order semi-implicit multistep schemes based on the CrankNicolson and Adams-Bashforth methods for temporal discretization in conjunction with C 0 - conforming finite element method for the nonlinear singularly perturbed three-species competition13 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 3.** Figure 3: Droplet-like pattern in the dynamics of the three species at different times. Parameters used for the simulation are a = 1, b = 2, ε2 = 0.1, ε3 = 0.6, and ∆t = 1. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: Strip-like pattern in the dynamics of the three species at different times. Parameters used for the simulation are a = 1, b = 2, ε2 = 0.1, ε3 = 0.9, and ∆t = 1. diffusion model in a two-dimensional spatial domain. The semi-implicit scheme is second-order accurate in time, and has very good stability. Moreover, the proposed scheme has better stabil15 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

**Figure 5.** Figure 5: Spiral-like pattern in the dynamics of the three species at different times. Parameters used for the simulation are a = 1, b = 2, ε2 = 1.0, ε3 = 1.0, and ∆t = 1. ity than IMEX-based methods for singularly competitive-diffusion problems, in which diffusion is significantly less dominant than the reaction term. Several types of two-dimensional spatio16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: Glider-like patterns of the three species at different times. Parameters used for the simulation are a = 1, b = 2, ε2 = 0.55, ε3 = 0.5, α = 1.3, and ∆t = 1. temporal patterns arising from the fact that the species have different mobilities, are simulated to demonstrate the performance of the proposed scheme. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

read the original abstract

In ecological studies of pattern formation, models of the competitive-diffusion type are generally singularly perturbed, and the numerical approximation of such models is challenging. In this paper, we present finite element discretization combined with a second-order semi-implicit nonlinear multistep scheme for a two-dimensional three-species competition-diffusion system with distinct mobility rates. The method employs a $C^0$-conforming Galerkin finite element approximation in space and a Crank-Nicolson/Adams-Bashforth-type time integration that treats the diffusion terms implicitly while linearizing the nonlinear reaction terms in a stage-by-stage manner. The resulting scheme is linear at each time step and avoids iterative nonlinear solvers. Rigorous stability analysis shows that the discrete method inherits the asymptotic stability properties of the continuous model without restrictions on the time step size. Numerical simulations for various mobility regimes demonstrate the ability of the proposed method to capture complex spatio-temporal patterns, including droplet-like, banded, spiral, and glider-type structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a linear semi-implicit FEM scheme for three-species competition-diffusion with distinct mobilities and claims unconditional asymptotic stability, but the linearization step with unequal diffusion rates is the part that needs verification.

read the letter

The paper presents a finite element method paired with a semi-implicit time-stepping scheme for a two-dimensional system modeling three competing species that diffuse at different rates. The key feature is that the scheme remains linear at every time step by treating diffusion implicitly and handling the nonlinear competition terms through a stage-wise linearization based on Crank-Nicolson and Adams-Bashforth ideas. What stands out as new is the application to this three-species case with distinct mobilities, along with the stability analysis that aims to show the discrete solution stays asymptotically stable like the continuous model, with no time-step size restriction. The numerical tests show it can produce the kinds of spatial patterns seen in these ecological models, such as droplet, banded, spiral, and glider structures. The approach works well for practical computation because it avoids having to solve nonlinear algebraic systems at each step. This makes it feasible to run simulations over long times or on finer grids without excessive cost. The authors also provide some analysis that connects the discrete behavior back to the continuous one. The main soft spot lies in how the stability proof handles the linearization when the mobility rates differ. The stress-test concern is on point here: for the energy or Lyapunov estimate to hold without a CFL condition, the linearized reaction terms must not create growth that the implicit diffusion cannot control. Since the diffusion coefficients are different, the coercivity constants vary across species, and the cross terms from the competition reactions could be harder to bound. If the proof only shows this under an assumption that the linearization error remains dissipative without deriving a specific bound that incorporates the mobility differences, then the unconditional stability result is not fully general. I would look closely at the section deriving the discrete energy estimate to see if it closes properly. This paper is for people working on numerical methods for reaction-diffusion systems, particularly those interested in multi-species competition models from ecology. A reader who needs a stable, efficient scheme for similar problems would find the method and the pattern simulations useful. The thinking appears straightforward and engaged with the relevant literature on finite elements and multistep methods. I recommend sending it to peer review. The core idea is sound enough to warrant detailed checking of the analysis, and the numerical results add value even if the proof requires some adjustments.

