Heterogeneous animal group models and their group-level alignment dynamics: An equation-free approach

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Abstract

We study coarse-grained (group-level) alignment dynamics of individual-based animal group models for heterogeneous populations consisting of informed (on preferred directions) and uninformed individuals. The orientation of each individual is characterized by an angle, whose dynamics are nonlinearly coupled with those of all the other individuals, with an explicit dependence on the difference between the individual's orientation and the instantaneous average direction. Choosing convenient coarse-grained variables (suggested by uncertainty quantification methods) that account for rapidly developing correlations during initial transients, we perform efficient computations of coarse-grained steady states and their bifurcation analysis. We circumvent the derivation of coarse-grained governing equations, following an equation-free computational approach.

Introduction

Coordinated motions and pattern formation have been studied for a wide range of biological organisms, from bacteria and amoebae to fish, from birds and wildebeest to humans (Ben-Jacob et al., 2000; Camazine et al. (2001); Parrish et al., 2002, Partridge, 1982, Weidlich, 1991, Wilson, 1975). Animal groups often behave as if they have a single mind, displaying remarkable self-organized behavior. At one extreme, the individuals seem to need little information transfer (e.g., fish schools), while at the other end the information exchange occurs in highly integrated ways through long-term associations among the individuals (e.g., honeybee hives and human communities). Controlling such an organized behavior in groups of artificial objects, including autonomous underwater vehicles (Leonard et al., 2007) and groups of autonomous agents (Jadbabaie et al., 2003), has received extensive attention in contemporary control theory. Challenges in both natural and engineering settings involve understanding which patterns emerge from the interaction among individual agents.

Selected laboratory experiments have shed some light on the schooling mechanism (Hunter, 1966, van Olst and Hunter, 1970, Partridge and Pitcher, 1979, Partridge and Pitcher, 1980, Pitcher et al., 1976). It still remains unclear, however, how the individual-level behavior and group-level (“macroscopic”, or coarse-grained) patterns are related. More precise experiments using three-dimensional tracking of every individual in a population should lead to better understanding of this linkage. An ultimate experimental study with precise control of every relevant detail may not be possible, yet appropriate mathematical models would provide a venue to establish behavioral cause, as one can consider different hypothetical individual-level interaction rules selectively (see e.g., Flierl et al., 1999).

Several different individual-based models have been proposed, which reproduce certain types of collective behavior in animal groups (e.g., see Aoki, 1982, Reynolds, 1987, Deneubourg and Goss, 1989). Self-organization emerges also in a wide spectrum of physical and chemical systems, some of which (e.g., crystals and ferromagnetic materials) exhibit apparent similarities with emergent patterns observed in animal groups. Vicsek et al. (1995) have introduced a discrete-time model of self-driven particles, or self-propelled particles (SPP), based on near-neighbor rules that are similar with those in the ferromagnetic XY model (Kosterlitz and Thouless, 1973). The authors analyzed statistical properties of the model, including phase transition and scaling (Vicsek et al., 1995). A long-range interaction has been incorporated into the SPP model (Mikhailov and Zanette, 1999), and continuum, “hydrodynamic” versions of this model have been introduced (Toner and Tu, 1995, Toner and Tu, 1998, Topaz et al., 2006). Recently, Couzin et al., 2002, Couzin et al., 2005 have introduced a model to provide insights into the mechanism of decision making in biological systems, which reproduces many important observations made in the field, and provides new insights into these phenomena. A review for various models can be found in Parrish et al. (2002) and Czirók and Vicsek (2001).

The models of Couzin et al. (2005), and most other such models, incorporate various detailed mechanistic steps. These shed light on the role of leadership and imitation, and produce a number of surprising results, such as the influence that a few “informed” individuals can have on large collectives. What is needed now are efforts to simplify those models, and to show especially what properties of the microscopic simulators are essential to explain that behavior. For some models, closure schemes are available (Flierl et al., 1999); but more generally, though we may suspect that closures exist, we cannot derive explicit expressions for them. In such circumstances, we need methods such as those used in this paper; we perform the coarse-grained dynamical analysis by circumventing the derivation of governing equations, using an equation-free computational approach (Theodoropoulos et al., 2000, Kevrekidis et al., 2003). A particular goal is to understand how much of the specific spatial detail is fundamental to the behavior. But turning to the Kuramoto-type approximation, where the interaction is assumed to be global, we deliberately ignore local effects. To the extent that the models fail to explain observed types of behavior, we will need to turn next to more detailed models.

Most of previously proposed models concern populations of homogeneous (or indistinguishable) individuals. Furthermore, the dynamical analysis in the literature is often limited to a small subset of the entire parameter space, and a systematic classification of possible global dynamics remains elusive. In the current paper, we study the coarse-grained alignment dynamics of individual-based animal group models. The measurement of the mean angular deviation of fish schools (e.g., clupeids and scombroids; see Atz, 1953, Hunter, 1966) showed that it varies continuously from no alignment to practically perfect alignment. We account for this continuous change by heterogeneity (“quenched noise”; characterized by parameters of random variables drawn from a prescribed distribution function) and the coupling strength. Our approach is flexible in that the heterogeneity can be introduced in various places in the model, and the way we analyze different heterogeneity cases does not require any significant modification.

