Consider the differential-difference system
where A travelling wave with speed for (dds) is a solution of the form and the paper is concerned with the ones satisfying
Under the assumption
the authors prove that
(i) Wave profiles of a given speed are unique up to a translation.
(ii) Any wave profile is monotone, i.e. on
(iii) Any solution of (twe) satisfies
where is a nonnegative real root of the characteristic equation
and is the negative real root of the characteristic equation
In addition, is the smallest root when and the largest root when This improves and completes earlier results of X. Chen and J. S. Guo.