speed of propagation towards positive steady states of , we will assume and throughout the rest of the paper, for to hold.

Supersolutions in the moving framework {#sec4}
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In this section we construct positive supersolutions to moving at appropriate speeds in the $x-$direction. This will be the key to find an upper bound for the asymptotic speed of propagation of Theorem \[thmain\] (see Section \[sec6\]). Observe that solutions of the linearized problem $$\label{41}
  \left\{ \begin{array}{lll}
  u_t(x,t)-Du_{xx}(x,t)=\nu v(x,L,t)-\m u(x,t) & \text{for } x\in{{\mathbb{R}}}, &\!\! t>0 \\
  v_t(x,y,t)-d\D v(x,y,t)=f'(0)v(x,y,t) & \text{for } (x,y)\in\O, &\!\! t>0  \\ 
  dv_y(x,L,t)=\m u(x,t)-\nu v(x,L,t) & \text{for } x\in{{\mathbb{R}}}, &\!\! t>0 \\
  v(x,0,t)=0 & \text{for } x\in{{\mathbb{R}}}, &\!\! t>0, \end{array}\right.$$ provide us with supersolutions to , thanks to the second condition of . We start looking for solutions of of the form $$\label{42}
(u(x,t),v(x,y,t))=e^{\a(x+ct)}(1,\g \sin (\b y))$$ with positive $\a,\g, c$ and $\b\in(0,\frac{\pi}{L})$, in order for $u$ and $v$ to be positive in $\O$. Notice that is a solution of if and only if $$\label{43}
  \left\{ \begin{array}{l}
  -D\a^2+c\a=\n\g\sin(\b L)-\m \\
  -d\a^2+d\b^2+c\a=f'(0) \\
  \g=\frac{\m}{d\b \cos(\b L)+\n \sin (\b L)}. \end{array}\right.$$ In order for $\g$ to be positive, $\b$ must lie in $(0,\ov{\b})$, where $\ov{\b}\in(\frac{\pi}{2L},\frac{\pi}{L})$ is the first positive 