Modelling nematode movement using time-fractional dynamics.

We use a correlated random walk model in two dimensions to simulate the movement of the slug parasitic nematode Phasmarhabditis hermaphrodita in homogeneous environments. The model incorporates the observed statistical distributions of turning angle and speed derived from time-lapse studies of individual nematode trails. We identify strong temporal correlations between the turning angles and speed that preclude the case of a simple random walk in which successive steps are independent. These correlated random walks are appropriately modelled using an anomalous diffusion model, more precisely using a fractional sub-diffusion model for which the associated stochastic process is characterised by strong memory effects in the probability density function.

[1]  R. Haydock,et al.  Vector continued fractions using a generalized inverse , 2003, math-ph/0310041.

[2]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[3]  Bryan S. Griffiths,et al.  Nematode movement along a chemical gradient in a structurally heterogeneous environment. 1. Experiment , 1997 .

[4]  Magnus Wiktorsson,et al.  Irregular walks and loops combines in small-scale movement of a soil insect: implications for dispersal biology. , 2004, Journal of theoretical biology.

[5]  Mingzhou Ding,et al.  Processes with long-range correlations : theory and applications , 2003 .

[6]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[7]  Laure Coutin,et al.  Operators associated with stochastic differential equations driven by fractional Brownian motions , 2005 .

[8]  Magnus Wiktorsson,et al.  Modelling the movement of a soil insect. , 2004, Journal of theoretical biology.

[9]  E. Teramoto,et al.  Mathematical Topics in Population Biology, Morphogenesis and Neurosciences , 1987 .

[10]  Simon A. Levin,et al.  Ecological and Evolutionary Aspects of Dispersal , 1987 .

[11]  E. Barkai.  CTRW pathways to the fractional diffusion equation , 2001, cond-mat/0108024.

[12]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .

[13]  V. V. Krishnan,et al.  The Effects of Edge Permeability and Habitat Geometry on Emigration from Patches of Habitat , 1987, The American Naturalist.

[14]  F. Bigler,et al.  An individual-based model of Trichogramma foraging behaviour : parameter estimation for single females , 1996 .

[15]  M. Levandowsky,et al.  Random movements of soil amebas , 1997 .

[16]  H. Larralde,et al.  Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi) , 2003, Behavioral Ecology and Sociobiology.

[17]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[18]  D. Macdonald,et al.  Scale‐free dynamics in the movement patterns of jackals , 2002 .

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

[20]  D. Glen,et al.  Mass cultivation and storage of the rhabditid nematode Phasmarhabditis hermaphrodita, a biocontrol agent for slugs , 1993 .

[21]  W. H. Neill,et al.  Modelling animal movement as a persistent random walk in two dimensions: expected magnitude of net displacement , 2000 .

[22]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[23]  D. Glen,et al.  The rhabditid nematode Phasmarhabditis hermaphrodita as a potential biological control agent for slugs , 1993 .

[24]  Nathanaël Enriquez.  A simple construction of the fractional Brownian motion , 2004 .

[25]  H. Preisler,et al.  Modeling animal movements using stochastic differential equations , 2004 .

[26]  P. A. Prince,et al.  Lévy flight search patterns of wandering albatrosses , 1996, Nature.

[27]  Peter J. Gregory,et al.  A general random walk model for the leptokurtic distribution of organism movement: Theory and application , 2007 .

[28]  H. T. Banks,et al.  Parameter estimation techniques for interaction and redistribution models: a predator-prey example , 1987, Oecologia.

[29]  J. M. Thomas,et al.  Introduction à l'analyse numérique des équations aux dérivées partielles , 1983 .

[30]  P. Kareiva,et al.  Analyzing insect movement as a correlated random walk , 1983, Oecologia.

[31]  I. Glazer,et al.  Comparison of Entomopathogenic Nematode Dispersal from Infected Hosts Versus Aqueous Suspension , 1996 .

[32]  W. J. Bell.  Searching Behaviour: The Behavioural Ecology of Finding Resources , 1991 .

[33]  T. J. Roper,et al.  Non-random dispersal in the butterfly Maniola jurtina: implications for metapopulation models , 2000, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[34]  R. Metzler,et al.  Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation , 2002 .

[35]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[36]  M. Chitwood,et al.  Plant Parasitic Nematodes , 1981 .