A Mathematical Biologist’s Guide to Absolute and Convective Instability

Sherratt, Jonathan A.; Dagbovie, Ayawoa S.; Hilker, Frank M.

doi:10.1007/s11538-013-9911-9

Cited by 21 publications

(26 citation statements)

References 70 publications

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“…It is important to emphasize that unstable patterns are not necessarily irrelevant for real instances of banded vegetation. Unstable solutions of a partial differential equation subdivide according to whether the instability is 'convective' or 'absolute' [59][60][61]. In the former case, the solutions can occur as persistent spatio-temporal transients [62,63].…”

Section: Discussionmentioning

confidence: 99%

Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Dagbovie

Sherratt

2014

J. R. Soc. Interface.

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Banded vegetation is a characteristic feature of semi-arid environments. It occurs on gentle slopes, with alternating stripes of vegetation and bare ground running parallel to the contours. A number of mathematical models have been proposed to investigate the mechanisms underlying these patterns, and how they might be affected by changes in environmental conditions. One of the most widely used models is due to Rietkerk and co-workers, and is based on a water redistribution hypothesis, with the key feedback being that the rate of rainwater infiltration into the soil is an increasing function of plant biomass. Here, for the first time, we present a detailed study of the existence and stability of pattern solutions of the Rietkerk model on slopes, using the software package WAVETRAIN (www. ma.hw.ac.uk/wavetrain). Specifically, we calculate the region of the rainfall-migration speed parameter plane in which patterns exist, and the sub-region in which these patterns are stable as solutions of the model partial differential equations. We then perform a detailed simulation-based study of the way in which patterns evolve when the rainfall parameter is slowly varied. This reveals complex behaviour, with sudden jumps in pattern wavelength, and hysteresis; we show that these jumps occur when the contours of constant pattern wavelength leave the parameter region giving stable patterns. Finally, we extend our results to the case in which a diffusion term for surface water is added to the model equations. The parameter regions for pattern existence and stability are relatively insensitive to small or moderate levels of surface water diffusion, but larger diffusion coefficients significantly change the subdivision into stable and unstable patterns.

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Section: Discussionmentioning

confidence: 99%

Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Dagbovie

Sherratt

2014

J. R. Soc. Interface.

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“…These newly demonstrated patterns of travelling waves with spatially regular structure can be interpreted in terms of the definition of convective stability introduced by [42]. Indeed, the developing regular travelling structures are convectively stable since they emerge as a result of complex spatio-temporal interactions.…”

Section: Discussionmentioning

confidence: 85%

Revealing new dynamical patterns in a reaction–diffusion model with cyclic competition via a novel computational framework

et al. 2018

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Understanding how patterns and travelling waves form in chemical and biological reaction-diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction-diffusion models, such as Lotka-Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction-diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.

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“…If the operator T has essential spectrum in the right half plane in the unweighted space, weights of interest are those that move this essential spectrum into the open left half plane. If such weights ν exist (and if there is no point spectrum in the right half plane), we say the travelling wave solution is spectrally stable in H 1 ν (R) and it is referred to as being transiently unstable [33,38]. Since the order of the spatial eigenvalues is not changed, the absolute spectrum is unaffected by weighting the function space and the presence of absolute spectrum in the right half plane indicates an absolute instability.…”

Section: 32mentioning

confidence: 99%

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

2017

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Abstract. We investigate the spectral stability of travelling wave solutions in a Keller-Segel model of bacterial chemotaxis with a logarithmic chemosensitivity function and a constant, sublinear, and linear consumption rate. Linearising around the travelling wave solutions, we locate the essential and absolute spectrum of the associated linear operators and find that all travelling wave solutions have essential spectrum in the right half plane. However, we show that in the case of constant or sublinear consumption there exists a range of parameters such that the absolute spectrum is contained in the open left half plane and the essential spectrum can thus be weighted into the open left half plane. For the constant and sublinear consumption rate models we also determine critical parameter values for which the absolute spectrum crosses into the right half plane, indicating the onset of an absolute instability of the travelling wave solution. We observe that this crossing always occurs off of the real axis.

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A Mathematical Biologist’s Guide to Absolute and Convective Instability

Cited by 21 publications

References 70 publications

Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Revealing new dynamical patterns in a reaction–diffusion model with cyclic competition via a novel computational framework

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

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