A framework for generating and analyzing movement paths on ecological landscapes

Abstract

The movement paths of individuals over landscapes are basically represented by sequences of points (x(i), y(i)) occurring at times t(i). Theoretically, these points can be viewed as being generated by stochastic processes that in the simplest cases are Gaussian random walks on featureless landscapes. Generalizations have been made of walks that (i) take place on landscapes with features, (ii) have correlated distributions of velocity and direction of movement in each time interval, (iii) are Lévy processes in which distance or waiting-time (time-between steps) distributions have infinite moments, or (iv) have paths bounded in space and time. We begin by demonstrating that rather mild truncations of fat-tailed step-size distributions have a dramatic effect on dispersion of organisms, where such truncations naturally arise in real walks of organisms bounded by space and, more generally, influenced by the interactions of physiological, behavioral, and ecological factors with landscape features. These generalizations permit not only increased realism and hence greater accuracy in constructing movement pathways, but also provide a biogeographically detailed epistemological framework for interpreting movement patterns in all organisms, whether tossed in the wind or willfully driven. We illustrate the utility of our framework by demonstrating how fission-fusion herding behavior arises among individuals endeavoring to satisfy both nutritional and safety demands in heterogeneous environments. We conclude with a brief discussion of potential methods that can be used to solve the inverse problem of identifying putative causal factors driving movement behavior on known landscapes, leaving details to references in the literature.

Fig. 1.

The logarithm of mean-square displacements (msd) ψ versus the logarithm of time t average over 10,000 simulations are plotted for a random walk with step lengths drawn from modified Pareto distributions (Upper Left, q = 2; Upper Right, q = 1.5; Lower Left, q = 1) and directions for each step completely random. From the lines inserted by “eye” (red, small t; blue, large t), Upper Left represents diffusion (P = 1 for large t), and Upper Right and Lower Left represent super diffusion (respectively, P = 1.15 > 1 and P = 2 for large t). In Lower Right, for the case q = 1, the length of these excursions are truncated at a step size of 100 (biologically, an upper bound is set by the maximum velocity of the organism multiplied by the length of the time interval), which is far out in the tails of the distribution (two orders of magnitude beyond the mode; see Fig. S3). In this case, the noisy superdiffusive behavior is completely tamed even though, initially, it looks superdiffusive (ψ ∼ t^1.4 for t ≤ 2).

Fig. 2.

Each of the five panels is an extract from a much larger mapping of the values of elements in the landscape modifier matrices (LMMs) for the vegetation and safety landscapes of the realized discretized movement distributions constructed from these matrices. The focal individual is represented by the small red squares in each of the five panels, with the positions of its conspecifics represented by other small squares in Upper Left and Lower. In the distributions represented in Lower, the most attractive areas are the lighter areas (but ignoring the small dark blue squares, which are just conspecific position markers for reference). The relative weighting of safety over resources ranges from safety being the only consideration (Lower Left), safety and resources being equally important (Lower Center), and resources being the only consideration (Lower Right). Imposed upon Lower is a circle representing the maximum possible movement displacement in one time step.

Fig. 3.

Simulation of fission–fusion behavior as a function of vegetation quality. Open squares, high quality; filled diamonds, low quality. Population defined to be in a two-herd (one herd) state when the fission index is <0.5 (>0.5). See main text and SI Text for details.