Diffusion in Fractcal Landscapes: Simulations and Experimental Studies of Tenebrionid Beetle Movements

Organismal movement is frequently treated as a diffusion process arising from a simple random walk in a spatially uniform environment. However, anomalous diffusion may arise due to: (1) intrinsic departures from random movements of individuals; or (2) the effects of barriers that impede, or corridors that facilitate, movement. We examine anomalous diffusion in field studies of Eleodes beetles (Coleoptera: Tenebrionidae) and in simulations of correlated random walks on maps of real and artificial landscapes. We show how diffusion alternates between ordinary and anomalous diffusion depending on movement rules, landscape pattern, and the spatial and temporal scales of observation. Recent theories of diffusion in spatially complex media predict power law relations for anomalous diffusion. Over time scales of 5-500 s, Eleodes exhibited power laws for: (1) mean squared displacement with time; and (2) the mean time to travel from the center to the perimeter of circles of various radii. In grasslands, diffusion exponents changed significantly at a radius of 42 cm, which characterized the size of grass and bare soil patches. A second change in diffusion at scales of 24-600 h characterized home range activity. Marked discrepancies between the dynamics of beetles in the field and in simulations suggest a needmore » for more comprehensive models of individual movement that use different rules for various domains of space and time. Studies of anomalous diffusion identify the relative effects of environment vs. innate behavior and reveal a range of scales over which the effects pertain. 41 refs., 12 figs., 4 tabs.« less

[1]  Simon A. Levin,et al.  The Spread of a Reinvading Species: Range Expansion in the California Sea Otter , 1988, The American Naturalist.

[2]  George Marsaglia,et al.  Toward a universal random number generator , 1987 .

[3]  B. Mandelbrot.  Fractal Geometry of Nature , 1984 .

[4]  J. Wiens.  Spatial Scaling in Ecology , 1989 .

[5]  R. Rammal,et al.  Random walk statistics on fractal structures , 1984 .

[6]  J. F. Mccarthy,et al.  Effective permeability of sandstone-shale reservoirs by a random walk method , 1990 .

[7]  Robert Tibshirani,et al.  Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy , 1986 .

[8]  H. Berg.  Random Walks in Biology , 2018 .

[9]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[10]  S. Alexander,et al.  Density of states on fractals : « fractons » , 1982 .

[11]  Stanley,et al.  Random-walk approach to the two-component random-conductor mixture: Perturbing away from the perfect random resistor network and random superconducting-network limits. , 1986, Physical review. B, Condensed matter.

[12]  S. Benhamou,et al.  Spatial analysis of animals' movements using a correlated random walk model* , 1988 .

[13]  Bruce J. West,et al.  Fractal dimensionality of Lévy processes. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[14]  R. Parmenter,et al.  Factors influencing microhabitat partitioning in arid-land darkling beetles (Tenebrionidae): temperature and water conservation , 1989 .

[15]  C. McCulloch,et al.  Analyzing Discrete Movement Data as a Correlated Random Walk , 1989 .

[16]  J. G. Skellam.  Random dispersal in theoretical populations , 1951, Biometrika.

[17]  R. Parmenter,et al.  Factors Influencing Microhabitat Partitioning among Coexisting Species of Arid-Land Darkling Beetles (Tenebrionidae): Behavioral Responses to Vegetation Architecture , 1989 .

[18]  J. L. Jackson,et al.  Dissipative structure: an explanation and an ecological example. , 1972, Journal of theoretical biology.

[19]  M. Mimura,et al.  On a diffusive prey--predator model which exhibits patchiness. , 1978, Journal of theoretical biology.

[20]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[21]  Dietrich Stauffer,et al.  Diffusion on random systems above, below, and at their percolation threshold in two and three dimensions , 1984 .

[22]  S. Redner.  Recent progress and current puzzles in percolation , 1983 .

[23]  Rob Hengeveld,et al.  Dynamics of Biological Invasions , 1989 .

[24]  M. Mimura,et al.  SOME DIFFUSIVE PREY AND PREDATOR SYSTEMS AND THEIR BIFURCATION PROBLEMS , 1979 .

[25]  R. Orbach,et al.  Dynamics of Fractal Networks , 1986, Science.