Mathematical Biology

Murray, J. D.

doi:10.1007/978-3-662-08539-4

Cited by 2,383 publications

(785 citation statements)

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“…Whether these other models exhibit qualitatively similar behaviour in response to growth is the subject of current investigation. Murray (1993) and Goodwin et al (1993) have discussed other scenarios in which the pattern selection problem may be overcome. If patterning is initiated from one sub-region of a domain, then a global pattern may form behind a moving front, generally increasing reliability for the selection of the pattern wavelength.…”

Section: 2mentioning

confidence: 99%

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

Crampin¹,

Gaffney²,

Maini³

1999

Bulletin of Mathematical Biology

333

388

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We investigate the sequence of patterns generated by a reaction-diffusion system on a growing domain. We derive a general evolution equation to incorporate domain growth in reaction-diffusion models and consider the case of slow and isotropic domain growth in one spatial dimension. We use a self-similarity argument to predict a frequency-doubling sequence of patterns for exponential domain growth and we find numerically that frequency-doubling is realized for a finite range of exponential growth rate. We consider pattern formation under different forms for the growth and show that in one dimension domain growth may be a mechanism for increased robustness of pattern formation.

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Section: 2mentioning

confidence: 99%

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

Crampin¹,

Gaffney²,

Maini³

1999

Bulletin of Mathematical Biology

333

388

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“…For example, Meinhardt (1982) showed that diverse stripe and spot patterns, observed in developmental pattern formation, could be generated by a simple system of chemical interaction between two factors. Moreover, Murray (1989) explained various phenomena of biological pattern formation, including pattern formation in animal pigmentation. Many animal species have characteristic coat patterns on the skin.…”

Section: Introductionmentioning

confidence: 99%

Origin of directionality in the fish stripe pattern

Shoji

Mochizuki

Iwasa

et al. 2003

Developmental Dynamics

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The formation of stripe patterns in animal skin has been explained by the reaction-diffusion (RD) system, a hypothetical chemical reaction proposed by A. Turing. Although animal stripes usually have directionality, the RD model alone cannot explain how the direction is specified. To investigate the mechanism regulating the direction of stripes, we studied stripe pattern formation in two species of Genicanthus during sexual conversion. These species share almost identical morphologic properties, except for their stripe direction. In both species, spots transiently arise at random positions and then combine and rearrange to form directional stripes. Computational analysis has shown that diffusion anisotropy is very effective at specifying the direction of stripes formed by the RD system. Model simulations reproduce the transient dynamics of directional pattern formation observed in fish as well as the resulting stripes. In cases where the magnitude and direction of diffusion anisotropy of the substances are identical, the resulting stripes are not directional. However, if they differ, stripes become directional. As only a small difference in anisotropy is required for this effect, any kind of structure with directional conformation might cause a marked change in stripe direction. Scales are the most likely candidate structure for generating anisotropic interactions in skin. Developmental Dynamics 226:627-633, 2003.

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“…There are three main types of interspecific interaction: predator-prey, competition, and mutualism. In a predator-prey relationship, the predator species benefits from killing and consuming the prey species, and the prey population size may be regulated as a result (Murray, 1989).…”

Section: Introductionmentioning

confidence: 99%

“…Traveling wave fronts are solutions to partial differential equations (PDEs), which have a fixed shape and translate at a constant speed c as time evolves. (See Murray, 1989 for a detailed discussion. )…”

Section: Introductionmentioning

confidence: 99%

Continuous Traveling Waves for Prey-Taxis

2008

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Spatially moving predators are often considered for biological control of invasive species. The question arises as to whether introduced predators are able to stop an advancing pest or foreign population. In recent studies of reaction-diffusion models, it has been shown that the prey invasion can only be stopped if the prey dynamics observes an Allee effect.In this paper, we include prey-taxis into the model. Prey-taxis describe the active movement of predators to regions of high prey density. This effect leads to the observation that predators are drawn away from the leading edge of a prey invasion where its density is low. This leads to counterintuitive result that prey-taxis can actually reduce the likelihood of effective biocontrol.

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Mathematical Biology

Cited by 2,383 publications

References 0 publications

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

Origin of directionality in the fish stripe pattern

Continuous Traveling Waves for Prey-Taxis

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