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.