In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to homogeneous Neumann boundary condition. system, are introduced in such way that prey runs away from predator, while predator moves large group preys. Firstly, establish priori estimate for positive solutions. Secondly, study non-existence re...

The aim of this paper is to contribute to the understanding of the pattern formation phenomenon in reaction-diffusion equations coupled with ordinary differential equations. Such systems of equations arise, for example, from modeling of interactions between cellular processes such as cell growth, differentiation or transformation and diffusing signaling factors. We focus on stability analysis o...

A global feedback control of a system that exhibits a subcritical monotonic instability at a non-zero wavenumber (short-wave, or Turing instability) in the presence of a zero mode is investigated using a Ginzburg-Landau equation coupled to an equation for the zero mode. The method based on a variational principle is applied for the derivation of a low-dimensional evolution model. In the framewo...

We present a theory of the frequency comb generation in high-Q ring microresonators with quadratic nonlinearity and normal dispersion demonstrate that naturally large difference repetition rates at fundamental 2nd harmonic frequencies supports family bright soliton combs providing parametric gain is moderated by tuning index-matching parameter to exceed rate significant factor. This factor equa...

In these notes, I discuss in more detail the nonlinear evolution equations and their linear stability analysis for chemicals that react and diffuse in solutions. Historically, such a linear stability analysis of a uniform state was first carried out in 1952 by Alan Turing [9]. He suggested the radical and highly stimulating idea that reaction and diffusion of chemicals in an initially uniform s...

It is shown that the toy Turing Tumble, suitably extended with an infinitely long game board and unlimited supply of pieces, Turing-Complete. This achieved via direct simulation a machine. Unlike previously informally presented constructions, we do not encode finite control many times, need only one trigger/ball-hopper pair, prove our construction correct. We believe this first natural extensio...