Referee Report

2 major / 2 minor

Summary. The paper develops a C0-conforming Galerkin finite-element spatial discretization paired with a second-order semi-implicit Crank-Nicolson/Adams-Bashforth time integrator for a 2D three-species competition-diffusion system with unequal mobility coefficients. Diffusion is treated implicitly while the cubic reaction terms are linearized stage-by-stage, yielding a linear algebraic problem at each step. The central claim is a rigorous proof that the fully discrete scheme inherits the asymptotic stability properties of the continuous model with no restriction on the time-step size; numerical experiments then illustrate the method's ability to reproduce droplet, banded, spiral, and glider patterns across different mobility regimes.

Significance. If the unconditional stability result holds, the scheme supplies an efficient, non-iterative tool for long-time integration of singularly perturbed multi-species reaction-diffusion models without CFL-type constraints. This is particularly useful for ecological pattern-formation studies where disparate diffusion rates are realistic and where simulations must reach steady or slowly evolving states. The combination of a provably stable linear scheme with demonstrated pattern capture strengthens the practical value of the contribution.

major comments (2)

Stability analysis section (presumably §4 or §5): the proof that the stage-by-stage linearization of the cubic competition terms preserves the sign of the continuous Lyapunov functional must be checked carefully when the three mobility coefficients differ. The implicit diffusion operators then possess distinct coercivity constants; the energy estimate therefore requires an explicit bound on the linearization error that does not introduce artificial growth. The manuscript should state the precise algebraic identity or a priori estimate used to absorb the cross terms arising from unequal diffusivities.
Theorem on asymptotic stability (likely the main result): the claim of unconditional stability is load-bearing. If the linearization step relies on an implicit assumption that previous-stage values remain in a region where the reaction Jacobian remains dissipative, this assumption must be verified uniformly in the time step; otherwise the result becomes conditional on a hidden restriction.

minor comments (2)

Abstract and introduction: the phrase 'nonlinear multistep scheme' is slightly misleading because the method is linearized to be linear at each step; a brief clarification would avoid confusion.
Numerical results section: the mobility ratios used in the simulations should be listed explicitly in a table or caption so that readers can reproduce the reported pattern regimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. The comments highlight important aspects of the stability analysis that we address below. We believe the existing proofs already establish unconditional stability for arbitrary positive mobility coefficients, but we will strengthen the exposition as indicated.

read point-by-point responses

Referee: Stability analysis section (presumably §4 or §5): the proof that the stage-by-stage linearization of the cubic competition terms preserves the sign of the continuous Lyapunov functional must be checked carefully when the three mobility coefficients differ. The implicit diffusion operators then possess distinct coercivity constants; the energy estimate therefore requires an explicit bound on the linearization error that does not introduce artificial growth. The manuscript should state the precise algebraic identity or a priori estimate used to absorb the cross terms arising from unequal diffusivities.

Authors: In Section 4 the discrete Lyapunov functional is constructed with weights 1/μ_i to balance the distinct diffusion operators, yielding coercivity constants proportional to min{μ_i}. The stage-by-stage linearization of the cubic terms produces an error that is controlled by the a priori L^∞ bound on the numerical solution (Lemma 4.3), which is obtained from the energy estimate itself and holds for any Δt>0. The cross terms generated by unequal diffusivities are absorbed via Young's inequality with constants depending only on the mobilities and the reaction coefficients; the precise algebraic identity is the expansion of the difference between the linearized and full cubic form, followed by integration by parts on the diffusion terms. We will add an explicit paragraph after the energy estimate in the revised manuscript stating this identity and the absorption constants. revision: yes
Referee: Theorem on asymptotic stability (likely the main result): the claim of unconditional stability is load-bearing. If the linearization step relies on an implicit assumption that previous-stage values remain in a region where the reaction Jacobian remains dissipative, this assumption must be verified uniformly in the time step; otherwise the result becomes conditional on a hidden restriction.

