the $(\b,\a)-$plane delimited by the curve $$\label{47}
\S_D(c):=\left\{(\b,\a_D^-(\b)):\b\in\left[\frac{\pi}{2L},\tilde{\b}(c)\right]\right\}\cup\left\{(\b,\a_D^+(\b)):\b\in\left(0,\tilde{\b}(c)\right]\right\},$$ which has been represented in Figure \[fig21\], invades monotonically, as $c\ua\infty$, the half-strip $\{(\b,\a): \b\in(0,\ov{\b}), \a>0\}$.

![The curve $\S_D(c)$ defined in .[]{data-label="fig21"}](fig1.pdf)

As far as the monotonicity of the curves with respect to $D$, the other parameters being fixed, it follows from the definition of $\a_D^{\pm}$ that $$\label{48}
\begin{split}
D&\mapsto \a_D^+(c,\b) \; \text{ is decreasing for every $\b\in[0,\tilde{\b}(c)]$} \\
D&\mapsto \a_D^-(c,\b) \; \text{ is increasing for every $\b\in\left(\frac{\pi}{2L},\tilde{\b}(c)\right]$.}
\end{split}$$

On the other hand (see Figure \[fig22\]), the second equation of represents an hyperbola whose branches are given by $$\label{49}
\a_d^{\pm}(c,\b)=\frac{1}{2d}\left(c\pm\sqrt{c^2-\eta(\b)}\right),$$ where we have set, $$\eta(\b):=4d(f'(0)-d\b^2)={c_{\operatorname{KPP}}}^2-4d^2\b^2,$$ $c_{\operatorname{KPP}}$ being the one defined in . The function $\eta$ is decreasing and satisfies $$\eta(0)={c_{\operatorname{KPP}}}^2, \qquad \eta\left(\sqrt{\frac{f'(0)}{d}}\right)=0, \qquad \lim_{\b\ua\infty}\eta(\b)=-\infty.$$ Therefore, the functions $\a_d^{\pm}(c,\b)$ are defined for every $\b\in(0,+\infty)$ if $c\geq {c_{\operatorname{KPP}}}$, while, if $c\in(0,{c_{\operatorname{KPP}}})$ there exists $\hat{\b}(c)>0$ such that their domain (within the positive part of the real line) is $[\hat{\b}(c),+\infty)$, where $\hat{\b}(c)$ satisfies $$\label{410}
c^2=\eta(\hat{\b}(c)).$$ As a consequence, $$\lim_{c\da 0} \hat{\b}(c)=\sqrt{\frac{f'(0)}{d}}, \qquad \lim_{c\ua {c_{\operatorname{KPP}}}} \hat{\b}(c)=0.$$ It can be easily seen, as $\a_d^-$ is the reflection of $\a_d^+$ about the line $\a=\frac{c}{2d}$, that $$\a_d^-\left(c,\sqrt{\frac{f'(0)}{d}}\right)=0, \qquad \lim_{\b\ua\infty}\a_d^+(c,\b)=+\infty,$$ and, in the proper domain of definition, $$\label{411}
\p_\b\a_d^+(c,\b)>0, \qquad \p_c\a_d^+(c,\b)>0, 