Numerical study of two-species chemotaxis models

Kurganov, Alexander; Lukáčová-Medviďová, Mária

doi:10.3934/dcdsb.2014.19.131

Cited by 25 publications

(42 citation statements)

References 26 publications

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“…The initial conditions for each experiment are given in 5.3. We note that although the first component is expanding, there is evidence that it nonetheless blows up in the L ∞ norm [22]. as a bump function supported on an ellipse centered at (0.1,0) with axes 2a and a/2, with the major axis parallel to the y-axis.…”

Section: Expanding Mpks With Blow-upmentioning

confidence: 80%

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Coalescing particle systems and applications to nonlinear Fokker–Planck equations

Zhelezov

2018

Communications in Mathematical Sciences

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We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre-and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.

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Section: Expanding Mpks With Blow-upmentioning

confidence: 80%

“…We expect that the MPKS can be regularized past blow-up times using a singular perturbation limit, as was done in [30,31] for the PKS, and proposed in [22] for the MPKS. In this case, the presented method is well-suited for the investigation of this regularization.…”

Section: Formation Of Singularities In the Mpksmentioning

confidence: 99%

Coalescing particle systems and applications to nonlinear Fokker–Planck equations

Zhelezov

2018

Communications in Mathematical Sciences

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“…In Section 3, we extend the numerical method to the Keller-Segel equations with degenerate diffusions. Several numerical examples are given in the last section to verify the claimed properties and demonstrate its application in various challenging cases, including blowup solutions, degenerate diffusions with large m (see [13]) and two-species models with different blowup behavior (see [20]).…”

Section: Introductionmentioning

confidence: 93%

“…In this section, we test our scheme on a two-species model [20]:        ∂ t ρ 1 + χ 1 ∇ · (ρ 1 ∇c) = µ 1 ∆ρ 1 , ∂ t ρ 2 + χ 2 ∇ · (ρ 2 ∇c) = µ 2 ∆ρ 2 , εc t = D∆c + α 1 ρ 1 + α 2 ρ 2 − βc. Here ρ 1 and ρ 2 denote the cell densities of the first and second species.…”

Section: Two Speciesmentioning

confidence: 99%

“…The main issue there is that the Jacobian matrices coming from the advection part may have complex eigenvalues, which force the advection part to be solved together with the stabilizing diffusion terms, and result in complicated CFL conditions. Based on this formulation, Kurganov and his collaborators have conducted many extensions, including more general chemotaxis flux model, multi-species model and constructing an alternative discontinuous Galerkin method; see [11,15,20]. Very recently, Li et.…”

Section: Introductionmentioning

confidence: 99%

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Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations

2017

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We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we prove that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.

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A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

Mostajeran

Hosseini

2023

Math Methods in App Sciences

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In this study, we introduce a new growing neural network algorithm that is based on wavelet neural networks and call our algorithm a growing wavelet neural network (GWNN) method. We apply our proposed scheme to train a wavelet neural network to solve chemotaxis problems with blow‐up. These problems are highly nonlinear time‐dependent systems of partial differential equations, and it is a challenge to get the pattern of the solution accurately. The proposed structure is partial retraining of the network, which increases its capacity to catch the spiky pattern of the solution. Our neural network‐based algorithm allows us to solve the nonlinear chemotaxis problems without the use of linearization techniques and regularization techniques, most of which reduce the accuracy of the model. This mesh‐free‐based method can manage a variety of blow‐up models with curved boundaries without imposing an extra cost. By proving the consistency and stability of the method, we show the convergence of GWNN solutions to analytical solutions of the chemotaxis problem. Several illustrative examples and simulation results are provided to demonstrate the correctness of the results and the robust performance of the presented algorithm. Moreover, to illustrate the effectiveness of the GWNN method, we make a comparison with two other network‐based methods.

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Numerical study of two-species chemotaxis models

Cited by 25 publications

References 26 publications

Coalescing particle systems and applications to nonlinear Fokker–Planck equations

Coalescing particle systems and applications to nonlinear Fokker–Planck equations

Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

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