Narrow Results

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.

A weakly nonlinear analysis provides a system constituting amplitude equations and its related analysis is capable of predicting parameter regimes with different patterns expected to co-exist in dynamical circumstances that exhibit complex fractional-order system characteristics. The Turing mechanism of pattern formation as a result of diffusion-induced instability of the homogeneous steady state is concerned with unpredictable conditions. The Turing instability caused by fractional diffusion in a Human Immunodeficiency Virus model has been addressed in this study. It is important that the effect of the Human Immunodeficiency Virus to the immune system can be modeled by the interaction of uninfected cells, unhealthy cells, virus particles and antigen-specific. Initially, all potential equilibrium points are defined and the stability of the interior equilibrium point is then evaluated using the Routh–Hurwitz criteria. The conditions for Turing instability are obtained by local equilibrium points with stability analysis. In the neighborhood of the Turing bifurcation point, weakly nonlinear analysis is employed to deduce the amplitude equations. After applying amplitude equations, it is observed that this system has a very rich dynamical behavior. The constraints for the formation of the patterns like a hexagon, spot, mixed and stripe patterns are identified for the amplitude equations by dynamic analysis. Furthermore, by using the numerical simulations, the theoretical results are verified. Within this framework, this study through the dynamical behavior of the complex system perspective and bifurcation point based on the viral death rate can provide the basis for several researchers working on Human Immunodeficiency Virus model through various aspects. Accordingly, the Turing bifurcation point and weakly nonlinear analysis employed within the complex fractional-order dynamics addressed herein are highly relevant experimentally since the related effects can be studied and applied concerning different mathematical, physical, engineering and biological models.

This paper introduces some important dissipative problems that are recent and still of intermittent interest. The classical dynamics of Helmholtz and Kelvin–Helmholtz instability equations are modeled with the Riesz operator which incorporates the left- and right-sided of the Riemann–Liouville non-integer order operators to mimic naturally the physical patterns of these models arising in hydrodynamics and geophysical fluids. The Laplace and Fourier transform techniques are used to approximate the Riesz fractional operator in a spatial direction. The behaviors of the Helmholtz and Kelvin–Helmholtz equations are observed for some values of fractional power in the regimes, $0<\alpha \le 1$ and $1<\alpha \le 2$, using different boundary conditions on a square domain in 1D, 2D and 3D (spatial-dimensions). Numerical results reveal some astonishing and very impressive phenomena which arise due to the variations in the initial and source function, as well as fractional parameter $\alpha$, for subdiffusive and superdiffusive scenarios.

In this report we present the formation of necklace type optical beams from Gaussian beam by rotational symmetry masks. The detailed studies of space evolution of the optical beam formation from the near field to the far field by the use of 7-fold symmetry mask and a single mode laser beam were performed. Different types of necklace beams with different numbers of "pearls" periodically or quasi-periodically situated along the different radii rings were observed during the space evolution of the optical diffraction pattern.