Narrow Results

Spatial and temporal pattern formations of interacting species in ecological and chemical systems have become a central object of research in recent decades. We consider a two-competing-prey and one-predator Lotka–Volterra reaction–diffusion model. We first establish the sufficient conditions for the global attractivity of positive constant equilibrium solution. Then by taking cross-diffusion coefficients as the bifurcation parameter, we show that our model undergoes an inhomogeneous Hopf bifurcation around the positive constant equilibrium solution when the bifurcation parameter is varied through critical values. Furthermore, we prove that these inhomogeneous bifurcating periodic solutions are stable when diffusion coefficients lie in suitable ranges. A common feature in the most existing research work is that the bifurcation factor that induces Hopf bifurcation appears in the reaction terms rather than diffusion terms. We show that the inhomogeneous Hopf bifurcation can be triggered by the effect of cross-diffusion factors. Our results demonstrate that cross-diffusion plays a crucial role in the formation of spatially inhomogeneous periodic oscillatory patterns.

In this paper, we presented a predator–prey model with self diffusion as well as cross diffusion. By using theory on linear stability, we obtain the conditions on Turing instability. The results of numerical simulations reveal that oscillating Turing patterns with hexagons arise in the system. And the values of the parameters we choose for simulations are outside of the Turing domain of the no cross diffusion system. Moreover, we show that cross diffusion has an effect on the persistence of the population, i.e., it causes the population to run a risk of extinction. Particularly, our results show that, without interaction with either a Hopf or a wave instability, the Turing instability together with cross diffusion in a predator–prey model can give rise to spatiotemporally oscillating solutions, which well enrich the finding of pattern formation in ecology.

Increasingly sophisticated techniques are being developed for the manufacture of functional nanomaterials. A growing interest is also developing in magnetic nanofluid coatings which contain magnetite nanoparticles suspended in a base fluid and are responsive to external magnetic fields. These nanomaterials are “smart” and their synthesis features high-temperature environments in which radiative heat transfer is present. Diffusion processes in the extruded nanomaterial sheet also feature Soret and Dufour (cross) diffusion effects. Filtration media are also utilized to control the heat, mass and momentum characteristics of extruded nanomaterials and porous media impedance effects arise. Magnetite nanofluids have also been shown to exhibit hydrodynamic wall slip which can arise due to non-adherence of the nanofluid to the boundary. Motivated by the multi-physical nature of magnetic nanomaterial manufacturing transport phenomena, in this paper, we develop a mathematical model to analyze the collective influence of hydrodynamic slip, radiative heat flux and cross-diffusion effects on transport phenomena in ferric oxide (${\mathrm{\text{Fe}}}_3{\mathrm{\text{O}}}_4$-water) magnetic nanofluid flow from a nonlinear stretching porous sheet in porous media. Hydrodynamic slip is included. Porous media drag is simulated with the Darcy model. Viscous magnetohydrodynamic theory is used to simulate Lorentzian magnetic drag effects. The Rosseland diffusion flux model is employed for thermal radiative effects. A set of appropriate similarity transformation variables are deployed to convert the original partial differential boundary value problem into an ordinary differential boundary value problem. The numerical solution of the coupled, multi-degree, nonlinear problem is achieved with an efficient shooting technique in MATLAB symbolic software. The physical influences of Hartmann (magnetic) number, Prandtl number, Richardson number, Soret (thermo-diffusive) number, permeability parameter, concentration buoyancy ratio, radiation parameter, Dufour (diffuso-thermal) parameter, momentum slip parameter and Schmidt number on transport characteristics (e.g. velocity, nanoparticle concentration and temperature profiles) are investigated, visualized and presented graphically. Flow deceleration is induced with increasing Hartmann number and wall slip, whereas flow acceleration is generated with greater Richardson number and buoyancy ratio parameter. Temperatures are elevated with increasing Dufour number and radiative parameter. Concentration magnitudes are enhanced with Soret number, whereas they are depleted with greater Schmidt number. Validation of the MATLAB computations with special cases of the general model is included. Further validation with generalized differential quadrature (GDQ) is also included.

3D image is a kind of image closer to life, and 3D image has a stronger visual impact. In the past, researchers mostly protected flat images, but ignored the protection of 3D images. This paper combines the 2D chaotic system and uses random cross diffusion to propose a new encryption algorithm for 3D images which are STL format. Different from the traditional scrambling and diffusion operations, this paper uses only one iterative operation to achieve the effects of scrambling and diffusion, making the algorithm more secure. In the encryption algorithm, each value of the ciphertext is XORed with the value of the plaintext on the same coordinate axis and different from the coordinate where the ciphertext is located, the chaotic sequence, the ciphertext of different coordinate axes significantly reduces the correlation between the adjacent coordinate values. In addition, the proposed 3D image encryption algorithm is tested by statistical analysis. The experimental results show that this 3D image encryption algorithm has high security.

