P. van Heijster

Front interactions in a three-component system

The three-component reaction-diffusion system introduced in (C. P. Schenk et al., Phys. Rev. Lett., 78 (1997), pp. 3781-3784) has become a paradigm model in pattern formation. It exhibits a rich variety of dynamics of fronts, pulses, and spots. The front and pulse interactions range in type from weak, in which the localized structures interact only through their exponentially small tails, to strong interactions, in which they annihilate or collide and in which all components are far from equilibrium in the domains between the localized structures. Intermediate to these two extremes sits the semistrong interaction regime, in which the activator component of the front is near equilibrium in the intervals between adjacent fronts but both inhibitor components are far from equilibrium there, and hence their concentration profiles drive the front evolution. In this paper, we focus on dynamically evolving N -front solutions in the semistrong regime. The primary result is use of a renormalization group method to rigorously derive the system of N coupled ODEs that governs the positions of the fronts. The operators associated with the linearization about the N -front solutions have N small eigenvalues, and the N -front solutions may be decomposed into a component in the space spanned by the associated eigenfunctions and a component projected onto the complement of this space. This decomposition is carried out iteratively at a sequence of times. The former projections yield the ODEs for the front positions, while the latter projections are associated with remainders that we show stay small in a suitable norm during each iteration of the renormalization group method. Our results also help extend the application of the renormalization group method from the weak interaction regime for which it was initially developed to the semistrong interaction regime. The second set of results that we present is a detailed analysis of this system of ODEs, providing a classification of the possible front interactions in the cases of N =1 , 2, 3, 4, as well as how front solutions interact with the stationary pulse solutions studied earlier in (A. Doelman, P. van Heijster, and T. J. Kaper,J. Dynam. Differential Equations, 21 (2009), pp. 73-115; P. van Heijster, A. Doelman, and T. J. Kaper, Phys. D, 237 (2008), pp. 3335-3368). Moreover, we present some results on the general case of N -front interactions.

Existence of Traveling Wave Solutions for a Model of Tumor Invasion

The existence of traveling wave solutions to a haptotaxis dominated model is analyzed. A version of this model has been derived in Perumpanani et al. [Phys. D, 126 (1999), pp. 145--159] to describe tumor invasion, where diffusion is neglected as it is assumed to play only a small role in the cell migration. By instead allowing diffusion to be small, we reformulate the model as a singular perturbation problem, which can then be analyzed using geometric singular perturbation theory. We prove the existence of three types of physically realistic traveling wave solutions in the case of small diffusion. These solutions reduce to the no-diffusion solutions in the singular limit as diffusion as is taken to zero. A fourth traveling wave solution is also shown to exist, but that is physically unrealistic as it has a component with negative cell population. The numerical stability, in particular the wavespeed of the traveling wave solutions, is also discussed.

A Geometric Approach to Stationary Defect Solutions in One Space Dimension

In this manuscript, we consider the impact of a small jump-type spatial heterogeneity on the existence of stationary localized patterns in a system of partial differential equations in one spatial dimension, i.e., defined on

Numerical computation of an Evans function for travelling waves

\mathbb{R}

NOVEL SOLUTIONS FOR A MODEL OF WOUND HEALING ANGIOGENESIS

. This problem corresponds to analyzing a discontinuous and non-autonomous

Absolute instabilities of travelling wave solutions in a Keller-Segel model

n

Spectral stability of travelling wave solutions in a Keller–Segel model

-dimensional system,

A geometric construction of travelling wave solutions to the Keller–Segel model

\scriptsize\dot{u}=\left\{ \begin{array}{ll} f(u),& t\leq0,\\ f(u)+\varepsilon g(u),& t>0, \end{array}\right.

Front Interactions in a Three-Component System

under the assumption that the unperturbed system, i.e., the

Traveling wave solutions in a model for tumor invasion with the acid-mediation hypothesis.

\varepsilon \to 0