Cross diffusion-induced pattern in an SI model

In this paper, a spatial SI epidemic model with self and cross diffusion is investigated. For the no cross diffusion system, there is no Turing pattern in the spatially extended model with equal self diffusion coefficients. While considering cross diffusion of the susceptible, we obtain typical Turing patterns. The results well enrich the finding in the epidemic model and may well explain the field observed in other areas.     

Numerical approach to chaotic pattern formation in diffusive predator–prey system with Caputo fractional operator

This paper is primarily concern with the formulation and analysis of a reliable numerical method based on the novel alternating direction implicit finite difference scheme for the solution of the fractional reaction–diffusion system. In the work, the integer first‐order derivative in time is replaced with the Caputo fractional derivative operator. As a case study, the dynamics of predator–prey model is considered. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction–diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve both self‐diffusion and cross‐diffusion problems in two‐dimensions. We observed in the experimental results a range of spatiotemporal and chaotic structures that are related to Turing pattern. It was also discovered in the simulations that cross‐diffusive case gives rise to spatial patterns faster than the diffusive case. Apart from chaotic spiral‐like structures obtained in this work, it should also be mentioned that Turing patterns such as stationary spots and stripes are obtainable, depending on the initial and parameters choices.     

Spatiotemporal dynamics of a predator-prey model incorporating a prey refuge

In this paper, we investigate the spatiotemporal dynamics of a ratio-dependent predator-prey model with cross diffusion incorporating proportion of prey refuge. First we get the critical lines of Hopf and Turing bifurcations in a spatial domain by using mathematical theory. More specifically, the exact Turing region is given in a two parameter space. Also we perform a series of numerical simulations. The obtained results reveal that this system has rich dynamics, such as spotted, stripe and labyrinth patterns which show that it is useful to use the predator-prey model to reveal the spatial dynamics in the real world.     

Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity

In this paper, we have investigated the phenomena of Turing pattern formation in a predator-prey model with habitat complexity in presence of cross diffusion. Using the linear stability analysis, the conditions for the existence of stationary pattern and the existence of Hopf bifurcation are obtained. It is shown analytically that the presence of cross diffusion in the system supports the formation of Turing pattern. Two parameter bifurcation analysis are done analytically and corresponding bifurcation diagrams are presented numerically. A series of simulation results are plotted for different biologically meaningful parameter values. Effects of variation of habitat complexity and the predator mortality rate and birth rate of prey on pattern formation are also reported. It is shown that cross-diffusion can lead to a wide variety of spatial and spatiotemporal pattern formation. It is found that the model exhibits spot and stripe pattern, and coexistence of both spot and strip patterns under the zero flux boundary condition. It is observed that cross-diffusion, habitat complexity, birth rate of prey and predator’s mortality rate play a significant role in the pattern formation of a distributed population system of predator-prey type.     

Analysis of spatiotemporal patterns in a single species reaction–diffusion model with spatiotemporal delay

Employing the theories of Turing bifurcation in the partial differential equations, we investigate the dynamical behavior of a single species reaction–diffusion model with spatiotemporal delay. The linear stability and the conditions for the occurrence of Turing bifurcation in this model are obtained. Moreover, the amplitude equations which represent different spatiotemporal patterns are also obtained near the Turing bifurcation point by using multiple scale method. In Turing space, it is found that the spatiotemporal distributions of the density of this researched species have spots pattern and stripes pattern. Finally, some numerical simulations corresponding to the different spatiotemporal patterns are given to verify our theoretical analysis.     

Spatiotemporal dynamics of reaction–diffusion models of interacting populations

The present investigation deals with the necessary conditions for Turing instability with zero-flux boundary conditions that arise in a ratio-dependent predator–prey model involving the influence of logistic population growth in prey and intra-specific competition among predators described by a system of non-linear partial differential equations. The prime objective is to investigate the parametric space for which Turing spatial structure takes place and to perform extensive numerical simulation from both the mathematical and the biological points of view in order to examine the role of diffusion coefficients in Turing instability. Various spatiotemporal distributions of interacting species through Turing instability in two dimensional spatial domain are portrayed and analyzed at length in order to substantiate the applicability of the present model.     

Spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge

In this paper, we investigate the spatiotemporal dynamics of a two-dimensional predator–prey model, which is based on a modified version of the Leslie–Gower scheme incorporating a prey refuge. We establish a Lyapunov function to prove the global stability of the equilibria with diffusion and determine the Turing space in the spatial domain. Furthermore, we perform a series of numerical simulations and find that the model dynamics exhibits complex Turing pattern replication: stripes, cold/hot spots-stripes coexistence and cold/hot spots patterns. The results indicate that the effect of the prey refuge for pattern formation is tremendous. This may enrich the dynamics of the effect of refuge on the predator-prey systems.     

