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In this paper by using upper-lower solution method, under appropriate assumptions on f and g the existence of travelling wave front solutions for the following reaction-diffusion system is proved: u_t - u_xx, = f(u,v) v_t - v_xx = g(u, v) As an application, the necessary and sufficient condition of the existence of monotone solutions for the boundary value problem u" + cu’ + u(1 - u- rv) = 0 v" + cv’ - buv = 0 u(-&#8734) = v(+&#8734) = 0 u(+&#8734) = v(-&#8734) = 1 where 0 < r < 1, 0 < b < 1 - rr are known constants and c is unknown constant to be obtained., number=3, journal=Journal of Partial Differential Equations, year=1991, month=Apr., pages=1–14