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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

Perturbation growth rate.Perturbation growth rate as a function of the wavenumber, Eq (8), for different values of the parameters b and c. Rs = 0.05, Rl = 0.1, D0 = 10−4 and a = 1.
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pone.0132261.g005: Perturbation growth rate.Perturbation growth rate as a function of the wavenumber, Eq (8), for different values of the parameters b and c. Rs = 0.05, Rl = 0.1, D0 = 10−4 and a = 1.

Mentions: We continue with the analytical approach performing a linear stability analysis of Eq (4). We note that the homogeneous distribution of the N individuals in the box of size L, i.e. ρ(r, t) = ρ0 = N/L2 always provides a stationary solution to such equation. The stability of this homogeneous distribution is checked by adding a small perturbation to it, so that ρ(r, t) = ρ0 + ϵψ(r, t) (ϵ ≪ 1). Inserting this into Eq (4) we find that the perturbation growth rate of ) is given byλ(k)=-D02(1+tanhγ+2cρ0G^l(k)-2bρ0G^s(k)cosh2γ)k2,(7)where γ = 2(a−bρ0+cρ0)−1. and are the Fourier transforms of the short-range and the long-range kernels, respectively. Given the choice made for the kernels (Eq (6)), the Fourier transforms areG^μ(k)=2J1(kRμ)/k/Rμ,(8)where μ = s or μ = l, and J1 is the first order Bessel function. The homogeneous distribution is unstable and then patterns would appear if the maximum of the growth rate (i.e., of the most unstable mode), λ(kc), is positive, which means that the perturbation of periodicity 2π//kc/ grows with time. λ is showed for different values of the parameters b and c in Fig 5. Depending on the value of b and c the model shows two different types of instabilities. Instability A has stable low wavenumbers (green curve in Fig 5, see inset) that prevent the clusters to grow. The characteristic wavelength of the pattern is well defined around kc = 49.52. On the other hand an instability of type B has a band of unstable modes starting at k = 0, which could allow the clusters to experience some coarsening in time. We observe that labyrinthic structures are formed by this type B instability.


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)

Perturbation growth rate.Perturbation growth rate as a function of the wavenumber, Eq (8), for different values of the parameters b and c. Rs = 0.05, Rl = 0.1, D0 = 10−4 and a = 1.
© Copyright Policy
Related In: Results  -  Collection

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

pone.0132261.g005: Perturbation growth rate.Perturbation growth rate as a function of the wavenumber, Eq (8), for different values of the parameters b and c. Rs = 0.05, Rl = 0.1, D0 = 10−4 and a = 1.
Mentions: We continue with the analytical approach performing a linear stability analysis of Eq (4). We note that the homogeneous distribution of the N individuals in the box of size L, i.e. ρ(r, t) = ρ0 = N/L2 always provides a stationary solution to such equation. The stability of this homogeneous distribution is checked by adding a small perturbation to it, so that ρ(r, t) = ρ0 + ϵψ(r, t) (ϵ ≪ 1). Inserting this into Eq (4) we find that the perturbation growth rate of ) is given byλ(k)=-D02(1+tanhγ+2cρ0G^l(k)-2bρ0G^s(k)cosh2γ)k2,(7)where γ = 2(a−bρ0+cρ0)−1. and are the Fourier transforms of the short-range and the long-range kernels, respectively. Given the choice made for the kernels (Eq (6)), the Fourier transforms areG^μ(k)=2J1(kRμ)/k/Rμ,(8)where μ = s or μ = l, and J1 is the first order Bessel function. The homogeneous distribution is unstable and then patterns would appear if the maximum of the growth rate (i.e., of the most unstable mode), λ(kc), is positive, which means that the perturbation of periodicity 2π//kc/ grows with time. λ is showed for different values of the parameters b and c in Fig 5. Depending on the value of b and c the model shows two different types of instabilities. Instability A has stable low wavenumbers (green curve in Fig 5, see inset) that prevent the clusters to grow. The characteristic wavelength of the pattern is well defined around kc = 49.52. On the other hand an instability of type B has a band of unstable modes starting at k = 0, which could allow the clusters to experience some coarsening in time. We observe that labyrinthic structures are formed by this type B instability.

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