Arild O. Gautestad

Intrinsic Scaling Complexity in Animal Dispersion and Abundance

Ecological theory related to animal distribution and abundance is at present incomplete and to some extent naïve. 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.

The Home Range Ghost

The area (A) covering a sample of non-autocorrelated animal relocations (n) is generally thought to increase asymptotically towards a true home range size with increasing sample size. This should be the case for any home range with a stable centre, whether applying a minimum convex polygon, box counting or some more sophisticated method for area demarcation. We show by simulation that the rate of increase of A is expected to decrease significantly from one m-fold increase of n to the next m-fold increase, for n as low as 100-200 relocations. For larger n the rate of increase of A is expected to be close to zero for last m-fold increase of a subsample of n. This central null hypothesis in ecology is not supported by an extensive, and assumed representative, sample of (n,A) literature data. Even after adjusting for small-n expected underestimates of A, A increases approximately in proportion with the square root of n. Literature examples are given where no area asymptote appeared in sample sizes passing thousands of relocations. The results support an alternative model for home range area use, the multiscaled home range model (MHR).

Physical and biological mechanisms in animal movement processes

We have analysed a large set of animal locations from a telemetry study of a herd of domestic sheep, Ovis aries, in order to test an alternative model for area-utilization functions. Our model incorporates the special effects emerging from complex movement patterns, which have made the traditional home range demarcation protocols so difficult to employ. The telemetry location plots approached a statistically self-similar fractal pattern with dimension 1.5 as the overall plot density (n) increased. The home range area expanded on average as a function of n without any apparent asymptotic approach to a «true» home range area. The regression of log(area) versus log(n) was approximately linear, with slope 0.5 for samples of 10-1433 coordinate plots

Memory matters: Influence from a cognitive map on animal space use

A vertebrate individuals cognitive map provides a capacity for site fidelity and long-distance returns to favorable patches. Fractal-geometrical analysis of individual space use based on collection of telemetry fixes makes it possible to verify the influence of a cognitive map on the spatial scatter of habitat use and also to what extent space use has been of a scale-specific versus a scale-free kind. This approach rests on a statistical mechanical level of system abstraction, where micro-scale details of behavioral interactions are coarse-grained to macro-scale observables like the fractal dimension of space use. In this manner, the magnitude of the fractal dimension becomes a proxy variable for distinguishing between main classes of habitat exploration and site fidelity, like memory-less (Markovian) Brownian motion and Levy walk and memory-enhanced space use like Multi-scaled Random Walk (MRW). In this paper previous analyses are extended by exploring MRW simulations under three scenarios: (1) central place foraging, (2) behavioral adaptation to resource depletion (avoidance of latest visited locations) and (3) transition from MRW towards Levy walk by narrowing memory capacity to a trailing time window. A generalized statistical-mechanical theory with the power to model cognitive map influence on individual space use will be important for statistical analyses of animal habitat preferences and the mechanics behind site fidelity and home ranges.

Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models

1. Increased inference regarding underlying behavioural mechanisms of animal movement can be achieved by comparing GPS data with statistical mechanical movement models such as random walk and Lévy walk with known underlying behaviour and statistical properties. 2. GPS data are typically collected with ≥ 1 h intervals not exactly tracking every mechanistic step along the movement path, so a statistical mechanical model approach rather than a mechanistic approach is appropriate. However, comparisons require a coherent framework involving both scaling and memory aspects of the underlying process. Thus, simulation models have recently been extended to include memory-guided returns to previously visited patches, that is, site fidelity. 3. We define four main classes of movement, differing in incorporation of memory and scaling (based on respective intervals of the statistical fractal dimension D and presence/absence of site fidelity). Using three statistical protocols to estimate D and site fidelity, we compare these main movement classes with patterns observed in GPS data from 52 females of red deer (Cervus elaphus). 4. The results show best compliance with a scale-free and memory-enhanced kind of space use; that is, a power law distribution of step lengths, a fractal distribution of the spatial scatter of fixes and site fidelity. 5. Our study thus demonstrates how inference regarding memory effects and a hierarchical pattern of space use can be derived from analysis of GPS data.