Authors: Theorem 5.1 establishes unconditional asymptotic stability without any restriction on Δt. The dissipativity of the reaction Jacobian is guaranteed by the positivity and uniform boundedness of the numerical solution, which are proved independently of Δt via the discrete maximum principle (Lemma 4.3) and the energy decay. The bootstrap argument in the proof closes for arbitrary time-step sizes because the linearization error is absorbed using the same a priori bounds that are already available at each step. No hidden CFL-type condition appears. We will insert a clarifying sentence in the statement of Theorem 5.1 and in the proof to emphasize that the Jacobian dissipativity holds uniformly thanks to the maximum principle. revision: partial

Circularity Check

0 steps flagged

No significant circularity; stability claim rests on independent energy analysis

full rationale

The paper presents a finite-element spatial discretization combined with a Crank-Nicolson/Adams-Bashforth-type semi-implicit time scheme for a three-species competition-diffusion system. Its strongest claim is a rigorous stability analysis proving that the fully discrete scheme inherits the continuous model's asymptotic stability without time-step restrictions. This is a direct mathematical estimate on the discrete operator (implicit diffusion plus stage-wise linearized reactions) relative to the continuous Lyapunov structure, not a fitted parameter, self-definition, or renaming of an empirical pattern. No load-bearing self-citation chain or ansatz smuggled via prior work is indicated; the derivation chain remains self-contained against the continuous model.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard well-posedness and asymptotic stability of the continuous competition-diffusion PDE system plus the approximation properties of the chosen finite-element space; no free parameters, ad-hoc constants, or new postulated entities are introduced in the abstract.

axioms (2)

domain assumption The continuous three-species competition-diffusion system possesses well-defined asymptotic stability properties.
Invoked when claiming that the discrete scheme inherits those properties.
standard math The finite-element space is C0-conforming and the diffusion operator is treated implicitly in a manner compatible with the continuous energy estimates.
Standard assumption for Galerkin methods on diffusion problems.

pith-pipeline@v0.9.0 · 5486 in / 1379 out tokens · 70938 ms · 2026-05-10T15:08:20.346575+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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A.N. Kolmogorov, I. Petrovsky and N. Piskunov, Investi gation of the equation of diﬀusion com- bined with increasing of the substance and its application t o a biology problem, Moscow University Mathematics Bulletin 1 (6) (1937) 1-25

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Mimura and M

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Petrovskii, K

S. Petrovskii, K. Kawasaki, F. Takasu and N. Shigesada, Diﬀusive waves, dynamical stabilization and spatiotemporal chaos in a community of three competitiv e species, Japan Journal of Industrial and Applied Mathematics 18 (2) (2001) 459-481

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A. Tandon and K. Jyotsna, Modeling the eﬀects of environm ental pollution intensiﬁed by ur- banization on human population, International Journal of Modeling, Simulation, and Scienti ﬁc Computing 7(3) (2016) 1650013

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[1] [1]

Adamson and A.Y

M.W. Adamson and A.Y. Morozov, Revising the role of speci es mobility in maintaining biodi- versity in communities with cyclic competition, Bulletin of Mathematical Biology 74 (9) (2012) 2004-2031

work page 2012

[2] [2]

Boscarino, F

S. Boscarino, F. Filbet and G. Russo, High order semi-imp licit schemes for time dependent partial diﬀerential equations, Journal of Scientiﬁc Computing 68 (3) (2016) 975-1001

work page 2016

[3] [3]

Cangiani, E.H

A. Cangiani, E.H. Georgoulis, A.Y. Morozov and O.J. Sutt on, Revealing new dynamical patterns in a reaction-diﬀusion model with cyclic competition via a no vel computational framework, Proc. R. Soc. A.47420170608 (2018)

work page 2018

[4] [4]

Cia and Z

X. Cia and Z. Cen, Numerical Simulation Technique for Non linear Singularly Perturbed Predator- Prey Reaction Diﬀusion System in Biomathematics, Natural Computation, ICNC , 5 (2007) 44-48

work page 2007

[5] [5]

Chen, L.-C

C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact tra velling wave solutions of three- species competition-diﬀusion systems, Discrete & Continuous Dynamical Systems-Series B 17(8) (2012) 2653-2669

work page 2012

[6] [6]

Ciarlet, The Finite Element Method for Elliptic Problems , North-Holland, Amsterdam, 1978

P.G. Ciarlet, The Finite Element Method for Elliptic Problems , North-Holland, Amsterdam, 1978

work page 1978

[7] [7]

Contento, M

L. Contento, M. Mimura and M. Tohma, Two-dimensional tra velling waves arising from planar front interaction in a three-species competition-diﬀusion system, Japan Journal of Industrial and Applied Mathematics 32 (3) (2015) 707-747

work page 2015

[8] [8]

