Bounds on the Wave Speed

Abstract

   In this paper, we will investigate the structure ofbounds for the wave speed cpresented in [1]. By constructing appropriate sub- and super-solutions to this system

−cu′ = u′′ + f(u, v),

−cv′ = ϵ²v′′ + g(u, v),

 (u, v)(−∞) = S_−,  

(u, v)(∞) = S₊     (1)

Where, we are interested in component-wise monotone travelling wave solutions of the system of equations

u_t= u_xx+ f(u, v),

v_t= ϵ²v_xx+ g(u, v),                   (2)

for (x, t ) ∈R × R⁺ for which the asymptotic conditions

(u, v)(−∞, t) = S_−, (u, v)(∞, t) = S₊, t >0             (3)

are satisfied. Similar to those introduced in [3] and using essentially identical arguments, itcan be shown that

−K ≤ c ≤ Lϵ,(4)

whereK and L are positive constants independent of ϵ. One immediate consequenceof this result is that in the limit ϵ → 0 only left travelling waves exist.We investigate the sharpness of these bounds in the special case of CLV kinetics.We show that:  the bounds of the wave speed given in [4] are optimal for the given leftand right solutions (sub-solutions and super-solutions).