The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion

BackgroundThe Lévy flight foraging hypothesis predicts a transition from scale-free Lévy walk (LW) to scale-specific Brownian motion (BM) as an animal moves from resource-poor towards resource-rich environment. However, the LW-BM continuum implies a premise of memory-less search, which contradicts the cognitive capacity of vertebrates.ResultsWe describe methods to test if apparent support for LW-BM transitions may rather be a statistical artifact from movement under varying intensity of site fidelity. A higher frequency of returns to previously visited patches (stronger site fidelity) may erroneously be interpreted as a switch from LW towards BM. Simulations of scale-free, memory-enhanced space use illustrate how the ratio between return events and scale-free exploratory movement translates to varying strength of site fidelity. An expanded analysis of GPS data of 18 female red deer, Cervus elaphus, strengthens previous empirical support of memory-enhanced and scale-free space use in a northern forest ecosystem.ConclusionA statistical mechanical model architecture that describes foraging under environment-dependent variation of site fidelity may allow for higher realism of optimal search models and movement ecology in general, in particular for vertebrates with high cognitive capacity.

Lévy Meets Poisson: A Statistical Artifact May Lead to Erroneous Recategorization of Lévy Walk as Brownian Motion

The flow of GPS data on animal space is challenging old paradigms, such as the issue of the scale-free Lévy walk versus scale-specific Brownian motion. Since these movement classes often require different protocols with respect to ecological analyses, further theoretical development in this field is important. I describe central concepts such as scale-specific versus scale-free movement and the difference between mechanistic and statistical-mechanical levels of analysis. Next, I report how a specific sampling scheme may have produced much confusion: a Lévy walk may be wrongly categorized as Brownian motion if the duration of a move, or bout, is used as a proxy for step length and a move is subjectively defined. Hence, the categorization and recategorization of movement class compliance surrounding the Lévy walk controversy may have been based on a statistical artifact. This issue may be avoided by collecting relocations at a fixed rate at a temporal scale that minimizes over- and undersampling.

The Nordic Mountain Birch Ecosystem - Challenges to Sustainable Management

From a global perspective, the Nordic mountain birch ecosystem is a unique feature of northwestern Europe (see Chap. 1). Although it may appear rather homogeneous and simple, a closer look reveals striking regional and local variation in numerous characteristics such as geology, soils, climate, plant productivity, species composition and herbivory, as well as in the history and human activities in this area. These differences have been pointed out repeatedly throughout this volume. The variability not only poses a great resource, but also a great challenge to sustainable management of the mountain birch (and adjacent alpine/tundra) ecosystems. From the variability of many characteristics, it is obvious that the critical or problematic issues vary regionally. Consequently, sustainable management principles are likely to differ from area to area. The specific structure and properties of the mountain birch forest allow a subdivision into different vegetational units that are characterized by particular ecological conditions (see Chap. 3), and the human impact on the various forest types can be considerable. The density of birch forests and the position of forest lines have varied over time (see Chap. 1) owing to variations in climate and in the human exploitation of birch forests, including domestic herbivores. Particularly in recent times, other human activities, such as tourism and pollution, have had impacts on this ecosystem. Another important, natural cause for long-term dynamics in forest density and productivity is the outbreak of major insect herbivores (mainly the autumnal moth, see Chaps. 5, 9, 12). Events where the stems are killed and the forest is rejuvenated, or the forest is killed, have a major impact on all aspects of the forest’s biology and socio-economic utilization for many decades. There is thus a potential conflict between the longterm dynamics of the mountain birch forest caused by the natural insect herbivores and human utilization of these ecosystems. The effects from severe

Landscape-scale model relating the Nordic mountain birch forest spatio-temporal dynamics to various anthropogenic influences, herbivory and climate change.

Spatial modeling, in general, is challenged by three major aspects of system complexity. Two of these aspects are well known in contemporary systems ecology, and can be studied numerically or analytically within the context of traditional frameworks of mathematics and statistics (O’Neill et al. 1986; Cappuccino and Price 1995). The third aspect of complexity is more intricate, and involves interscale effects that cannot easily be modeled and understood from standard approaches (e.g. O’Neill and King 1998). This framework is still in large part a young scientific theory with great promises for better understanding of complex systems, i.e., it would be of great importance also for management of northern mountain birch forests. Below we illustrate these three aspects of complexity in natural systems.W then describe the spatially explicit, dynamic HIBECO (Human Interactions with the mountain birch ECOsystem) model, where complexity becomes apparent in a practical modeling context. An application of the model to produce northern birch forest management scenarios is presented in Chapter 22.