to be examined is the one in which all the four curves touch for the first time, being tangent, at $\b=0$ for $c={c_{\operatorname{int}}}$, where ${c_{\operatorname{int}}}$ is $$\label{420}
{c_{\operatorname{int}}}^2={c_{\operatorname{int}}}^2(D)=\frac{\left((d+\nu L)D{c_{\operatorname{KPP}}}^2+ 4\mu d^3\right)^2}{4d(D-d)(d+\nu L)\left((d+\nu L){c_{\operatorname{KPP}}}^2 + 4\mu d^2\right)}$$ (see figure \[fig26\](C)). In this case, thanks to and , we have that $\S_D(c)$ and $\S_d(c)$ intersect for every $c>{c_{\operatorname{int}}}$ and a certain $\b>0$. Therefore we set $c^*={c_{\operatorname{int}}}$.

The monotonicity of the function $D\mapsto c^*(D)$ follows from the monotonicity of the curves $\a_D^{\pm}$ and $\tilde{\a}_D^+$, given by and the last relation of respectively.

\[remark42\]

-   The proof of Proposition \[pr41\] shows indeed that it is possible to construct a supersolution to of type or not only for every $c>c^*$, but also for $c=c^*$, except for the case represented in Figure \[fig26\](C). Actually it is possible to construct a supersolution also in this case, when $c^*={c_{\operatorname{int}}}$, by taking $$\label{421}
    (u(x,t),v(x,y,t))=e^{\a(x+ct)}\left(1,\frac{\mu y}{d+\nu L}\right).$$ This can be heuristically seen by taking the limit of or as $\b\da 0$ (notice that $\g=\g(\b)\to 0$), while a formal proof consists in plugging into and observing that the resulting algebraic system in $\a,c$ has a solution for $c={c_{\operatorname{int}}}$.

-   From the last part of proof of Proposition \[pr41\], it arises the 