Related papers: Turing Instabilities and Patterns Near a Hopf Bifu…
The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267].…
Patterns in reaction-diffusion systems often contain two spatial scales; a long scale determined by a typical wavelength or domain size, and a short scale pertaining to front structures separating different domains. Such patterns naturally…
In the present work, we explore the influence of habitat complexity on the activities of prey and predator of a spatio-temporal system by incorporating self diffusion. First we modify the Rosenzweig-MacArthur predator-prey model by…
We consider boundary value problems for semilinear hyperbolic systems of the type $$ \partial_tu_j + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with smooth coefficient functions $a_j$ and $b_j$ such that…
We set up a rigorous thermodynamic description of reaction-diffusion systems driven out of equilibrium by time-dependent space-distributed chemostats. Building on the assumption of local equilibrium, nonequilibrium thermodynamic potentials…
We study analytically and numerically a bistable reaction-diffusion equation on an arbitrary finite network. We prove that stable fixed points (multi-fronts) exist for any configuration as long as the diffusion is small. We also study fold…
We analyze the asymptotic stability of a nonlinear system of two differential equations with delay describing the dynamics of blood cell production. This process takes place in the bone marrow where stem cells differentiate throughout…
Pattern formation in the classical and fractional Schnakenberg equations is studied to understand the nonlocal effects of anomalous diffusion. Starting with linear stability analysis, we find that if the activator and inhibitor have the…
We describe the resulting spatiotemporal dynamics when a homogeneous equilibrium loses stability in a spatially extended system. More precisely, we consider reaction-diffusion systems, assuming only that the reaction kinetics undergo a…
Reaction-diffusion processes on networked systems have received mounting attention in the past two decades, and the corresponding theory of network dynamics has been continuously enriched with the advancement of network science. Recently,…
Numerical continuation is used to compute solution branches in a two-component reaction-diffusion model of Leslie--Gower type. %in the vicinity of a Turing-Hopf interaction. Two regimes are studied in detail. In the first, the homogeneous…
We analyze rate-dependent tipping in a fast/slow system with an equilibrium near the fold of a critical manifiold. We find a Hopf bifurcation as the rate parameter increases in the reduced co-moving system. This implies the growth of a…
Classical models of pattern formation are based on diffusion-driven instability (DDI) of constant stationary solutions of reaction-diffusion equations, which leads to emergence of stable, regular Turing patterns formed around that…
We consider the Taylor-Couette problem in an infinitely extended cylindrical domain. There exist modulated front solutions which describe the spreading of the stable Taylor vortices into the region of the unstable Couette flow. These…
In order to investigate the emergence of periodic oscillations of rimming flows, we study analytically the stability of steady states for the model of (Benilov, Kopteva, O'Brien, 2005), which describes the dynamics of a thin fluid film…
Turing patterns in reaction-diffusion (RD) systems have classically been studied only in RD systems which do not explicitly depend on independent variables such as space. In practise, many systems for which Turing patterning is important…
We study the behavior of vortex filaments subject to a uniform density of phase twist in oscillatory media described by the complex Ginzburg-Landau equation. The first instability is a supercritical Hopf bifurcation to stable propagating…
Following the approach pioneered by Eckhaus, Mielke, Schneider, and others for reaction diffusion systems [E, M1, M2, S1, S2, SZJV], we systematically derive formally by multiscale expansion and justify rigorously by Lyapunov-Schmidt…
We study time-periodic forcing of spatially-extended patterns near a Turing-Hopf bifurcation point. A symmetry-based normal form analysis yields several predictions, including that (i) weak forcing near the intrinsic Hopf frequency enhances…
In their way to/from turbulence, plane wall-bounded flows display an interesting transitional regime where laminar and turbulent oblique bands alternate, the origin of which is still mysterious. In line with Barkley's recent work about the…