Pattern formation of a predator–prey model

In this paper, we analyze the spatial pattern of a predator–prey system. We get the critical line of Hopf and Turing bifurcation in a spatial domain. In particular, the exact Turing domain is given. Also we perform a series of numerical simulations. The obtained results reveal that this system has rich dynamics, such as spotted, stripe and labyrinth patterns, which shows that it is useful to use the reaction–diffusion model to reveal the spatial dynamics in the real world.     

Spatiotemporal Dynamic Analysis in a Time-space Discrete Brusselator Mode

In this paper, we study the spatiotemporal patterns of a Brusselator model with discrete time-space by using the coupled mapping lattice (CML) model. The existence and stability conditions of the equilibrium point are obtained by using linear stability analysis. Then, applying the center manifold reduction theorem and the bifurcation theory, the parametric conditions of the flip and the Neimark-Sacker bifurcation are described respectively. Under space diffusion, the model admits the Turing instability at stable homogeneous solutions under some certain conditions. Two nonlinear mechanisms, including flip-Turing instability and Neimark-Sacker-Turing instability, are presented. Through numerical simulation, periodic windows, invariant circles, chaotic phenomenon and some interesting spatial patterns are found.     

Turing bifurcation in a system with cross diffusion

This paper treats the conditions for the existence and stability properties of stationary solutions of reaction–diffusion equations subject to Neumann boundary data. Hence, we assume that there are two substances in a two-dimensional bounded spatial domain where they are diffusing according to Fick's law: the velocity of the flow of diffusing substance is directed opposite to the (spatial) gradient of the density and is proportional to its modulus, but the spatial flow of each substance is influenced not only by its own but also by the other one's density (cross diffusion). The domains in which the substances are diffusing are of three type: a regular hexagon, a rectangle and an isosceles rectangular triangle. It will be assumed that there is no migration across the boundary of these domains. Considering one of the relevant features of the domains as a bifurcation parameter it will be shown that at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion) but becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises.     

Turing patterns and spatiotemporal patterns in a tritrophic food chain model with diffusion

In this paper, the temporal, spatial, and spatiotemporal patterns of a tritrophic food chain reaction–diffusion model with Holling type II functional response are studied. Firstly, for the model with or without diffusion, we perform a detailed stability and Hopf bifurcation analysis and derive criteria for determining the direction and stability of the bifurcation by the center manifold and normal form theory. Moreover, diffusion-driven Turing instability occurs, which induces spatial inhomogeneous patterns for the reaction–diffusion model. Then, the existence of positive non-constant steady-states of the reaction–diffusion model is established by the Leray–Schauder degree theory and some a priori estimates. Finally, numerical simulations are presented to visualize the complex dynamic behavior.     

Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

In this paper, we concentrate on the spatiotemporal patterns of a delayed reaction‐diffusion Holling‐Tanner model with Neumann boundary conditions. In particular, the time delay that is incorporated in the negative feedback of the predator density is considered as one of the principal factors to affect the dynamic behavior. Firstly, a global Turing bifurcation theorem for τ = 0 and a local Turing bifurcation theorem for τ > 0 are given. Then, further considering the degenerated situation, we derive the existence of Bogdanov‐Takens bifurcation and Turing‐Hopf bifurcation. The normal form method is used to study the explicit dynamics near the Turing‐Hopf singularity. It is shown that a pair of stable nonconstant steady states (stripe patterns) and a pair of stable spatially inhomogeneous periodic solutions (spot patterns) could be bifurcated from a positive equilibrium. Moreover, the Turing‐Turing‐Hopf–type spatiotemporal patterns, that is, a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and explained theoretically. Our results imply that the interaction of Turing and Hopf instabilities can be considered as the simplest mechanism for the appearance of complex spatiotemporal dynamics.     

Patrones de Turing sobre superficies sometidas a deformación: un acercamiento desde el método lagrangiano total

In this work we have developed several numerical examples of reaction-diffusion equations with growing domain. For this purpose we have used the Schnakenberg reaction model with parameters in space Turing. Therefore numerical tests are performed on the appearance of Turing patterns on surfaces that have high deformation rate. For the solution of reaction diffusion equations is presented a solution method on surfaces in three dimensions using the finite element method under the use of the total Lagrangian formulation. The results show that the formation of Turing patterns depends on the features of surface deformation and the rate at which change in position of each point of the domain. These results can explain some phenomena of change of pattern on the surface of the skin of animals that exhibit characteristic spots.     

