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From Birds to Bacteria: Generalised Velocity Jump Processes with Resting States.

Taylor-King JP, van Loon EE, Rosser G, Chapman SJ - Bull. Math. Biol. (2015)

Bottom Line: The resulting system of integro-partial differential equations is tumultuous, and therefore, it is necessary to both simplify and derive summary statistics.Finally, a large time diffusive approximation is considered via a Cattaneo approximation (Hillen in Discrete Continuous Dyn Syst Ser B, 5:299-318, 2003).This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution.

View Article: PubMed Central - PubMed

Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK, jake.taylor-king@sjc.ox.ac.uk.

ABSTRACT
There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long-distance movement state and a resting or local movement state. Here, a mathematical description of this process is formulated, adapted from Friedrich et al. (Phys Rev E, 74:041103, 2006b). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of integro-partial differential equations is tumultuous, and therefore, it is necessary to both simplify and derive summary statistics. An expression for the mean squared displacement is derived, which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally, a large time diffusive approximation is considered via a Cattaneo approximation (Hillen in Discrete Continuous Dyn Syst Ser B, 5:299-318, 2003). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution.

No MeSH data available.


Related in: MedlinePlus

Cross-sectional comparisons along the line  between the diffusion equation and Gillespie simulations of the VJ process. In the (solid) black line, we see the Gaussian solution to the heat equation with diffusion constant , in the (dashed) black line, we see the same solution but for diffusion constant . For the  Gillespie simulations, red asterisks denote the case where  and , and blue plusses show the case  and . Green crosses () show the Gillespie simulation for  and  which corresponds to  (Color figure online)
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Fig5: Cross-sectional comparisons along the line between the diffusion equation and Gillespie simulations of the VJ process. In the (solid) black line, we see the Gaussian solution to the heat equation with diffusion constant , in the (dashed) black line, we see the same solution but for diffusion constant . For the Gillespie simulations, red asterisks denote the case where and , and blue plusses show the case and . Green crosses () show the Gillespie simulation for and which corresponds to (Color figure online)

Mentions: We now carry out a comparison between the underlying differential equation and Gillespie simulation. In Fig. 5, we see a comparison between slices of the solution to the diffusion equation on the plane () for a delta function initial condition6 compared with data simulated using the algorithm given in Sect. 2.Fig. 5


From Birds to Bacteria: Generalised Velocity Jump Processes with Resting States.

Taylor-King JP, van Loon EE, Rosser G, Chapman SJ - Bull. Math. Biol. (2015)

Cross-sectional comparisons along the line  between the diffusion equation and Gillespie simulations of the VJ process. In the (solid) black line, we see the Gaussian solution to the heat equation with diffusion constant , in the (dashed) black line, we see the same solution but for diffusion constant . For the  Gillespie simulations, red asterisks denote the case where  and , and blue plusses show the case  and . Green crosses () show the Gillespie simulation for  and  which corresponds to  (Color figure online)
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4548017&req=5

Fig5: Cross-sectional comparisons along the line between the diffusion equation and Gillespie simulations of the VJ process. In the (solid) black line, we see the Gaussian solution to the heat equation with diffusion constant , in the (dashed) black line, we see the same solution but for diffusion constant . For the Gillespie simulations, red asterisks denote the case where and , and blue plusses show the case and . Green crosses () show the Gillespie simulation for and which corresponds to (Color figure online)
Mentions: We now carry out a comparison between the underlying differential equation and Gillespie simulation. In Fig. 5, we see a comparison between slices of the solution to the diffusion equation on the plane () for a delta function initial condition6 compared with data simulated using the algorithm given in Sect. 2.Fig. 5

Bottom Line: The resulting system of integro-partial differential equations is tumultuous, and therefore, it is necessary to both simplify and derive summary statistics.Finally, a large time diffusive approximation is considered via a Cattaneo approximation (Hillen in Discrete Continuous Dyn Syst Ser B, 5:299-318, 2003).This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution.

View Article: PubMed Central - PubMed

Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK, jake.taylor-king@sjc.ox.ac.uk.

ABSTRACT
There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long-distance movement state and a resting or local movement state. Here, a mathematical description of this process is formulated, adapted from Friedrich et al. (Phys Rev E, 74:041103, 2006b). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of integro-partial differential equations is tumultuous, and therefore, it is necessary to both simplify and derive summary statistics. An expression for the mean squared displacement is derived, which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally, a large time diffusive approximation is considered via a Cattaneo approximation (Hillen in Discrete Continuous Dyn Syst Ser B, 5:299-318, 2003). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution.

No MeSH data available.


Related in: MedlinePlus