Intrinsic Scaling Complexity in Animal Dispersion and Abundance

Abstract

Ecological theory related to animal distribution and abundance is at present incomplete and to some extent naive. We suggest that this may partly be due to a long tradition in the field of model development for choosing mathematical and statistical tools for convenience rather than applicability. Real population dynamics are influenced by nonlinear interactions, nonequilibrium conditions, and scaling complexity from system openness. Thus, a coherent theory for individual-, population-, and community-level processes should rest on mathematical and statistical methods that explicitly confront these issues in a manner that satisfies principles from statistical mechanics for complex systems. Instead, ecological theory is traditionally based on premises from simpler statistical mechanical theory for memory-free, scale-specific, random-walk, and diffusion processes, while animals from many taxa generally express strategic homing, site fidelity, and conspecific attraction in direct violation of primary model assumptions. Thus, the main challenge is to generalize the theory for memory-free physical, many-body systems to include a more realistic memory-influenced framework that better satisfies ecological realism. We describe, simulate, and discuss three testable aspects of a model for multiscaled habitat use at the individual level: (1) scale-free distribution of movement steps under influence of self-reinforcing site fidelity, (2) fractal spatial dispersion of intra-home range relocations, and (3) nonasymptotic expansion of observed intra-home range patch use with increasing set of relocations. Examples of literature data apparently supporting the conjecture that multiscaled, strategic space use is widespread among many animal taxa are also described. We suggest that the present approach, which provides a protocol to test for influence from scale-free, memory-dependent habitat use at the individual level, may also point toward a guideline for development of a generalized theoretical framework for complex population kinetics and spatiotemporal population dynamics.

Spatial and Temporal Occurrence of Blue Whales off the U.S. West Coast, with Implications for Management

“…Home ranges created from tracking data have been used in a variety of wildlife studies to describe the geographically restricted area used by animals and how it relates to the distribution and abundance of a population [30], habitat selection [31], [32], and predator-prey dynamics [33] among others topics [34]. The modern statistical modelling of home ranges implemented in this study uses the location data to estimate a probability density function that describes the likelihood of an animal being present at a given point within the home range [34], [35], [36].…”

Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

“…[26] (see also Refs. [4,5]). In this case, the memory rule (ii) is equivalent to revisiting an already-visited site, say, n, with probability proportional to the total amount of time spent by the walker at this site since t = 0.…”

“…These walks have been used for the description of the displacements of ants or bacteria [2]. In ecology, they can also represent simple models of "site fidelity," a behavior observed in many animals in the wild [4,5]. Many reinforced walk models are defined through transition probabilities that depend on the number of visits (or crossings) received by the sites (or edges) and the resulting dynamics is thus strongly path dependent.…”

Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion

“…As t obs is increased beyond 100 t, the step-length distribution from the return step process-in combination with the step-length truncation-dominates the overall process. The power law aspect of the movement-when expressed through equation (1.1)-then becomes 'hidden' at finer temporal scales, as previously shown by Gautestad & Mysterud [15]. Hence, if an animal space-use process is both scale-free and memory-influenced, then it is crucial to estimate the level of spatial auto-correlation prior to testing for power-law compliance.…”

“…Reynolds & Rhodes [12] and Viswanathan et al [13]. However, it has been argued that theoretical progress in the context of vertebrate space use also depends on a realistic implementation of the memory aspect of movement [9][10][11], including a statistical mechanical system description in this regard [14][15][16]. Thus, on the one hand, the scaling property observed in real GPS data over some scale range [13,17,18]-even if the power-law fit has been questioned for some of the datasets [19,20]-requires a deeper understanding of the processes behind the emergence of scale-free movement.…”