Cross-diffusion Terms Research Articles

Induction of patterns through crowding in a cross-diffusion model

In this paper we focus on pattern formation in systems of interacting populations. We show that if one considers these populations to be “crowded” in a way that is defined below, then cross-diffusion terms appear naturally. Moreover, we show that these additional cross-diffusion terms can generate stable spatial patterns that are not manifest in the corresponding standard “dilute” formulation. This result demonstrates the need for care when choosing standard Fickian diffusion as the default in applications to population dynamics.

A free-energy based multiple-distribution-function lattice Boltzmann method for multi-component and multi-phase flows

This study presents the development of a multiple-distribution-function lattice Boltzmann model (MDF-LBM) for the accurate simulation of multi-component and multi-phase flow. The model is based on the diffuse interface theory and free energy model, which enable the derivation of hydrodynamic equations for the system. These equations comprise a Cahn-Hilliard (CH) type mass balance equation, which accounts for cross diffusion terms for each species, and a momentum balance equation. By establishing a relationship between the total chemical potential and the general pressure, the momentum balance equation is reformulated in a potential form. This potential form, together with the CH type mass balance equation, is then utilized to construct the MDF-LBM as a coupled convection–diffusion system. Numerical simulations demonstrate that the proposed MDF-LBM accurately captures phase behavior and ensures mass conservation. Additionally, the calculated interface tension exhibits good agreement with experimental data obtained from laboratory studies.

Instability of double-diffusive magnetoconvection in a non-Newtonian fluid layer with cross-diffusion effects

The study explores the initiation of two-dimensional double-diffusive convection in a horizontal layer of an electrically conducting non-Newtonian Navier–Stokes–Voigt fluid, subjected to a uniform vertical magnetic field and cross-diffusion effects. The numerical results are presented by obtaining the analytical solutions for both steady and oscillatory onset scenarios. The viscoelastic nature of the fluid either delays or hastens the onset of oscillatory convection depending on the strength of solute concentration. The analysis also uncovers contradictions in the linear instability characteristics with and without cross-diffusion terms, even when other input parameters are identical. Under specific conditions, three novel phenomena are observed that are not typically seen in double-diffusive cases: (i) an electrically conducting Navier–Stokes–Voigt fluid layer, initially linearly stable in the presence of a magnetic field, can become destabilized with the addition of a heavy solute to the fluid's bottom; (ii) a stable double-diffusive electrically conducting Navier–Stokes–Voigt fluid layer can be destabilized by the application of a magnetic field; and (iii) the requirement of three critical values of the thermal Rayleigh number to determine linear instability, as opposed to the usual single value owing to the existence of disconnected closed convex oscillatory neutral curves. The results are shown to align with previously published findings in the limiting cases.

Nonstandard finite difference methods for a convective predator-prey pursuit and evasion model

One standard and two nonstandard finite difference methods are used to solve a convective predator-prey model consisting of cross-diffusion terms for which no exact solution is known. The model involves a system of strongly coupled nonlinear parabolic partial differential equations. Through some numerical experiments, we observe that reasonable numerical solutions are obtained when the temporal step size satisfies k ≤ 0.0005 with spatial step size h = 0.1 . We construct a nonstandard method using ideas from Anguelov et al. and the scheme is denoted by NSFD1. We obtain four conditions for NSFD1 to replicate the positivity property of the continuous model, but these conditions consist of many parameters. Therefore, an unconditionally positivity-preserving scheme is considered which is denoted as NSFD2. However, we found that NSFD2 is in general not consistent. We obtain conditions under which NSFD2 is both consistent and bounded and some numerical results are presented when these conditions are satisfied.

Cross diffusion effects on MHD double diffusive viscous flow through Hermite wavelet method

AbstractDouble-diffusive convection is a form of fluid flow that occurs when two processes of molecular diffusion are active in a fluid at the same time, causing instabilities and also complicated behaviour. One chemical or biological species concentration can cause a flux of another species, either linearly or nonlinearly, a phenomenon known as cross-diffusion. The cross-diffusion effects on double-diffusive MHD fluid flow through the Hermite wavelet method is examined. The governing coupled partial differential equations of the problem under consideration are transformed to highly nonlinear ordinary differential equations over a finite domain with the help of similarity transformations. The results are obtained for the skin friction coefficient, as well as the velocity, temperature and the concentration profiles for some values of the governing parameters, namely, the cross diffusion terms, Hartmann number, thermophoresis parameter, squeeze number, Prandtl number and suction/injection parameter. The obtained results are validated against previously published results for special case of the problems.

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

A nonlinear crime model is generalized by introducing self- and cross-diffusion terms. The effect of diffusion on the stability of non-negative constant steady states is applied. In particular, the cross-diffusion-driven instability, called Turing instability, is analyzed by linear stability analysis, and several Turing patterns driven by the cross-diffusion are studied through numerical investigations. When the Turing–Hopf conditions are satisfied, the type of instability highlighted in the ODE model persists in the PDE system, still showing an oscillatory behavior.