Durrett and S

R. Durrett and S. Levin, Spatial aspects of interspeciﬁc competition, Theoretical Population Bi- ology 53 (1) (1998) 30-43. 18

work page 1998

[9] [9]

S.I. Ei, R. Ikota and M. Mimura, Segregating partition pr oblem in competition-diﬀusion systems, Interfaces and Free Boundaries 1 (1) (1999) 57-80

work page 1999

[10] [10]

Fisher, The wave of advance of advantageous genes, Annals of Human Genetics 7 (4) (1937) 355-369

R.A. Fisher, The wave of advance of advantageous genes, Annals of Human Genetics 7 (4) (1937) 355-369

work page 1937

[11] [11]

Gilpin, Limit cycles in competition communities, The American Naturalist 109 (965) (1975) 51-60

M.E. Gilpin, Limit cycles in competition communities, The American Naturalist 109 (965) (1975) 51-60

work page 1975

[12] [12]

Kishimoto, The diﬀusive Lotka-Volterra system with t hree species can have a stable non constant equilibrium solution, Journal of Mathematical Biology 16 (1) (1982) 103-112

K. Kishimoto, The diﬀusive Lotka-Volterra system with t hree species can have a stable non constant equilibrium solution, Journal of Mathematical Biology 16 (1) (1982) 103-112

work page 1982

[13] [13]

Kolmogorov, I

A.N. Kolmogorov, I. Petrovsky and N. Piskunov, Investi gation of the equation of diﬀusion com- bined with increasing of the substance and its application t o a biology problem, Moscow University Mathematics Bulletin 1 (6) (1937) 1-25

work page 1937

[14] [14]

May and W.J

R.M. May and W.J. Leonard, Nonlinear aspects of competi tion between three species, SIAM Journal of Applied Mathematics 29 (2) (1975) 243-253

work page 1975

[15] [15]

Mimura and P.C

M. Mimura and P.C. Fife, A 3-component system of competi tion -diﬀusion system, Hiroshima Mathematical Journal 16 (1986) 189-207

work page 1986

[16] [16]

Mimura and M

M. Mimura and M. Tohma, Dynamic coexistence in a three-s pecies competition-diﬀusion system, Ecological Complexity 21 (2015) 215-232

work page 2015

[17] [17]

Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989

J.D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989

work page 1989

[18] [18]

Okubo, Diﬀusion and Ecological Problems: Mathematical Models , Springer-Verlag, Berlin, 1989

A. Okubo, Diﬀusion and Ecological Problems: Mathematical Models , Springer-Verlag, Berlin, 1989

work page 1989

[19] [19]

Pereira, J

T.A. Pereira, J. Menezes and L. Losano, Interface netwo rks in models of competing species, International Journal of Modeling, Simulation, and Scienti ﬁc Computing 9(5) (2018) 1850046

work page 2018

[20] [20]

Petrovskii, K

S. Petrovskii, K. Kawasaki, F. Takasu and N. Shigesada, Diﬀusive waves, dynamical stabilization and spatiotemporal chaos in a community of three competitiv e species, Japan Journal of Industrial and Applied Mathematics 18 (2) (2001) 459-481

work page 2001

[21] [21]

Reichenbach, M

T. Reichenbach, M. Mobilia and E. Frey, Mobility promot es and jeopardizes biodiversity in rock– paper–scissors games, Nature 448 (7157) (2007) 1046-1049

work page 2007

[22] [22]

Shigesada and K

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice , Oxford University Press, Oxford, 1997

work page 1997

[23] [23]

Tandon and K

A. Tandon and K. Jyotsna, Modeling the eﬀects of environm ental pollution intensiﬁed by ur- banization on human population, International Journal of Modeling, Simulation, and Scienti ﬁc Computing 7(3) (2016) 1650013

work page 2016

[24] [24]

Volterra, Variazioni e ﬂuttuazioni del numero d’ind ividui in specie animali conviventi, Memorie Accademia Nazionale Dei Lincei 6 (2) (1926) 31-113

V. Volterra, Variazioni e ﬂuttuazioni del numero d’ind ividui in specie animali conviventi, Memorie Accademia Nazionale Dei Lincei 6 (2) (1926) 31-113. 19

work page 1926