Dynamics and pattern formations in diffusive predator‐prey models with two prey‐taxis

A reaction‐diffusion two‐predator‐one‐prey system with prey‐taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing preys. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any small prey‐taxis sensitivity coefficient are investigated by the semigroup theory. The spatial pattern formation induced by the prey‐taxis is characterized by the Turing type linear instability of homogeneous state; it is shown that prey‐taxis can both compress and prompt the spatial patterns produced through diffusion‐induced instability in two‐predator‐one‐prey systems.     

Pattern formation driven by cross-diffusion in a 2D domain

In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.     

Numerical approximation of Turing patterns in electrodeposition by ADI methods

In this paper we study the numerical approximation of Turing patterns corresponding to steady state solutions of a PDE system of reaction–diffusion equations modeling an electrodeposition process. We apply the Method of Lines (MOL) and describe the semi-discretization by high order finite differences in space given by the Extended Central Difference Formulas (ECDFs) that approximate Neumann boundary conditions (BCs) with the same accuracy. We introduce a test equation to describe the interplay between the diffusion and the reaction time scales. We present a stability analysis of a selection of time-integrators (IMEX 2-SBDF method, Crank–Nicolson (CN), Alternating Direction Implicit (ADI) method) for the test equation as well as for the Schnakenberg model, prototype of nonlinear reaction–diffusion systems with Turing patterns. Eventually, we apply the ADI-ECDF schemes to solve the electrodeposition model until the stationary patterns (spots & worms and only spots) are reached. We validate the model by comparison with experiments on Cu film growth by electrodeposition.     

Turing Instabilities at Hopf Bifurcation

Turing–Hopf instabilities for reaction-diffusion systems provide spatially inhomogeneous time-periodic patterns of chemical concentrations. In this paper we suggest a way for deriving asymptotic expansions to the limit cycle solutions due to a Hopf bifurcation in two-dimensional reaction systems and we use them to build convenient normal modes for the analysis of Turing instabilities of the limit cycle. They extend the Fourier modes for the steady state in the classical Turing approach, as they include time-periodic fluctuations induced by the limit cycle. Diffusive instabilities can be properly considered because of the non-catastrophic loss of stability that the steady state shows while the limit cycle appears. Moreover, we shall see that instabilities may appear even though the diffusion coefficients are equal. The obtained normal modes suggest that there are two possible ways, one weak and the other strong, in which the limit cycle generates oscillatory Turing instabilities near a Turing–Hopf bifurcation point. In the first case slight oscillations superpose over a dominant steady inhomogeneous pattern. In the second, the unstable modes show an intermittent switching between complementary spatial patterns, producing the effect known as twinkling patterns.     

Pattern formation in reaction-diffusion neural networks with leakage delay

Due to the heterogeneity of the electromagnetic field in neural networks, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate pattern formation in a reaction-diffusion neural network with leakage delay. The existence of Hopf bifurcation, as well as the necessary and sufficient conditions for Turing instability, are studied by analyzing the corresponding characteristic equation. Based on the multiple-scale analysis, amplitude equations of the model are derived, which determine the selection and competition of Turing patterns. Numerical simulations are carried out to show the possible patterns and how these patterns evolve. In some cases, the stability performance of Turing patterns is weakened by leakage delay and synaptic transmission delay.     

Spatial Dynamics of a Diffusive Predator-prey Model with Leslie-Gower Functional Response and Strong Allee Effect

In this paper, spatial dynamics of a diffusive predator-prey model with Leslie-Gower functional response and strong Allee effect is studied. Firstly, we obtain the critical condition of Hopf bifurcation and Turing bifurcation of the PDE model. Secondly, taking self-diffusion coefficient of the prey as bi- furcation parameter, the amplitude equations are derived by using multi-scale analysis methods. Finally, numerical simulations are carried out to verify our theoretical results. The simulations show that with the decrease of self- diffusion coefficient of the prey, the preys present three pattern structures: spot pattern, mixed pattern, and stripe pattern. We also observe the transi- tion from spot patterns to stripe patterns of the prey by changing the intrinsic growth rate of the predator. Our results reveal that both diffusion and the intrinsic growth rate play important roles in the spatial distribution of species.     

Investigating the Turing Conditions for Diffusion-driven Instability in Predator-prey System with Hunting Cooperation Functional Response

In this paper, we focus on stability analysis of steady-state solutions of a predator-prey system with hunting cooperation functional response. The results show that the Turing instability can be affected not only the existence of hunting cooperation, but also the diffusion coefficients: (1) in the absence of predator diffusion, diffusion-driven instability can be induced by hunting cooperation, but no stable patterns appear; (2) the system can occur diffusion-driven instability and Turing patterns, when both predator and prey have diffusion, and the diffusion coefficient of prey is greater than that of the predator. The numerical simulations of two cases are presented to verify the validity of our theoretical results.