Unravelling the Turing bifurcation using spatially varying diffusion coefficients

Benson, Debbie L.; Maini, Philip K.; Sherratt, Jonathan A.

doi:10.1007/s002850050135

Cited by 56 publications

(55 citation statements)

References 29 publications

(32 reference statements)

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“…The Turing bifurcations for the Schnakenberg model with spatially varying diffusion coefficients are studied on a spatial two-dimensional square in [1]. It is shown how this spatial variation can be used to partially reduce the degeneracy in the Turing bifurcation.…”

Section: Previous Results On Peaked Solutionsmentioning

confidence: 99%

“…To establish Part II, the analysis used in Part I is not good enough for the eigenvalues with λ = o (1) since knowing that their limit is zero does not tell us anything about the signs of the real parts of the eigenvalues for small. So a refined analysis is needed which goes beyond the leading order O(1).…”

Section: Previous Results On Peaked Solutionsmentioning

confidence: 99%

“…In this section we study the small eigenvalues, i.e., we assume that λ = o (1). Small eigenvalues are important in the sense that they control the translational dynamics of the multi-spots.…”

Section: Stability Part Ii: Small Eigenvaluesmentioning

confidence: 99%

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Stationary multiple spots for reaction–diffusion systems

Wei

Winter

2007

J. Math. Biol.

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Abstract. In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reactiondiffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activatorsubstrate systems (such as the Gray-Scott system or the Schnakenberg model).The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature.We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator.Existence of multi-spots is proved using tools from nonlinear functional analysis such as LiapunovSchmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis.Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots.Eigenvalues converging to zero are investigated using a projection similar to Liapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green's function of the Laplacian plays a central role in the analysis.The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations.

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Section: Previous Results On Peaked Solutionsmentioning

confidence: 99%

Section: Previous Results On Peaked Solutionsmentioning

confidence: 99%

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Stationary multiple spots for reaction–diffusion systems

Wei

Winter

2007

J. Math. Biol.

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“…In [34] similar results are shown for two-dimensional domains. In [2] it is shown how the degeneracy of the Turing bifurcation can be lifted using spatially varying diffusion coefficients. In [17,18,19] spikes are considered rigorously for the shadow system.…”

Section: A Cooperative Consumer Chain Modelmentioning

confidence: 99%

Stability of cluster solutions in a cooperative consumer chain model

Wei

Winter

2012

J. Math. Biol.

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Abstract. We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in [9,22] for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir.In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green's function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and and the other one is unstable. The instability arises through large eigenvalues of order O(1). Further, there are small eigenvalues of order o(1) which do not cause any instabilities.Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order O(1) leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously.(ii) The analysis of the small eigenvalues of order o(1) needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant 2 2 .

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“…In particular, they found that if the parameter was chosen to be above its bifurcation value only in a sub-interval of the domain, then it was possible for patterns to propagate into the domain where linear analysis would predict stability of the uniform steady state-a situation they described as "environmental instability" (Benson et al, 1993). A detailed weakly nonlinear bifurcation analysis for the two-dimensional case showed how degenerate bifurcation points could be split using a spatially varying parameter (Benson et al, 1998). These results show that environmental inhomogeneity could be an important regulator of biological pattern formation.…”

Section: Previous Mathematical Modelsmentioning

confidence: 99%

Breast Cancer: Modelling and Detection

Gavaghan¹,

Brady

Behrenbruch

et al. 2001

Computational and Mathematical Methods in Medicine

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This paper reviews a number of the mathematical models used in cancer modelling and then chooses a specific cancer, breast carcinoma, to illustrate how the modelling can be used in aiding detection. We then discuss mathematical models that underpin mammographic image analysis, which complements models of tumour growth and facilitates diagnosis and treatment of cancer. Mammographic images are notoriously difficult to interpret, and we give an overview of the primary image enhancement technologies that have been introduced, before focusing on a more detailed description of some of our own recent work on the use of physics-based modelling in mammography. This theoretical approach to image analysis yields a wealth of information that could be incorporated into the mathematical models, and we conclude by describing how current mathematical models might be enhanced by use of this information, and how these models in turn will help to meet some of the major challenges in cancer detection.

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Unravelling the Turing bifurcation using spatially varying diffusion coefficients

Cited by 56 publications

References 29 publications

Stationary multiple spots for reaction–diffusion systems

Stationary multiple spots for reaction–diffusion systems

Stability of cluster solutions in a cooperative consumer chain model

Breast Cancer: Modelling and Detection

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