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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

Solutions of the continuous density equation.Long time solution of Eq (4) with a short interaction range Rs = 0.05 and a long interaction length Rl = 0.1. D0 = 10−4, a = 1 and density ρ0 = 104 in all the panels. An Euler algorithm was implemented and integration performed over a square area with lateral size L = 1 and periodic boundary conditions. Left panel: b = 8.5 × 10−4, c = 7 × 10−4 (labyrinth pattern). Right panel: b = 4.3 × 10−4, c = 3.9 × 10−4 (spot pattern).
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pone.0132261.g003: Solutions of the continuous density equation.Long time solution of Eq (4) with a short interaction range Rs = 0.05 and a long interaction length Rl = 0.1. D0 = 10−4, a = 1 and density ρ0 = 104 in all the panels. An Euler algorithm was implemented and integration performed over a square area with lateral size L = 1 and periodic boundary conditions. Left panel: b = 8.5 × 10−4, c = 7 × 10−4 (labyrinth pattern). Right panel: b = 4.3 × 10−4, c = 3.9 × 10−4 (spot pattern).

Mentions: A deeper understanding of the pattern formation dynamics can be addressed using the continuum description given by Eq (4). To corroborate the correspondence between the individual based description by Eq (1) and the continuous approach in terms of Eq (4), we numerically integrate Eq (4). Kernels are fixed as given by Eq (6) and the parameters take the same values as in Fig 2 to allow a direct comparison with the discrete simulations (see caption of Fig 3 for details). The laberynth and spot patterns showed in Fig 3 exhibit a good agreement with the distributions of Fig 2 resulting from the stochastic particle dynamics. In particular, details of hollow clusters for both micro and macro descriptions are plotted in Fig 4. The distribution of the particles within the clusters is a particularly interesting question that will be discussed later in this section.


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)

Solutions of the continuous density equation.Long time solution of Eq (4) with a short interaction range Rs = 0.05 and a long interaction length Rl = 0.1. D0 = 10−4, a = 1 and density ρ0 = 104 in all the panels. An Euler algorithm was implemented and integration performed over a square area with lateral size L = 1 and periodic boundary conditions. Left panel: b = 8.5 × 10−4, c = 7 × 10−4 (labyrinth pattern). Right panel: b = 4.3 × 10−4, c = 3.9 × 10−4 (spot pattern).
© Copyright Policy
Related In: Results  -  Collection

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

pone.0132261.g003: Solutions of the continuous density equation.Long time solution of Eq (4) with a short interaction range Rs = 0.05 and a long interaction length Rl = 0.1. D0 = 10−4, a = 1 and density ρ0 = 104 in all the panels. An Euler algorithm was implemented and integration performed over a square area with lateral size L = 1 and periodic boundary conditions. Left panel: b = 8.5 × 10−4, c = 7 × 10−4 (labyrinth pattern). Right panel: b = 4.3 × 10−4, c = 3.9 × 10−4 (spot pattern).
Mentions: A deeper understanding of the pattern formation dynamics can be addressed using the continuum description given by Eq (4). To corroborate the correspondence between the individual based description by Eq (1) and the continuous approach in terms of Eq (4), we numerically integrate Eq (4). Kernels are fixed as given by Eq (6) and the parameters take the same values as in Fig 2 to allow a direct comparison with the discrete simulations (see caption of Fig 3 for details). The laberynth and spot patterns showed in Fig 3 exhibit a good agreement with the distributions of Fig 2 resulting from the stochastic particle dynamics. In particular, details of hollow clusters for both micro and macro descriptions are plotted in Fig 4. The distribution of the particles within the clusters is a particularly interesting question that will be discussed later in this section.

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