A velocity–diffusion method for a Lotka–Volterra system with nonlinear cross and self-diffusion

“…In [26,27] the authors prove the convergence of a semidiscretization in time scheme in one space dimension by introducing a transformation of the unknowns. Several results in one spatial dimension using a deterministic particle method are presented in [28]. In [6], the authors propose a fully discrete finite element scheme for a cross-diffusion system using an entropy inequality, and also present one-dimensional examples.…”

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confidence: 99%

Analysis of a Finite Volume Method for a Cross-Diffusion Model in Population Dynamics

Andreïanov

Bendahmane

Ruiz–Baier

2011

Math. Models Methods Appl. Sci.

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Abstract. The main goal of this work is to propose a convergent finite volume method for a reaction-diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time L 1 compactness argument that mimics the compactness lemma due to S.N. Kruzhkov. The proofs of these results are given in the Appendix.

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“…Recently [59] investigated the chaotic behavior of diffusive Allee model. More recently, some researchers investigated the Turing instability, the phenomena of pattern formation, travelling front solutions, stable spatially inhomogeneous periodic solution in the ecological system [34,61,78,95,96] . Therefore, different types of mathematical models (ODE, PDE, DDE, SDE, etc.)…”

Section: Introductionmentioning

confidence: 99%

Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator

Saifuddin

Biswas

Samanta

et al. 2016

Chaos, Solitons & Fractals

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A velocity–diffusion method for a Lotka–Volterra system with nonlinear cross and self-diffusion

Cited by 46 publications

References 30 publications

Competitive spatially distributed population dynamics models: Does diversity in diffusion strategies promote coexistence?

Competitive spatially distributed population dynamics models: Does diversity in diffusion strategies promote coexistence?

Analysis of a Finite Volume Method for a Cross-Diffusion Model in Population Dynamics

Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator

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