  Standard diffusion equation is based on Brownian motion of the dispersing
species without considering persistence in the movement of the individuals.
This description allows for the instantaneous spreading of the transported
species over an arbitrarily large distances from their original location
predicting infinite velocities. This feature is unrealistic particularly while
considering biological invasion dynamics and a better description needs the
consideration of dispersal with inertia. We here examine the behavior of
non-infinitesimal perturbation on the steady state of an one-dimensional
reaction-Cattaneo system with a cubic polynomial source term describing
population dynamics or flame propagation models. It has been shown analytically
that while linear analysis predicts stability of the homogeneous state,
consideration of nonlinear contribution leads to a growth of spatiotemporal
perturbation as a traveling wave. We show that the presence of a small finite
relaxation time of the diffusive flux modifies the speed of the traveling wave.
Specifically, we find that the wave speed decays with an increase of a finite
relaxation time of flux. Our analytical predictions are well corroborated with
the numerical results.