Triple diffusive free magneto-convection of electro-conductive nanofluid from a vertical stretching sheet

In the present investigation, an analysis is carried out to study the MHD triple diffusive free thermo-solutal convection boundary layer flow of an electro-conductive nanofluid flow over a vertical stretching sheet. This problem is relevant to magnetic nanomaterials fabrication operations in which multiple species in addition to nanoparticles are present. In addition to the nanoparticle diffusion, two different salts (species) having different properties are considered. A variable magnetic field is applied transverse to the vertical sheet. It is assumed that the surface is in contact with the hot magnetic nanofluid at a temperature which provides a variable heat transfer coefficient. Buongiorno’s model is employed for the nanofluid. It is also assumed that the Oberbeck-Boussinesq approximation is valid and the mixture of nanofluid and salts is homogenous and is in local thermal equilibrium. In addition, the thermal energy equation features cross-diffusion (Soret and Dufour) terms for both components of salts having different concentration. Appropriate similarity transformations are deployed to render the model non-dimensional. The emerging transformed dimensionless non-linear non-dimensional ordinary differential boundary value problem is solved with the robust bvp4c method in MATLAB. Validation with previous studies has been included for special cases of the general model. The simulations show that the addition of nanoparticles and salts, strongly modifies Temperature and nanoparticle and salt 1 and 2 concentrations. With stronger magnetic field the velocity is suppressed as is momentum boundary layer thickness whereas temperatures are boosted.

Cross-diffusion induced instability on networks

Abstract The concept of Turing instability, namely that diffusion can destabilize the homogenous steady state, is well known either in the context of partial differential equations (PDEs) or in networks of dynamical systems. Recently, reaction–diffusion equations with non-linear cross-diffusion terms have been investigated, showing an analogous effect called cross-diffusion induced instability. In this article, we consider non-linear cross-diffusion effects on networks of dynamical systems, showing that also in this framework the spectrum of the graph Laplacian determines the instability appearance, as well as the spectrum of the Laplace operator in reaction–diffusion equations. We extend to network dynamics a particular network model for competing species, coming from the PDEs context, for which the non-linear cross-diffusion terms have been justified, e.g. via a fast-reaction limit. In particular, the influence of different topology structures on the cross-diffusion induced instability is highlighted, considering regular rings and lattices, and also small-world, Erdős–Réyni, and Barabási–Albert networks.

A nonlinear repair technique for the MPFA-D scheme in single-phase flow problems and heterogeneous and anisotropic media

A novel Flux Limited Splitting (FLS) non-linear Finite Volume (FV) method for families of linear Control Volume Distributed Multi Point Flux Approximation (CVD-MPFA) schemes is presented. The new formulation imposes a local discrete maximum principal (LDMP) which ensures that the discrete solution is free of spurious oscillations. The FLS scheme can be seen as a natural extension of the M-Matrix Flux Splitting method that splits the MPFA flux components in terms of the Two-Point Flux Approximation (TPFA) flux and Cross Diffusion Terms (CDT), with the addition of a dynamically computed relaxation parameter to the CDT that identifies and locally corrects the regions where the LDMP is violated. Moreover, the whole non-linear procedure was devised as a series of simple straightforward matrix operations. The methodology is presented considering the Multi-Point Flux Approximation with a Diamond (MPFA-D) in what we call the FLS + MPFA-D formulation which is tested using a series of challenging benchmark problems. For all test cases, the FLS repair technique imposes the LDMP and eliminates the spurious oscillations induced by the original MPFA-D method.

Improving the bioactive content in honeydew melon by impregnation with Hibiscus extract/sucrose solutions: A coupled mass transfer analysis

This study investigated the bioactive content enrichment of honeydew melon cuts (hemispheres) immersed into Hibiscus extract/sucrose solutions (HESS). A face-centered design (FCD) was initially conducted to explore the effect of time (30–210 min), solution concentration (20–60°Bx) and temperature (40–60 °C) on water loss (WL), solids gain (SG), as well as in the increase of anthocyanins (ANT), phenolics (PHE), and antioxidant capacity (AXC) of melon. WL and SG were also investigated with pure sucrose syrups (SS) for comparison purposes. Processing conditions were further optimized by a Pareto approach to maximize both WL and bioactive content enrichment with minimal SG. Then, impregnation kinetics, conducted around the optimal experimental conditions, were analyzed with a 2D diffusive model, considering the anisotropic shrinkage of melon and solved under four complexity levels to examine coupled mass transfer (including constant diffusivities, main plus cross diffusion terms or nonlinear diffusivity functions of local water content). HESS favored a higher WL and lower SG than SS during FCD. The optimal condition occurred at 30 min, 20°Bx and about 50 °C, where melon gained ANT not originally present in it, and PHE and AXC increased by factors of 4.5 and 6.1 in comparison with untreated fruit, respectively. No evidence of significant coupling among WL, SG and bioactive transfer was found (p>0.05), with diffusivities for ANT, PHE, AXC, water and solids ranging from 3.26 to 3.45, 12.8–14.4, 1.24–5.19, 0.82–1.36, and 1.19–4.26 (×10−10) m2/s, respectively.

