Narrow Results

We study the Gierer–Meinhardt model of reaction-diffusion on a site-disordered square lattice. Let p be the site occupation probability of the square lattice. For p greater than a critical value p_c, the steady state consists of stripe-like patterns with long-range connectivity. For p < p_c, the connectivity is lost. The value of p_c is found to be much greater than that of the site percolation threshold for the square lattice. In the vicinity of p_c, the cluster-related quantities exhibit power-law scaling behavior. The method of finite-size scaling is used to determine the values of the fractal dimension d_f, the ratio, γ/ν, of the average cluster size exponent γ and the correlation length exponent ν. The values appear to indicate that the disordered GM model belongs to the universality class of ordinary percolation.

We study the discrete Gierer–Meinhardt model of reaction–diffusion on three different types of networks: regular, random and scale-free. The model dynamics lead to the formation of stationary Turing patterns in the steady state in certain parameter regions. Some general features of the patterns are studied through numerical simulation. The results for the random and scale-free networks show a marked difference from those in the case of the regular network. The difference may be ascribed to the small world character of the first two types of networks.

Cellular neural networks have found applications both as a platform for high speed image processing, as well as a model for complex spatiotemporal dynamical phenomena. This paper describes spatial pattern formation in a CMOS VLSI implementation of a CNN. With four layers of 32×64 cells, this CNN contains the most recurrently interconnected 2D layers of any VLSI CNN implementation to date. With over 8000 cells, it is also one of the largest in terms of the total number of cells. The spatial patterns generated by this network are bands of activity with a preferred scale and orientation, both of which can be adjusted via external bias voltages. The VLSI implementation provides a physical substrate for experimenting with complex spatiotemporal phenomena in real time, while also being amenable to theoretical analysis. Our experimental characterization of the observed patterns are in excellent agreement with our theoretical predictions.

Among reaction–diffusion systems showing Turing patterns, the diffusive Gray–Scott model [Pearson, 1993], stands out by showing self-replicating patterns (spots), which makes it the ideal simple model for developmental research. A first study of the influence of noise in the Gray–Scott model was performed by Lesmes et al. [2003] concluding that there exists an optimal noise intensity for which spot multiplication is maximal. Here we show in detail the transition from nonspotlike to spotlike pattern, with the identification of a wide range of noise intensities instead of an optimal value for which this transition occurs. Additional studies also reveal that noise produces a shift and a shrinkage of the regions of spatial patterns in the parameters space, without introducing qualitative changes to the diagram.

We consider three-dimensional Turing patterns and their isoconcentration surfaces corresponding to the equilibrium concentration of the reaction kinetics. We call these surfaces equilibrium concentration surfaces (EC surfaces). They are the interfaces between the regions of "high" and "low" concentrations in Turing patterns. We give alternate characterizations of EC surfaces by means of two variational principles, one of them being that they are optimal for diffusive transport. Several examples of EC surfaces are considered. Remarkably, they are often very well approximated by certain minimal surfaces. We give a dynamical explanation for the emergence of Scherk's surface in certain cases, a structure that has been observed numerically previously in [De Wit et al., 1997].

Bazykin proposed a Lotka–Volterra-type ecological model that accounts for simplified territoriality, which neither depends on territory size nor on food availability. In this study, we describe the global dynamics of the Bazykin model using analytical and numerical methods. We specifically focus on the effects of mutual predator interference and the prey carrying capacity since the variability of each could have especially dramatic ecological repercussions. The model displays a broad array of complex dynamics in space and time; for instance, we find the coexistence of a limit cycle and a steady state, and bistability of steady states. We also characterize super- and subcritical Poincaré–Andronov–Hopf bifurcations and a Bogdanov–Takens bifurcation. To illustrate the system's ability to naturally shift from stable to unstable dynamics, we construct bursting solutions, which depend on the slow dynamics of the carrying capacity. We also consider the stabilizing effect of the intraspecies interaction parameter, without which the system only shows either a stable steady state or oscillatory solutions with large amplitudes. We argue that this large amplitude behavior is the source of chaotic behavior reported in systems that use the MacArthur–Rosenzweig model to describe food-chain dynamics. Finally, we find the sufficient conditions in parameter space for Turing patterns and obtain the so-called "back-eye" pattern and localized structures.

Reaction–diffusion (RD) equation behavior has been studied in various fields, including biology, bioengineering and chemistry. Their solution leads to the formation of patterns which are stable in time and unstable in space, especially when RD system parameters are in the Turing space. Such patterns can be changed due to the growth of the domain where the reaction takes place. This article presents RD equations concerning growing domains in 2D and 3D. Several numerical examples have been solved using different geometries to study the effect of growth on pattern formation. The finite element method was used in conjunction with the Newton–Raphson method for the numerical solution to approximate nonlinear partial differential equations. It was found that growth affected Turing pattern formation, thereby generating complex structures in the domain.

We investigate the stability of Turing-type patterns in one spatial dimension in a system of reaction–diffusion equations with a term depending linearly on the spatial position. The system is a generic model of two interacting chemical species where production rates are dependent on a linear external gradient. This is motivated by mathematical models in developmental biology. In a previous paper, we found analytic approximations of Turing-like steady state patterns. In the present article, we derive conditions for the stability of these patterns and show bifurcation diagrams in two small parameters related to the slope of the external gradient and the ratio of the diffusion coefficients.

