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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

Phase diagram of the continuous approach.Parameter space of the continuous model where the regions of patterns and homogeneous solutions have been identified using the perturbation growth. Rs = 0.05, Rl = 0.1, D0 = 10−4 and a = 1. The yellow dashed line shows the transition from Instability A (above the line) to Instability B (below the line).
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pone.0132261.g006: Phase diagram of the continuous approach.Parameter space of the continuous model where the regions of patterns and homogeneous solutions have been identified using the perturbation growth. Rs = 0.05, Rl = 0.1, D0 = 10−4 and a = 1. The yellow dashed line shows the transition from Instability A (above the line) to Instability B (below the line).

Mentions: Evaluating the perturbation growth rate in Eq (7) with Eq (8), we may compute the phase diagram of the model (see Fig 6) for parameters b and c that gives information about the final spatial distribution of the system, homogeneous or patterned. The reduction of the diffusivity at high short-range densities is the responsible of the formation of patterns since clusters appear when b > c, that is when ρs is more relevant for the dynamics than ρl. On the other hand, considering the effect of the long-range density alone on the diffusivity, the system shows homogenous distributions regardless of the value of c when b = 0. These are expected results since high values of the short-range density reduce the mobility of the individuals promoting clustering, while high values of the long-range density enhance longer displacements in the population, thus leading to homogenous distributions. However, the instability caused by a diffusivity reduction is enhanced by the presence of the ρl dependence because animals in-between two aggregations make longer displacements that allow them to reach a group. A similar argument has been used to explain the formation of clusters of species [44] and vegetation [45] in systems that only present long-range competitive interactions.


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)

Phase diagram of the continuous approach.Parameter space of the continuous model where the regions of patterns and homogeneous solutions have been identified using the perturbation growth. Rs = 0.05, Rl = 0.1, D0 = 10−4 and a = 1. The yellow dashed line shows the transition from Instability A (above the line) to Instability B (below the line).
© Copyright Policy
Related In: Results  -  Collection

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

pone.0132261.g006: Phase diagram of the continuous approach.Parameter space of the continuous model where the regions of patterns and homogeneous solutions have been identified using the perturbation growth. Rs = 0.05, Rl = 0.1, D0 = 10−4 and a = 1. The yellow dashed line shows the transition from Instability A (above the line) to Instability B (below the line).
Mentions: Evaluating the perturbation growth rate in Eq (7) with Eq (8), we may compute the phase diagram of the model (see Fig 6) for parameters b and c that gives information about the final spatial distribution of the system, homogeneous or patterned. The reduction of the diffusivity at high short-range densities is the responsible of the formation of patterns since clusters appear when b > c, that is when ρs is more relevant for the dynamics than ρl. On the other hand, considering the effect of the long-range density alone on the diffusivity, the system shows homogenous distributions regardless of the value of c when b = 0. These are expected results since high values of the short-range density reduce the mobility of the individuals promoting clustering, while high values of the long-range density enhance longer displacements in the population, thus leading to homogenous distributions. However, the instability caused by a diffusivity reduction is enhanced by the presence of the ρl dependence because animals in-between two aggregations make longer displacements that allow them to reach a group. A similar argument has been used to explain the formation of clusters of species [44] and vegetation [45] in systems that only present long-range competitive interactions.

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