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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

Formation of ring-like structures in the 1D model.Evolution of the 1D version of the model starting from an initial condition for ρ(x, t = 0) consisting on a pulse of height unity and length 2Rl (displayed in the upper panel). The legend indicates the quantity represented by the different lines. The two vertical lines indicate the minima of the function g(x) (i.e. the diffusion coefficient in units of D0) at t = 0 where particles will initially tend to accumulate. The bottom panel represent the same quantities (although ρ(x, t) has been divided by a factor 5 to fit into the same scale as the other curves) after a very long integration time (t = 6 × 105). A double-peak structure has developed. The inset displays this long-time configuration in logarithmic scale, showing that g(x) ≈ 4 × 10−6 in the central region. Parameters: a = 1, b = 3.33, c = 2.67, D0 = 10−4, Rl = 0.75, Rs = 0.4, N = ∫ dxρ(x, t = 0) = 1.5
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pone.0132261.g008: Formation of ring-like structures in the 1D model.Evolution of the 1D version of the model starting from an initial condition for ρ(x, t = 0) consisting on a pulse of height unity and length 2Rl (displayed in the upper panel). The legend indicates the quantity represented by the different lines. The two vertical lines indicate the minima of the function g(x) (i.e. the diffusion coefficient in units of D0) at t = 0 where particles will initially tend to accumulate. The bottom panel represent the same quantities (although ρ(x, t) has been divided by a factor 5 to fit into the same scale as the other curves) after a very long integration time (t = 6 × 105). A double-peak structure has developed. The inset displays this long-time configuration in logarithmic scale, showing that g(x) ≈ 4 × 10−6 in the central region. Parameters: a = 1, b = 3.33, c = 2.67, D0 = 10−4, Rl = 0.75, Rs = 0.4, N = ∫ dxρ(x, t = 0) = 1.5

Mentions: As it was stated before, the ring-like shape of the clusters deserves further consideration. To go deeper into this question we use the one-dimensional version of the model starting from an initial condition consisting of a single pulse of height unity (top panel of Fig 8). The mean nonlocal densities (dashed red line) and (dashed green line) can be easily obtained and lead to a diffusivity which in units of D0 is the function g, with two minima where particles will tend initially to clump (magenta vertical dashed lines in Fig 8). As time advances a two-peak distribution establishes, which is the one-dimensional analogue of the two-dimensional rings observed before. This double peak, of a spatial size close to Rs, persists for extremely long times. However the inset in the bottom panel of Fig 8 shows that the diffusion coefficient in between the two peaks takes a nearly constant value which is very small but not zero (g(x) = D(x)/D0 ≈ 4 × 10−6). This implies that at still longer times (of the order of after the time displayed in the bottom panel of Fig 8) particles will diffuse between the two peaks, replacing them by a homogeneous distribution. The same will occur in two dimensions, since as showed in Fig 9, the diffusion coefficient in the two-dimensional system is also homogeneous (but very small) inside the clusters so that at extremely long times the pattern of hollow clusters of Fig 3 will be replaced by homogeneous clusters. Thus the ring structures seem to be a very-long lived transient state. They will disappear faster if the prescription in Eq (3) for g is changed by another functional form with higher minimum values. Alternatively, for a choice such that g(x) is strictly zero for then the rings will persists for infinite time as stationary structures.


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)

Formation of ring-like structures in the 1D model.Evolution of the 1D version of the model starting from an initial condition for ρ(x, t = 0) consisting on a pulse of height unity and length 2Rl (displayed in the upper panel). The legend indicates the quantity represented by the different lines. The two vertical lines indicate the minima of the function g(x) (i.e. the diffusion coefficient in units of D0) at t = 0 where particles will initially tend to accumulate. The bottom panel represent the same quantities (although ρ(x, t) has been divided by a factor 5 to fit into the same scale as the other curves) after a very long integration time (t = 6 × 105). A double-peak structure has developed. The inset displays this long-time configuration in logarithmic scale, showing that g(x) ≈ 4 × 10−6 in the central region. Parameters: a = 1, b = 3.33, c = 2.67, D0 = 10−4, Rl = 0.75, Rs = 0.4, N = ∫ dxρ(x, t = 0) = 1.5
© Copyright Policy
Related In: Results  -  Collection

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

pone.0132261.g008: Formation of ring-like structures in the 1D model.Evolution of the 1D version of the model starting from an initial condition for ρ(x, t = 0) consisting on a pulse of height unity and length 2Rl (displayed in the upper panel). The legend indicates the quantity represented by the different lines. The two vertical lines indicate the minima of the function g(x) (i.e. the diffusion coefficient in units of D0) at t = 0 where particles will initially tend to accumulate. The bottom panel represent the same quantities (although ρ(x, t) has been divided by a factor 5 to fit into the same scale as the other curves) after a very long integration time (t = 6 × 105). A double-peak structure has developed. The inset displays this long-time configuration in logarithmic scale, showing that g(x) ≈ 4 × 10−6 in the central region. Parameters: a = 1, b = 3.33, c = 2.67, D0 = 10−4, Rl = 0.75, Rs = 0.4, N = ∫ dxρ(x, t = 0) = 1.5
Mentions: As it was stated before, the ring-like shape of the clusters deserves further consideration. To go deeper into this question we use the one-dimensional version of the model starting from an initial condition consisting of a single pulse of height unity (top panel of Fig 8). The mean nonlocal densities (dashed red line) and (dashed green line) can be easily obtained and lead to a diffusivity which in units of D0 is the function g, with two minima where particles will tend initially to clump (magenta vertical dashed lines in Fig 8). As time advances a two-peak distribution establishes, which is the one-dimensional analogue of the two-dimensional rings observed before. This double peak, of a spatial size close to Rs, persists for extremely long times. However the inset in the bottom panel of Fig 8 shows that the diffusion coefficient in between the two peaks takes a nearly constant value which is very small but not zero (g(x) = D(x)/D0 ≈ 4 × 10−6). This implies that at still longer times (of the order of after the time displayed in the bottom panel of Fig 8) particles will diffuse between the two peaks, replacing them by a homogeneous distribution. The same will occur in two dimensions, since as showed in Fig 9, the diffusion coefficient in the two-dimensional system is also homogeneous (but very small) inside the clusters so that at extremely long times the pattern of hollow clusters of Fig 3 will be replaced by homogeneous clusters. Thus the ring structures seem to be a very-long lived transient state. They will disappear faster if the prescription in Eq (3) for g is changed by another functional form with higher minimum values. Alternatively, for a choice such that g(x) is strictly zero for then the rings will persists for infinite time as stationary structures.

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