in this work, we analyse a pair of two-dimensional coupled reaction-diusion equations known as the gray-scott model, in which spot patterns have been observed. we focus on stationary patterns, and begin by deriving the asymptotic scaling of the parameters and variables necessary for the analysis of these patterns. a complete bifurcation study of these solutions is presented. the main mathematic...

In this paper, we rigorously prove the existence and stability of asymmetric spotty patterns for the Gray-Scott model in a bounded two dimensional domain. We show that given any two positive integers k 1 ; k 2 , there are asymmetric solutions with k 1 large spots (type A) and k 2 small spots (type B). We also give conditions for their location and calculate their heights. Most of these asymmetr...

Some 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω1. Here, I show that assuming the Proper Forcing Axiom (PFA), every proper Scott set is the standard system of a model of PA. I define that a Scott set X is proper if it is arithmetically closed and the quotie...

We performed a comprehensive study of the spike autosolitons: self-sustained solitary inhomogeneous states, in the classical reaction-diffusion system — the GrayScott model. We developed singular perturbation techniques based on the strong separation of the length scales to construct asymptotically the solutions in the form of a one-dimensional static autosolitons, higher-dimensional radially-s...

SCOTT (Semantically Constrained Otter) is a resolution-based automatic theorem prover for first order logic. It is based on the high performance prover OTTER by W. McCune and also incorporates a model generator. This finds finite models which SCOTT is able to use in a variety of ways to direct its proof search. Clauses generated by the prover are in turn used as axioms of theories to be modelle...

We performed a comprehensive study of spike autosolitons: highly localized solitary states, in the classical reaction-diffusion system — the Gray-Scott model. We developed singular perturbation techniques based on the strong separation of length scales to construct asymptotically the solutions in the form of one-dimensional and higher-dimensional radially-symmetric static autosolitons, and two ...