Optimal Distributed Control for a Viscous Non-local Tumour Growth Model

In this paper, we address an optimal distributed control problem for a non-local model of phase-field type, describing the evolution of tumour cells in presence of a nutrient. The model couples a non-local and viscous Cahn–Hilliard equation for the phase parameter with a reaction-diffusion equation for the nutrient. The optimal control problem aims at finding a therapy, encoded as a source term in the system, both in the form of radiotherapy and chemotherapy, which could lead to the evolution of the phase variable towards a desired final target. First, we prove strong well-posedness for the system of non-linear partial differential equations. In particular, due to the presence of a viscous regularisation, we can also consider double-well potentials of singular type and cross-diffusion terms related to the effects of chemotaxis. Moreover, the particular structure of the reaction terms allows us to prove new regularity results for this kind of system. Then, turning to the optimal control problem, we prove the existence of an optimal therapy and, by studying Fréchet-differentiability properties of the control-to-state operator and the corresponding adjoint system, we obtain the first-order necessary optimality conditions.

Excitable FitzHugh-Nagumo model with cross-diffusion: long-range activation instabilities

AbstractIn this paper, we shall study a spatially extended version of the FitzHugh-Nagumo model, where one describes the motion of the species through cross-diffusion. The motivation comes from modeling biological species where reciprocal interaction influences spatial movement. We shall focus our analysis on the excitable regime of the system. In this case, we shall see how cross-diffusion terms can destabilize uniform equilibrium, allowing for the formation of close-to-equilibrium patterns; the species are out-of-phase spatially distributed, namely high concentration areas of one species correspond to a low density of the other (cross-Turing patterns). Moreover, depending on the magnitude of the inhibitor’s cross-diffusion, the pattern’s development can proceed in either case of the inhibitor/activator diffusivity ratio being higher or smaller than unity. This allows for spatial segregation of the species in both cases of short-range activation/long-range inhibition or long-range activation/short-range inhibition.

Excitable FitzHugh-Nagumo model with cross-diffusion: close and far-from-equilibrium coherent structures

In this paper, we shall study the formation of stationary patterns for a reaction-diffusion system in which the FitzHugh-Nagumo (FHN) kinetics, in its excitable regime, is coupled to linear cross-diffusion terms. In (Gambino et al. in Excitable Fitzhugh-Nagumo model with cross-diffusion: long-range activation instabilities, 2023), we proved that the model supports the emergence of cross-Turing patterns, i.e., close-to-equilibrium structures occurring as an effect of cross-diffusion. Here, we shall construct the cross-Turing patterns close to equilibrium on 1-D and 2-D rectangular domains. Through this analysis, we shall show that the species are out-of-phase spatially distributed and derive the amplitude equations that govern the pattern dynamics close to criticality. Moreover, we shall classify the bifurcation in the parameter space, distinguishing between super-and sub-critical transitions. In the final part of the paper, we shall numerically investigate the impact of the cross-diffusion terms on large-amplitude pulse-like solutions existing outside the cross-Turing regime, showing their emergence also in the case of lateral activation and short-range inhibition.

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path.

Numerical Solution of Radiative and Dissipative Flow on Non-Newtonian Casson Fluid Model via Infinite Vertical Plate with Thermo-Diffusion and Diffusion-Thermo Effects

This research presents mathematically developed model to examine non-Newtonian Casson fluid flow in the existence of radiation, Ohmic dissipation, thermo-diffusion and diffusion-thermo over infinite vertical plate domain. Using similarity transformations, the governing partial derivative related to fluid model is transmuted to ordinary derivative equations and then solved computationally by adopting Runge-Kutta method via shooting quadrature in mathematical software MAPLE. The impacts of various considered effects were assed and solutions for momentum velocity profiles, heat transfer energy and mass transfer concentration profiles are investigated via graphical presentation. The outcomes show that radiation and magnetic field increased heat distribution and improvement in yield stress through an enhancement in Casson term reduces the flow speed. Presence of Cross diffusion terms has remarkable impact on thermal and solutal profiles. Further, numerical significances of engineering quantities such as skin friction, Nusselt number and Sherwood number are provided in tabular form. Finally, to justify the outcomes of this study, a resemblance is taken with earlier published works and found there is good correlation.