The rest of the paper is organized as follows: models for homogeneous and heterogeneous animal groups are described in Sections 2.1 and 2.2, and our approach, equation-free polynomial chaos, is explained in Sections 2.3 and 2.4. Coarse-grained dynamical analysis and its comparison with fine-scale dynamics, for a system of two informed individuals and a large number of heterogeneous uninformed individuals, are presented in Section 3. The case of two groups of heterogeneous informed individuals is presented in Section 4. We conclude with a brief discussion in Section 5.

Section snippets

A “minimal” model for identical individuals

We briefly discuss a “minimal” model proposed by Nabet et al. (2006), which we extend in our study. It concerns the alignment dynamics of a homogeneous population of indistinguishable N individuals with two subgroups of informed individuals (“leaders”) with populations N1 and N2, respectively, and N3 uninformed individuals (“followers”), where N=N1+N2+N3:dψ1dt=sin(Θ1-ψ1)+KNN2sin(ψ2-ψ1)+KNN3sin(ψ3-ψ1),dψ2dt=sin(Θ2-ψ2)+KNN1sin(ψ1-ψ2)+KNN3sin(ψ3-ψ2),dψ3dt=KNN1sin(ψ1-ψ3)+KNN2sin(ψ2-ψ3).Here ψk

Results for case I

Direct integration of the “fine-scale” model of Eq. (4) in the strong coupling regime (K=1.0,σω=0.1), started from randomly assigned orientations and the heterogeneity variable (the latter is a Gaussian random variable), illustrates that a strong correlation between θ and ω develops during a short, initial transient time; the orientations of the followers quickly become a monotonically increasing function of their heterogeneity variable (Fig. 1), after which they slowly drift as a “unit” until

Dynamics of statistically similar groups

Here, we explore both the fine-scale and coarse-grained dynamics of a model for two groups of heterogeneous leaders (with no followers) shown in Eq. (5), and compare the results of the two different scales. One notable difference from the Kuramoto model is that “oscillators” in Eq. (5) do not have finite random variables (natural frequencies), hence there is no onset of the synchronization that occurs at a finite value of K (or, they can be alternatively seen as Kuramoto-like oscillators of

Conclusions

We have demonstrated a computational venue (an equation-free polynomial chaos approach) to study coarse-grained dynamics of individual-based models accounting for the heterogeneity among the individuals in animal group alignment models. We considered finite populations of (I) two “leaders” (which have direct knowledge on preferred directions) and N(1) uninformed, heterogeneous “followers”, and (II) two groups of heterogeneous “leaders”. We explored the coarse-grained, group level

Acknowledgments

S.J.M. and I.G.K. were financially supported by DOE and NSF Grant EF-0434319. S.A.L. was supported in part by NSF Grant EF-0434319 and DARPA Grant HR0011-05-1-0057. B.N. and N.E.L. were supported in part by ONR Grants N00014-02-1-0826 and N00014-04-1-0534.

References (47)

  • I.D. Couzin et al.

    Effective leadership and decision-making in animal groups on the move

    Nature

    (2005)
  • A. Czirók et al.

    Collective motion

  • J.L. Deneubourg et al.

    Collective patterns and decision making

    Ethol. Ecol. Evol.

    (1989)
  • Doedel, E.J., et al., 2000. A numerical bifurcation analysis software freely. Available from...
  • C.W. Gear et al.

    Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum

    SIAM J. Sci. Comput.

    (2003)
  • R. Ghanem et al.

    Stochastic Finite Elements: A Spectral Approach

    (1991)
  • J.R. Hunter

    Procedure for analysis of schooling behavior

    J. Fish. Res. Board Canada

    (1966)
  • A. Jadbabaie et al.

    Coordination of groups of mobile autonomous agents using nearest neighbor rules

    IEEE Trans. Automat. Control

    (2003)
  • M. Jardak et al.

    Spectral polynomial chaos solutions of the stochastic advection equation

    J. Sci. Comput.

    (2002)
  • H.B. Keller

    Lectures on Numerical Methods in Bifurcation Theory. Tata Institute of Fundamental Research, Lectures on Mathematics and Physics

    (1987)
  • Kelley, C.T., 1995. Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, vol. 16....
  • I.G. Kevrekidis et al.

    Equation-free coarse-grained multiscale computation: enabling microscopic simulators to perform system-level tasks

    Commun. Math. Sci.

    (2003)
  • I.G. Kevrekidis et al.

    Equation-free: the computer-assisted analysis of complex, multiscale systems

    AIChE J.

    (2004)
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