Journal of Applied Probability

Biased movement at a boundary and conditional occupancy times for diffusion processes

Otso Ovaskainen and Stephen J. Cornell

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Motivated by edge behaviour reported for biological organisms, we show that random walks with a bias at a boundary lead to a discontinuous probability density across the boundary. We continue by studying more general diffusion processes with such a discontinuity across an interior boundary. We show how hitting probabilities, occupancy times and conditional occupancy times may be solved from problems that are adjoint to the original diffusion problem. We highlight our results with a biologically motivated example, where we analyze the movement behaviour of an individual in a network of habitat patches surrounded by dispersal habitat.

Article information

J. Appl. Probab. Volume 40, Number 3 (2003), 557-580.

First available in Project Euclid: 24 July 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 92D50: Animal behavior

Random walk bias at boundary diffusion approximation hitting probability exit time conditional exit time occupancy time conditional occupancy time


Ovaskainen, Otso; Cornell, Stephen J. Biased movement at a boundary and conditional occupancy times for diffusion processes. J. Appl. Probab. 40 (2003), no. 3, 557--580. doi:10.1239/jap/1059060888.

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  • Abramowitz, M. and Stegun, I. A. (1972). A Handbook of Mathematical Functions. Dover, New York.
  • Cantrell, R. S. and Cosner, C. (1998). Skew Brownian motion: a model for diffusion with interfaces? In Proc. Internat. Conf. Math. Models Medical Health Sci., Vanderbilt University Press, pp. 74--78.
  • Cantrell, R. S. and Cosner, C. (1999). Diffusion models for population dynamics incorporating individual behaviour at boundaries: applications to refuge design. Theoret. Pop. Biol. 55, 189--207.
  • Fagan, W. F., Cantrell, R. S. and Cosner, C. (1999). How habitat edges change species interactions. Amer. Naturalist 153, 165--182.
  • Hanski, I. (1999). Metapopulation Ecology. Oxford University Press.
  • Hanski, I., Alho, J. and Moilanen, A. (2000). Estimating the parameters of survival and migration of individuals in metapopulations. Ecology 81, 239--251.
  • Harrison, J. M. and Shepp, L. A. (1981). On skew Brownian motion. Ann. Prob. 9, 309--313.
  • Kaiser, H. (1983). Small spatial scale heterogeneity influences predation success in an unexpected way: model experiments on the functional response of predatory mites (Acarina). Oecologia 56, 249--256.
  • Okubo, A. and Levin, S. A. (2001). Diffusion and Ecological Problems: Modern Perspectives. Springer, New York.
  • Redner, S. (2001). A Guide to First-Passage Processes. Cambridge University Press.
  • Ricketts, T. (2001). The matrix matters: effective isolation in fragmented landscapes. Amer. Naturalist 158, 87--99.
  • Ries, L. and Debinski, D. M. (2001). Butterfly responses to habitat edges in the highly fragmented prairies of Central Iowa. J. Anim. Ecol. 70, 840--852.
  • Turchin, P. (1998). Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants. Sinauer Associates, Sunderland, MA.
  • Walsh, J. B. (1978). A diffusion with discontinuous local time. Astérisque 52--53, 37--45.