Heterogeneous, "aggregated" patterns in the spatial distributions of individuals are almost universal across living organisms, from bacteria to higher vertebrates. Whereas specific features of aggregations are often visually striking to human eyes, a heuristic analysis based on human vision is usually not sufficient to answer fundamental questions about how and why organisms aggregate. What are the individual-level behavioral traits that give rise to these features? When qualitatively similar spatial patterns arise from purely physical mechanisms, are these patterns in organisms biologically significant, or are they simply epiphenomena that are likely characteristics of any set of interacting autonomous individuals? If specific features of spatial aggregations do confer advantages or disadvantages in the fitness of group members, how has evolution operated to shape individual behavior in balancing costs and benefits at the individual and group levels? Mathematical models of social behaviors such as schooling in fishes provide a promising avenue to address some of these questions. However, the literature on schooling models has lacked a common framework to objectively and quantitatively characterize relationships between individual-level behaviors and grouplevel patterns. In this paper, we briefly survey similarities and differences in behavioral algorithms and aggregation statistics among existing schooling models. We present preliminary results of our efforts to develop a modeling framework that synthesizes much of this previous work, and to identify relationships between behavioral parameters and group-level statistics.
Polarity, group velocity, and inter-individual spacing are characteristics of fish schools that strongly affect individual school members. However, these characteristics are group-level 'emergent properties': collective outcomes of behavioral interactions among members, not under direct control of any single member. The relationships between members' behaviors and the emergent group properties they produce are complex and poorly understood. In this study, we quantified 3D trajectories of all individual fish within 4-and 8-fish populations of Danio aequipinnatus, using stereo videography and a computerized tracking algorithm. We compared group polarity, group speed, and mean nearest-neighbor distances of schools within these populations to a simulation model that explored how fish responded to attraction/repulsion, alignment and random forces. Real fish exhibited a high degree of temporal variability in both polarity and group speed. Polarity and speed of simulated schools depended very strongly on the strength of the alignment force. Time-averaged polarity of real fish schools was most similar to simulated schools when alignment force was 1 to 5% of the attraction/repulsion force. For both real and simulated fish, a clear relationship existed between group speed and polarity: polarized groups were faster than non-polarized groups. We propose a multi-dimensional state space where several emergent property statistics are represented along the axes, and suggest certain 'preferred' ranges of state space within which animal groups tend to localize, and in which they can sustain distinct types of regular architecture.
Forecasting population decline to a certain critical threshold (the quasi-extinction risk) is one of the central objectives of population viability analysis (PVA), and such predictions figure prominently in the decisions of major conservation organizations. In this paper, we argue that accurate forecasting of a population's quasi-extinction risk does not necessarily require knowledge of the underlying biological mechanisms. Because of the stochastic and multiplicative nature of population growth, the ensemble behaviour of population trajectories converges to common statistical forms across a wide variety of stochastic population processes. This paper provides a theoretical basis for this argument. We show that the quasi-extinction surfaces of a variety of complex stochastic population processes (including age-structured, density-dependent and spatially structured populations) can be modelled by a simple stochastic approximation: the stochastic exponential growth process overlaid with Gaussian errors. Using simulated and real data, we show that this model can be estimated with 20-30 years of data and can provide relatively unbiased quasi-extinction risk with confidence intervals considerably smaller than (0,1). This was found to be true even for simulated data derived from some of the noisiest population processes (density-dependent feedback, species interactions and strong age-structure cycling). A key advantage of statistical models is that their parameters and the uncertainty of those parameters can be estimated from time series data using standard statistical methods. In contrast for most species of conservation concern, biologically realistic models must often be specified rather than estimated because of the limited data available for all the various parameters. Biologically realistic models will always have a prominent place in PVA for evaluating specific management options which affect a single segment of a population, a single demographic rate, or different geographic areas. However, for forecasting quasi-extinction risk, statistical models that are based on the convergent statistical properties of population processes offer many advantages over biologically realistic models.
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