Models of dispersal in biological systems

Abstract

In order to provide a general framework within which the dispersal of cells or organisms can be studied, we introduce two stochastic processes that model the major modes of dispersal that are observed in nature. In the first type of movement, which we call the position jump or kangaroo process, the process comprises a sequence of alternating pauses and jumps. The duration of a pause is governed by a waiting time distribution, and the direction and distance traveled during a jump is fixed by the kernel of an integral operator that governs the spatial redistribution. Under certain assumptions concerning the existence of limits as the mean step size goes to zero and the frequency of stepping goes to infinity the process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The second major type of movement leads to what we call a velocity jump process. In this case the motion consists of a sequence of "runs" separated by reorientations, during which a new velocity is chosen. We show that under certain assumptions this process leads to a damped wave equation called the telegrapher's equation. We derive explicit expressions for the mean squared displacement and other experimentally observable quantities. Several generalizations, including the incorporation of a resting time between movements, are also studied. The available data on the motion of cells and other organisms is reviewed, and it is shown how the analysis of such data within the framework provided here can be carried out.

A user’s guide to PDE models for chemotaxis

“…A number of authors dating back to Patlak [86] in the 1950s have derived models for chemotaxis based on a more realistic description of individual cell migration (see also [2,76]). Rivero et al [92] (1 + κv) 2 for the case of the flagella bacteria Escherichia coli.…”

Analytic steady-state space use patterns and rapid computations in mechanistic home range analysis

“…In order for time evolution to take place at all, we must also take the limit τ → 0. It is well known [13,14] that the correct way to balance these two limits in order to reach a well-defined diffusion coefficient is to choose the variance of the kernel k to scale as τ . In the 1D case of smooth w, and a translation-invariant kernel (8), we have derived [10] that in this limit our MERSA model gives a Fokker-Planck PDE with known advection and diffusion coefficients.…”

The velocity of spatial population expansion

“…In the paper 'Models of dispersal in biological systems ', Othmer et al (1988) give an excellent overview of mechanistic models of individual dispersal behaviour. Such models can be used as submodels for the dispersal density, D. Here, we will use two mechanistic models for dispersal that lead to well-known probability densities with simple Laplace transforms.…”