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

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Abstract

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.

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Notes

  1. Here we are considering “dispersal” in a generic sense: movement along a certain path.

  2. The counterpart situation where the particles are destroyed at the boundaries is described by the condition of no outgoing density at the borders, which is satisfied if \(u_{-} \left( 0, \zeta \right) = u_{+} \left( 1, \zeta \right) = 0\). The Robin set of conditions \(u \left( 0, \zeta \right) = j \left( 0, \zeta \right) \) and \(u \left( 1, \zeta \right) = -j \left( 1, \zeta \right) \) is only marginally different from the one used in the text, thus generating similar results.

  3. The appearance of Eq. (45) can also be understood as a result of the boundary condition at \(z=0\) being implemented for the spatial and temporal parts independently: if we impose that \(U \left( 0\right) = J \left( 0\right) \), we obtain \(C_{1} = C_{2}\) and \(C_{4} = -C_{3}\); now the temporal parts must satisfy \(\Psi \left( \zeta \right) = - \Phi \left( \zeta \right) \), which generates equation (45).

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Acknowledgements

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.).

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Authors and Affiliations

  1. Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

    Paulo F. C. Tilles & Sergei V. Petrovskii

  2. Departamento de Matemática, Universidade Federal de Santa Maria, Santa Maria, RS, Brazil

    Paulo F. C. Tilles

  3. Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, Russian Federation, 117198

    Sergei V. Petrovskii

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  1. Paulo F. C. Tilles
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Correspondence to Sergei V. Petrovskii.

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Communicated by Irene Giardina.

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Tilles, P.F.C., Petrovskii, S.V. On the Consistency of the Reaction-Telegraph Process Within Finite Domains. J Stat Phys 177, 569–587 (2019). https://doi.org/10.1007/s10955-019-02379-0

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