High-order finite difference approximation of the Keller-Segel model with additional self- and cross-diffusion terms and a logistic source

<abstract><p>In this paper, we consider the Keller-Segel chemotaxis model with self- and cross-diffusion terms and a logistic source. This system consists of a fully nonlinear reaction-diffusion equation with additional cross-diffusion. We establish some high-order finite difference schemes for solving one- and two-dimensional problems. The truncation error remainder correction method and fourth-order Padé compact schemes are employed to approximate the spatial and temporal derivatives, respectively. It is shown that the numerical schemes yield second-order accuracy in time and fourth-order accuracy in space. Some numerical experiments are demonstrated to verify the accuracy and reliability of the proposed schemes. Furthermore, the blow-up phenomenon and bacterial pattern formation are numerically simulated.</p></abstract>

Turing pattern selection for a plant–wrack model with cross-diffusion

We investigate the Turing instability and pattern formation mechanism of a plant–wrack model with both self-diffusion and cross-diffusion terms. We first study the effect of self-diffusion on the stability of equilibrium. We then derive the conditions for the occurrence of the Turing patterns induced by cross-diffusion based on self-diffusion stability. Next, we analyze the pattern selection by using the amplitude equation and obtain the exact parameter ranges of different types of patterns, including stripe patterns, hexagonal patterns and mixed states. Finally, numerical simulations confirm the theoretical results.

Turing and wave instabilities in hyperbolic reaction–diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation

Hyperbolic reaction–diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities not present in classical two-species reaction–diffusion systems. We explore the onset of diffusive instabilities and resulting pattern formation for such systems. Starting with a rather general formulation of the problem, we obtain necessary and sufficient conditions for the Turing and wave instabilities in such systems, thereby classifying parameter spaces for which these diffusive instabilities occur. We find that the additional temporal terms do not strongly modify the Turing patterns which form or parameters which admit them, but only their regions of existence. This is in contrast to the case of additional space derivatives, where past work has shown that resulting patterned structures are sensitive to second-order cross-diffusion and first-order advection. We also show that additional temporal terms are necessary for the emergence of spatiotemporal patterns under the wave instability. We find that such wave instabilities exist for parameters which are mutually exclusive to those parameters leading to stationary Turing patterns. This implies that wave instabilities may occur in cases where the activator diffuses faster than the inhibitor, leading to routes to spatial symmetry breaking in reaction–diffusion systems which are distinct from the well studied Turing case.

A convergent finite volume scheme for dissipation driven models with volume filling constraint

In this paper we propose and study an implicit finite volume scheme for a general model which describes the evolution of the composition of a multi-component mixture in a bounded domain. We assume that the whole domain is occupied by the different phases of the mixture which leads to a volume filling constraint. In the continuous model this constraint yields the introduction of a pressure, which should be thought as a Lagrange multiplier for the volume filling constraint. The pressure solves an elliptic equation, to be coupled with parabolic equations, possibly including cross-diffusion terms, which govern the evolution of the mixture composition. Besides the system admits an entropy structure which is at the cornerstone of our analysis. More precisely, the main objective of this work is to design a two-point flux approximation finite volume scheme which preserves the key properties of the continuous model, namely the volume filling constraint and the control of the entropy production. Thanks to these properties, and in particular the discrete entropy-entropy dissipation relation, we are able to prove the existence of solutions to the scheme and its convergence. Finally, we illustrate the behavior of our scheme through different applications.

Spatiotemporal pattern selection in a nontoxic-phytoplankton - toxic-phytoplankton - zooplankton model with toxin avoidance effects

In this research, considering the toxin avoidance effects of zooplankton to toxic phytoplankton in space as cross-diffusion terms, a modified model is proposed on the complex spatiotemporal dynamics of non-toxic phytoplankton, toxic phytoplankton and zooplankton system. In order to explore the toxin avoidance effects on pattern formation and selection, Turing bifurcation analysis and amplitude analysis are performed. The results of Turing analysis show that patterns can be formed when diffusion coefficient is less than a critical value. Amplitude equations are calculated and amplitude analysis show that stripe patterns, hexagon patterns and mixed patterns of stripes and hexagons can be theoretically predicted. Focusing on toxin avoidance coefficient (cross-diffusion coefficient) and diffusion coefficient of zooplankton, parameter space for pattern formation and selection is plotted. Numerical simulations on pattern formation are carried out to validate the theoretical analysis with five groups of parameters. With the increase of toxin avoidance effects, simulated patterns can be obtained as hexagons, mixed patterns of stripes and hexagons, stripes, mixed patterns of stripes and hexagons, and hexagons. Finally, taking stripe pattern as an example, simulated pattern is compared with predicted pattern. Simulation results are all consistent with theoretical analysis.