In this paper we consider a model for the behavior of students in graduate programs at neighboring universities which is a modified form of the model proposed by [Scheurle & Seydel, 2000], and observe that the stationary solution of this two-component system becomes unstable in the presence of diffusion. We assume that both types of individuals are continuously distributed throughout a bounded two-dimensional spatial domain of two types (regular hexagon and rhombus), across whose boundaries there is no migration, and which simultaneously undergo simple (Fickian) diffusion but the spatial flow is influenced not only by its own but also by the other's density (cross diffusion). We will show that at a critical value of a parameter a Turing bifurcation takes place: a spatially nonhomogenous solution (pattern) arises.

The main goal of this paper is to continue the investigations of the important system proposed by [Scheurle & Seydel, 2000] and modified by [Sándor, 2003]. I consider spatio-temporal models for the behavior of students in graduate programs at neighboring universities as systems of ODE which describe two-identical patch-two-species systems linked by migration, where the phenomenon of the Turing bifurcation occurs. It is assumed in the model that the per capita migration rate of each individual is influenced not only by its own (Fickian) but also by the other densities, i.e. there is cross diffusion present. We study the conditions of the existence and stability properties of the equilibrium solutions in a kinetic model (no migration). We will show that analytically at a critical value of a parameter a Turing bifurcation takes place: a spatially nonhomogenous solution (pattern) arises. A numerical example is also included.

The mechanical tumor-growth model of Jackson and Byrne is analyzed. The model consists of nonlinear parabolic cross-diffusion equations in one space dimension for the volume fractions of the tumor cells and the extracellular matrix (ECM). It describes tumor encapsulation influenced by a cell-induced pressure coefficient. The global-in-time existence of bounded weak solutions to the initial-boundary-value problem is proved when the cell-induced pressure coefficient is smaller than a certain explicit critical value. Moreover, when the production rates vanish, the volume fractions converge exponentially fast to the homogeneous steady state. The proofs are based on the existence of entropy variables, which allows for a proof of the non-negativity and boundedness of the volume fractions, and of an entropy functional, which yields gradient estimates and provides a new thermodynamic structure. Numerical experiments using the entropy formulation of the model indicate that the solutions exist globally in time also for cell-induced pressure coefficients larger than the critical value. For such coefficients, a peak in the ECM volume fraction forms and the entropy production density can be locally negative.

A presentation of a special issue is delivered by this editorial paper on the derivation and related analytical problems, of cross diffusion models in complex environments. A brief introduction to motivations toward this research topic is presented in the first part. Subsequently, a concise description of the contents follows. Lastly, some ideas on possible research perspectives are proposed mainly focusing on modeling topics.

This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale.

We establish weak–strong uniqueness and stability properties of renormalized solutions to a class of energy-reaction-diffusion systems. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. The weak–strong uniqueness principle holds for dissipative renormalized solutions, which in addition to the renormalized formulation obey suitable dissipation inequalities consistent with previous existence results. We treat general entropy-dissipating reactions without growth restrictions, and certain models with a non-integrable diffusive flux.

The results also apply to a class of (isoenergetic) reaction-cross-diffusion systems.

Reaction-diffusion systems with cross-diffusion are analyzed here for modeling the population dynamics of epidemic systems. In this paper specific attention is devoted to the numerical analysis and simulation of such systems to show that, far from possible pathologies, the qualitative behaviour of the systems may well interpret the dynamics of real systems.

The growth of precipitates in a solution system is studied using the asymptotic method, taking into account the cross diffusion between solutes. The resulting asymptotic solution for the dynamic model of the precipitate growing in the solution reveals that the concentration distribution is significantly changed by the cross diffusion between solutes and the growth of precipitate depends not only on the self diffusion of solutes, but also on the diffusive interaction between solutes. The attractive diffusive interaction between solutes in the solution enhances the solute diffusion and facilitates the growth of precipitate, whereas the repulsive diffusive interaction between solutes depresses the solute diffusion and inhibits the growth of precipitate.

The concept of cross diffusion is applied to some biological systems. The conditions for persistence and Turing instability in the presence of cross diffusion are derived. Many examples including: predator-prey, epidemics (with and without delay), hawk–dove–retaliate and prisoner's dilemma games are given.

The purpose of this paper is to study the effect of cross diffusion in a competition model with stage structure, under homogeneous Neumann boundary condition. It will be shown that cross diffusion cannot only destabilize a uniform positive equilibrium, it can also help diffusion to induce instability of the uniform positive equilibrium. Moreover, stationary patterns can arise from the effect of cross diffusion.