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Pattern Formation in Populations with Density-Dependent Movement and Two Interaction Scales.

Martínez-García R, Murgui C, Hernández-García E, López C - PLoS ONE (2015)

Bottom Line: We perform stochastic simulations of a particle-level model of the system, and derive and analyze a continuous density description (a nonlinear diffusion equation).In both cases we show that this interplay of scale-dependent-behaviors gives rise to a rich formation of spatial patterns ranging from labyrinths to periodic cluster arrangements.In most cases these clusters have the very peculiar appearance of ring-like structures, i.e., organisms arranging in the perimeter of the clusters, which we discuss in detail.

View Article: PubMed Central - PubMed

Affiliation: IFISC (Instituto de Física Interdisciplinar y Sistemas Complejos), CSIC-UIB, Campus UIB, Palma de Mallorca, Spain; Department of Ecology and Evolutionary Biology, Princeton University, Princeton NJ, 08544-1003, United States of America.

ABSTRACT
We study the spatial patterns formed by a system of interacting particles where the mobility of any individual is determined by the population crowding at two different spatial scales. In this way we model the behavior of some biological organisms (like mussels) that tend to cluster at short ranges as a defensive strategy, and strongly disperse if there is a high population pressure at large ranges for optimizing foraging. We perform stochastic simulations of a particle-level model of the system, and derive and analyze a continuous density description (a nonlinear diffusion equation). In both cases we show that this interplay of scale-dependent-behaviors gives rise to a rich formation of spatial patterns ranging from labyrinths to periodic cluster arrangements. In most cases these clusters have the very peculiar appearance of ring-like structures, i.e., organisms arranging in the perimeter of the clusters, which we discuss in detail.

No MeSH data available.


Related in: MedlinePlus

Structure functions.Structure function of the patterns obtained with the continuous and the discrete model for the case of labyrinthic and spotted patterns.
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pone.0132261.g007: Structure functions.Structure function of the patterns obtained with the continuous and the discrete model for the case of labyrinthic and spotted patterns.

Mentions: The typical scale of the pattern, that is, the distance between aggregates, can be studied with the structure function (Fig 7). It can be computed for both the patterns of particles and the density distribution. In the first case it is , where rj is the position vector of particle j, k is a two-dimensional wave vector with modulus k, and the average indicates a spherical average over the wave vectors with modulus k and in time. In the continuous approach, the structure function is calculated as the modulus of the spatial Fourier transform of density field, averaged spherically and in time. Note that both quantities are related but not identical, and their first maximum, kc, allows to compute the typical distance between clusters d = 2π/kc. For the spotted patterns kc = 50.24 (discrete) and kc = 49.52 (continuum) so that d ≈ 0.125–0.126. Regarding the case of the labyrinth pattern (central panel of Figs 2 and 3), kc = 56.52 (discrete) and kc = 51.31 (continuum), so that the typical distance between aggregates is d ≈ 0.11.


Pattern Formation in Populations with Density-Dependent Movement and Two Interaction Scales.

Martínez-García R, Murgui C, Hernández-García E, López C - PLoS ONE (2015)

Structure functions.Structure function of the patterns obtained with the continuous and the discrete model for the case of labyrinthic and spotted patterns.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4493154&req=5

pone.0132261.g007: Structure functions.Structure function of the patterns obtained with the continuous and the discrete model for the case of labyrinthic and spotted patterns.
Mentions: The typical scale of the pattern, that is, the distance between aggregates, can be studied with the structure function (Fig 7). It can be computed for both the patterns of particles and the density distribution. In the first case it is , where rj is the position vector of particle j, k is a two-dimensional wave vector with modulus k, and the average indicates a spherical average over the wave vectors with modulus k and in time. In the continuous approach, the structure function is calculated as the modulus of the spatial Fourier transform of density field, averaged spherically and in time. Note that both quantities are related but not identical, and their first maximum, kc, allows to compute the typical distance between clusters d = 2π/kc. For the spotted patterns kc = 50.24 (discrete) and kc = 49.52 (continuum) so that d ≈ 0.125–0.126. Regarding the case of the labyrinth pattern (central panel of Figs 2 and 3), kc = 56.52 (discrete) and kc = 51.31 (continuum), so that the typical distance between aggregates is d ≈ 0.11.

Bottom Line: We perform stochastic simulations of a particle-level model of the system, and derive and analyze a continuous density description (a nonlinear diffusion equation).In both cases we show that this interplay of scale-dependent-behaviors gives rise to a rich formation of spatial patterns ranging from labyrinths to periodic cluster arrangements.In most cases these clusters have the very peculiar appearance of ring-like structures, i.e., organisms arranging in the perimeter of the clusters, which we discuss in detail.

View Article: PubMed Central - PubMed

Affiliation: IFISC (Instituto de Física Interdisciplinar y Sistemas Complejos), CSIC-UIB, Campus UIB, Palma de Mallorca, Spain; Department of Ecology and Evolutionary Biology, Princeton University, Princeton NJ, 08544-1003, United States of America.

ABSTRACT
We study the spatial patterns formed by a system of interacting particles where the mobility of any individual is determined by the population crowding at two different spatial scales. In this way we model the behavior of some biological organisms (like mussels) that tend to cluster at short ranges as a defensive strategy, and strongly disperse if there is a high population pressure at large ranges for optimizing foraging. We perform stochastic simulations of a particle-level model of the system, and derive and analyze a continuous density description (a nonlinear diffusion equation). In both cases we show that this interplay of scale-dependent-behaviors gives rise to a rich formation of spatial patterns ranging from labyrinths to periodic cluster arrangements. In most cases these clusters have the very peculiar appearance of ring-like structures, i.e., organisms arranging in the perimeter of the clusters, which we discuss in detail.

No MeSH data available.


Related in: MedlinePlus