This paper focuses on the Turing patterns in the general Gierer–Meinhardt model of morphogenesis. The stability analysis of the equilibrium for the associated ODE system is carried out and the stability conditions are obtained. Furthermore, we perform a detailed Hopf bifurcation analysis for this system. The results show that the equilibrium undergoes a supercritical Hopf bifurcation in certain parameter range and the bifurcated limit cycle is stable. With added diffusions, we then show that both the stable equilibrium and the Hopf periodic solution experience Turing instability with unequal spatial diffusions and obtain the instability conditions. Numerical simulations are given to illustrate the theoretical analysis, which show that the Turing patterns are of either spot or stripe type.

In this paper, a spatial model has been designed to study a damaged diffusive eco-epidemiological system of Tilapia and Pelican populations in Salton Sea, California, USA. The nature of different equilibrium points and the existence of Hopf bifurcations are obtained. Conditions for Turing instability caused by local random movement of populations are derived. Numerically, the presence/existence of the wave of chaos phenomena is reported. Further, we show that the contact rate between susceptible and infected Tilapia population plays an important role in the distribution of the infected Tilapia population. The results suggest that the removal of infected Tilapia at regular time duration and controlling salinity will help to restore the system which provides a perspective for conservation strategy.

In this paper, we have investigated a diffusive predator–prey model with herd behavior. Also, we considered that the mortality of predators is linear as well as quadratic. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of control parameters. Using extensive numerical simulations, we obtain non-Turing spatiotemporal patterns in the model with linear mortality of predators, and Turing pattern formation, namely, spotted pattern and mixed pattern (spots-stripes) in model with quadratic mortality of predators. The results focus on the effect of the changing mortality rates of predator in pattern dynamics of a diffusive predator–prey model and help us in the better understanding of the dynamics of the predator–prey interaction in real environment.

In this article, a study of long-term behavior of reaction–diffusion systems augmented with self- and cross-diffusion is reported, using an augmented Gray–Scott system as a generic example. The methodology remains general, and is therefore applicable to some other systems. Simulations of the temporal model (nonlinear parabolic system) reveal the presence of steady states, often associated with energy dissipation. A Newton method based on a mixed finite element method is provided, in order to directly evaluate the steady states (nonlinear elliptic system) of the temporal system, and validated against its solutions. Linear stability analysis using Fourier analysis is carried out around homogeneous equilibrium, and using spectral analysis around nonhomogeneous ones. For the latter, the spectral problem is solved numerically. A multiparameter bifurcation is reported. Original steady-state patterns are unveiled, not observable with linear diffusion only. Two key observations are: a dependency of the pattern with the initial condition of the system, and a dependency on the geometry of the domain.

In this work we propose an ecological model, which is a modified version of the Bazykin model. This model of predator–prey interaction emphasizes predator pairing that yields steady state, periodic and extinction stable solutions. Moreover, we find attractor coexistence between limit cycles, steady states, and the extinction solution, which is always a stable attractor. We also study this model as a spatially extended system in one and two dimensions and obtain Turing patterns such as stripes and spots as well as the so-called black-eye patterns, and, as in the homogeneous case, the spatial patterns coexist with the homogeneous extinction solution.

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.

We study the reaction-diffusion system

During fetal development the morphology and function of the organs and tissues is determined. An example occurs with the formation of the cerebral cortex. On the external surface of the brain there are numerous folds (gyri, sulci, and fissures) that determine brain function. The exact cause for the formation of patterns of these folds is unknown. This article proposes a reaction-diffusion model in conjunction with a process of surface mechanical strain to explain the morphogenesis of the superficial structure of the brain.The model is solved using finite elements. There have been tests done on the formation of brain patterns through the reaction-diffusion equations with parameters in the space of Turing and by random mechanical strain. Several numerical examples have been developed that show an acceptable correlation between the results and clinical reality. With the model we can represent, qualitatively, the formation of the cerebral cortex by the proposed model. The model can approximate, and explain, lissencephaly and polymicrogyria, diseases that develop in the cerebral cortex and lead to medical complications to sufferers.

In this paper, a strongly coupled diffusive predator–prey system with a modified Leslie–Gower term is considered. We will show that under certain hypotheses, even though the unique positive equilibrium is asymptotically stable for the dynamics with diffusion, Turing instability can produce due to the presence of the cross-diffusion. In particular, we establish the existence of non-constant positive steady states of this system. The results indicate that cross-diffusion can create stationary patterns.

The Belousov-Zhabotinsky reaction embedded into water-in-oil reverse microemulsions (BZ-AOT system) is a well-known prototype for studying pattern formation. The diversity of self-organized structures observed in the BZ-AOT system, most of them unobtainable without the confined media, lies in the tunability of the medium. These complex patterns not only depend upon changes in the concentration of the BZ reactants but also modifications in the size and concentration of the droplets. Firstly, in this chapter we will give a general description of the BZ-AOT system. Then, we will present several fascinating self-organized patterns and, finally, we will describe a parameter that characterizes the morphology of the patterns.