Bifurcation and Patterns Analysis for a Spatiotemporal Discrete Gierer-Meinhardt System

The Gierer-Meinhardt system is one of the prototypical pattern formation models. The bifurcation and pattern dynamics of a spatiotemporal discrete Gierer-Meinhardt system are investigated via the couple map lattice model (CML) method in this paper. The linear stability of the fixed points to such spatiotemporal discrete system is analyzed by stability theory. By using the bifurcation theory, the center manifold theory and the Turing instability theory, the Turing instability conditions in flip bifurcation and Neimark–Sacker bifurcation are considered, respectively. To illustrate the above theoretical results, numerical simulations are carried out, such as bifurcation diagram, maximum Lyapunov exponents, phase orbits, and pattern formations.

Dynamical analysis, optimal control and spatial pattern in an influenza model with adaptive immunity in two stratified population

<abstract><p>Consistently, influenza has become a major cause of illness and mortality worldwide and it has posed a serious threat to global public health particularly among the immuno-compromised people all around the world. The development of medication to control influenza has become a major challenge now. This work proposes and analyzes a structured model based on two geographical areas, in order to study the spread of influenza. The overall underlying population is separated into two sub populations: urban and rural. This geographical distinction is required as the immunity levels are significantly higher in rural areas as compared to urban areas. Hence, this paper is a novel attempt to proposes a linear and non-linear mathematical model with adaptive immunity and compare the host immune response to disease. For both the models, disease-free equilibrium points are obtained which are locally as well as globally stable if the reproduction number is less than 1 (<italic>R</italic>₀₁ &lt; 1 &amp; <italic>R</italic>₀₂ &lt; 1) and the endemic point is stable if the reproduction number is greater then 1 (<italic>R</italic>₀₁ &gt; 1 &amp; <italic>R</italic>₀₂ &gt; 1). Next, we have incorporated two treatments in the model that constitute the effectiveness of antidots and vaccination in restraining viral creation and slow down the production of new infections and analyzed an optimal control problem. Further, we have also proposed a spatial model involving diffusion and obtained the local stability for both the models. By the use of local stability, we have derived the Turing instability condition. Finally, all the theoretical results are verified with numerical simulation using MATLAB.</p></abstract>

Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities

In this paper, a diffusive predator–prey system with a functional response that increases in both predator and prey densities is considered. By analyzing the characteristic roots of the partial differential equation system, the Turing instability and Hopf bifurcation are studied. In order to consider the dynamics of the model where the Turing bifurcation curve and the Hopf bifurcation curve intersect, we chose the diffusion coefficients d1 and β as bifurcating parameters. In particular, the normal form of Turing–Hopf bifurcation was calculated so that we could obtain the phase diagram. For parameters in each region of the phase diagram, there are different types of solutions, and their dynamic properties are extremely rich. In this study, we have used some numerical simulations in order to confirm these ideas.

Delay-Induced Self-Organization Dynamics in a Prey-Predator Network with Diffusion

Time delays can induce the loss of stability and degradation of performance. In this paper, the pattern dynamics of a prey-predator network with diffusion and delay are investigated, where the inhomogeneous distribution of species in space can be viewed as a random network, and delay can affect the stability of the network system. Our results show that time delay can induce the emergence of Hopf and Turing bifurcations, which are independent of the network, and the conditions of bifurcation are derived by linear stability analysis. Moreover, we find that the Turing pattern can be related to the network connection probability. The Turing instability region involving delay and network connection probability is obtained. Finally, the numerical simulation verifies our results.

Pattern formation from spatially heterogeneous reaction–diffusion systems

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Dynamics of a Reaction Diusion Brucellosis Model

To understand the effects of animal movement on transmission and control of brucellosis infection, a reaction diffusion partial differential equation (PDE) brucellosis model that incorporates wild and domesticated animals under homogeneous Neumann boundary conditions is proposed and analysed. We computed the reproductive number for the brucellosis model in the absence of spatial movement and we established that, the associated model has a globally asymptotically stable disease-free equilibrium whenever the reproductive number is less or equal to unity. However, if the reproductive number is greater than unity an endemic equilibrium point which is globally asymptotically stable exists. We performed sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. For the model with spatial movement the disease threshold is studied by using the basic reproductive number. Additionally we investigate the existence of a Turing stability and travelling waves. Our results shows that incorporating diffusive spatial spread does not produce a Turing instability when the reproductive number R0ODE associated with the ODE model is less than unity. Finally the results suggest that minimizing interaction between buffalo and cattle population can be essential to manage brucellosis spillover between domesticated and wildlife animals. Numerical simulations are carried out to support analytical findings.