Turing Patterns in a Reaction-Diffusion System

We have further investigated Turing patterns in a reaction-diffusion system by theoretical analysis and numerical simulations. Simple Turing patterns and complex superlattice structures are observed. We find that the shape and type of Turing patterns depend on dynamical parameters and external periodic forcing, and is independent of effective diffusivity rate σ in the Lengyel Epstein model Our numerical results provide additional insight into understanding the mechanism of development of Turing patterns and predicting new pattern formations.

Study of Turing patterns in a SI reaction-diffusion propagation system based on network and non-network environments

The study of rumor propagation dynamics is of great significance to reduce.false news and ensure the authenticity of news information.In this paper,a SI reaction-diffusion rumor propagation model with nonlinear saturation incidence is studied.First,through stability analysis,we obtain the conditions for the existence and local stability of the positive equilibrium point.By selecting suitable variable as the control parameter,the critical value of Turing bifurcation and the existence theorem of Turing bifurcation are obtained.Then,using the above theorem and multi-scale standard analysis,the expression of amplitude equation around Turing bifurcation point is obtained.By analyzing the amplitude equation,different types of Turing pattern are divided such as uniform steady-state mode,hexagonal mode,stripe mode and mixed structure mode.Further,in the numerical simulation part,by observing different patterns corresponding to different values of control variable,the correctness of the theory is verified.Finally,the effects of different network structures on patterns are investigated.The results show that there are significant differences in the distribution of users on different network structures.

TURING PATTERNS OF A PREDATOR-PREY MODEL WITH A MODIFIED LESLIE-GOWER TERM AND CROSS-DIFFUSIONS

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.

CROSS-DIFFUSION INDUCED TURING PATTERNS IN A SEX-STRUCTURED PREDATOR-PREY MODEL

In this paper, we consider a sex-structured predator prey model with strongly coupled nonlinear reaction diffusion. Using the Lyapunov functional and Leray Schauder degree theory, the existence and stability of both homogenous and heterogenous steady-states are investigated. Our results demonstrate that the unique homogenous steady-state is locally asymptotically stable for the associated ODE system and PDE system with self-diffusion. With the presence of the cross-diffusion, the homogeneous equilibrium is destabilized, and a heterogenous steady-state emerges as a consequence. In addition, the conditions guaranteeing the emergence of Turing patterns are derived.

Lattice Boltzmann Modeling of Advection-Diffusion-Reaction Equations:Pattern Formation Under Uniform Differential Advection

A lattice Boltzmann model for the study of advection-diffusion-reaction(ADR)problems is proposed.Via multiscale expansion analysis,we derive from the LB model the resulting macroscopic equations.It is shown that a linear equilibrium distribution is sufficient to produce ADR equations within error terms of the order of the Mach number squared.Furthermore,we study spatially varying structures arising from the interaction of advective transport with a cubic autocatalytic reaction-diffusion process under an imposed uniform flow.While advecting all the present species leads to trivial translation of the Turing patterns,differential advection leads to flow induced instability characterized with traveling stripes with a velocity dependent wave vector parallel to the flow direction.Predictions from a linear stability analysis of the model equations are found to be in line with these observations.