University of Leicester
Browse
JOSS-D-19-00084_R1.pdf (903.38 kB)

On the Consistency of the Reaction-Telegraph Process Within Finite Domains

Download (903.38 kB)
journal contribution
posted on 2019-09-23, 10:23 authored by Paulo F. C. Tilles, Sergei V. Petrovskii
Reaction-telegraph equation (RTE) is a mathematical model that has often been used to describe natural phenomena, with specific applications ranging from physics to social sciences. In particular, in the context of ecology, it is believed to be a more realistic model to describe animal movement than the more traditional approach based on the reaction-diffusion equations. Indeed, the reaction-telegraph equation arises from more realistic microscopic assumptions about individual animal movement (the correlated random walk) and hence could be expected to be more relevant than the diffusion-type models that assume the simple, unbiased Brownian motion. However, the RTE has one significant drawback as its solutions are not positively defined. It is not clear at which stage of the RTE derivation the realism of the microscopic description is lost and/or whether the RTE can somehow be ‘improved’ to guarantee the solutions positivity. Here we show that the origin of the problem is twofold. Firstly, the RTE is not fully equivalent to the Cattaneo system from which it is obtained; the equivalence can only be achieved in a certain parameter range and only for the initial conditions containing a finite number of Fourier modes. Secondly, the Dirichlet type boundary conditions routinely used for reaction-diffusion equations appear to be meaningless if used for the RTE resulting in solutions with unrealistic properties. We conclude that, for the positivity to be regained, one has to use the Cattaneo system with boundary conditions of Robin type or Neumann type, and we show how relevant classes of solutions can be obtained.

Funding

This work was supported by The Royal Society (UK) through the Grant No. NF161377 (to P.F.C.T and S.V.P.). The publication has been prepared with the support of the “RUDN University Program 5-100” (to S.V.P.).

History

Citation

Journal of Statistical Physics, 2019

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

  • AM (Accepted Manuscript)

Published in

Journal of Statistical Physics

Publisher

Springer Verlag (Germany)

issn

0022-4715

eissn

1572-9613

Acceptance date

2019-08-27

Copyright date

2019

Publisher version

https://link.springer.com/article/10.1007/s10955-019-02379-0

Notes

The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.

Language

en

Usage metrics

    University of Leicester Publications

    Categories

    No categories selected

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC