Emergent properties of Animal Space Use - Part I

Animal space use is complex, not only due to being complicated to understand in model terms but also from the philosophical perspective of the term "emergence". Qualitatively, we are then flipping from complicatedness to true complexity. Emergence of a system's behaviour occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Recently I illustrated this fascinating topic in Frontiers in Ecology and Evolution (Gautestad 2022). Here the "parts" are represented by a large collection of an individual's relocations (sample of spatial fixes of movement) under influence of memory. Memory map utilization invites to study animal space use from two complementary perspectives, topologically and spatiotemporally. In Gautestad (2022) I cover both aspects; first, I use simulations involving memory-dependent site fidelity of an individual to explore in phenomenological terms the network-topological aspect of the emerging network of nodes (targets for return events). Secondly, I toggle from the topological aspect of networks to the spatiotemporal aspect of space use under this premise. In this Part I of the post I present an excerpt of the topological analysis.

Individual movement may be considered to be a mixture of exploratory moves and some occasional events of return, where the latter generate site fidelity but depend on spatial memory. Some locations will over time become more frequently revisited than others; a property that may be called non-random self-crossing of the individual’s movement path. In overall terms the animal’s home range becomes an emergent property of the tendency to revisit historic locations. Thus, memory map utilization is a key aspect of cognitive movement ecology. 

From a network theoretical perspective sampled locations along a movement path (fixes) may be said to represent potential nodes. Actual nodes will emerge from memory-dependent returns to a small subset of these historic locations. This kind of individual-centric network topology deviates conceptually and qualitatively from the geometrically explicit dispersion of patches the animal is attracted to and the paths the animal follows to commute between them. For example, the set of the closest patches in the network may be independent of the Euclidean distance between the network node and its neighbor nodes*. Independence between physical distance and closeness based on historic revisitation events has been supported empirically in American bison Bison bison (Merkle et al., 2014, 2017) and Fowler’s toads Anaxyrus fowleri (Marchand et al., 2017). For the present context of cognitive movement ecology I label the scenarios “Site Fidelity Network” (SFN).

Network graphs, produced in Gephi 0.8.2 and 0.9.2 (Bastian et al., 2009), illustrate how a site fidelity network (SFN) becomes increasingly complex by the emergence of an increasing number of hierarchical levels of connected nodes (re-used locations) over time. Each series in the upper part of the Figure above consists of 104 steps with a new potential or realized node creation every 10 step on average. Node degree ("popularity") is expressed by the size of the bullets, while colors only serve to distinguish the respective magnitudes of degree. In the upper image the center part of the network visualization represents Level 1, containing nodes with only one connecting link (smallest network fragments). The lower image shows that the link structure of the initial 104 return events of a series of 105 steps becomes more complex with more levels when studied as the initial sub-section of a 10 times larger series than in the upper image. In other words, the network is more “mature” owing to the longer time span of development of the initial part of the network.

In general terms we are surrounded by networks, both real and virtual (see references in my paper). On the WorldWide Web two Websites are connected if there is a URL pointing from one site to another. Statistically, most websites are referred to by a few other sites, while a few sites have a tremendous number or referring sites. Mathematically the distribution tends to self-organize into power law compliance: k times larger Website popularity is reduced by a factor 1/k^γ . The distribution P(k)=k^-γ is scale-free over the range of the part of P(k) where γ is stable, and is said to be complex over this range. Popular sites apparently grow in popularity in a self-reinforcing, positive feedback manner (“rich get richer”).

In short, networks are at the center of studying and ultimately understanding complex systems in very broad terms. On the other hand, a non-complex (“regular”) distribution would be expected to comply with an exponential rather than

a power law decline of popularity. In this case γ is not stable over a large range of k, and the frequency of ultralarge-k events becomes negligible in comparison to the power law range, which tends to enlarge the “fat tail” of the distribution. In the context of animal space use, while most locations have a low revisit probability, emergence of extreme patch “popularity,” albeit rare, are also expected.

The so-called degree distribution of node weights (connectivity) with inset showing log-log transformation (upper part of the Figure). Ten independent series were produced, and the respective replicates were averaged. The middle and lower panes show the log-log transformed degree distribution for medium and strong preferential connectivity, respectively (detals given in my paper).

For the first time the theory of network topology is applied to understand animal space use under memory utilization.  Both aspects of complex space use, network topology and dispersion of site fidelity (see upcoming Part II), provide in tandem important descriptors of behavioral ecology with relevance to habitat selection.

NOTE

*) In a virtual network, if the animal revisits a node from another node the topological distance between the two nodes is shortened in an incremental manner for each such revisit between the two. Such return events represent inter-node attachment growth. Short-cuts where for example the individual moves in the node sequence A-B-C-A contributes to increasing the connectivity strength (called the degree, illustrated as bullet size in the illustration) of the revisited nodes. When the animal returns directly to A from C, node A is advancing upwards in the hierarchy of node connectivity strength, which is shown by the new connecting line segment from C to A. This new connection means that A and C also move closer in network topological terms. In contrast, the physical (Euclidean) distance between nodes A, B and C (the “patches” A’, B’ and C’ in the lower part of the illustration) remains the same regardless of node degree and respective topological distances. Along the dotted path only A’, B’ and C’ belong to the

network due to previous revisits, while the rest of the path (dotted line) represents potential nodes with still unrealized connectivity to the network ofactual nodes.

REFERENCES

Bastian, M., Heymann, S., and Jackomy, M. (2009). “Gephi: an open source software for exploring and manipulating networks,” in Proceedings of the International AAAI Conference on Weblogs and Social Media (Menlo Park, CA:

Association for the Advancement of Artificial Intelligence), 361–362.

Gautestad A.O (2022) Individual Network Topology of Patch Selection Under Influence of Drifting Site Fidelity. Front. Ecol. Evol. 10:695854. doi: 10.3389/fevo.2022.695854

Merkle, J. A., Fortin, D., and Morales, J. M. (2014). A memory-based foraging tactic reveals an adaptive mechanism for restricted space use. Ecol. Lett. 17, 924–931. doi: 10.1111/ele.12294

Merkle, J. A., Potts, J. R., and Fortin, D. (2017). Energy benefits and emergent space use patterns of an empirically parameterized model of memory-based patch selection. Oikos 126, 185–195. doi: 10.1111/oik.03356

Marchand, P., Boenke, M., and Green, D. M. (2017). A stochastic movement model reproduces patterns of site fidelity and long-distance dispersal in a population of Fowler’s toads (Anaxyrus fowleri). Ecol. Model. 360, 63–69. doi:10.1016/j.ecolmodel.2017.06.025