and, together with the first and third relations of , this implies that there exists a value of $D$, denoted by $D_{\operatorname{KPP}}$, for which, for $c=c_{\operatorname{KPP}}$, $\a_{D_{\operatorname{KPP}}}^+$ and the straight line $\a_d^-$ are tangent (see Figure \[fig23\](C)). Observe that, for $D=2d$, we have $$\a_{D}^+(c_{\operatorname{KPP}},0)>\a_{D}^+\left(c_{\operatorname{KPP}},\frac{\pi}{2L}\right)=\frac{c_{\operatorname{KPP}}}{D}=\frac{c_{\operatorname{KPP}}}{2d}=\a_{d}^-(c_{\operatorname{KPP}},0),$$ which implies that $D_{\operatorname{KPP}}>2d$. Thanks to , for $d<D<D_{\operatorname{KPP}}$ and $c=c_{\operatorname{KPP}}$, $\a_D^+$ and $\a_d^-$ will be secant, therefore in this case the tangency will occur between $\a_D^+$ and $\a_d^-$ for $\un{c}<c^*<{c_{\operatorname{KPP}}}$, as represented in Figure \[fig23\](D).

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   ![Different configurations of the tangency between $\S_D(c)$ and $\S_d(c)$ (continuous lines) or $\tilde{\S}_D(c)$ and $\tilde{\S}_d(c)$ (dashed lines), for $c>c_{\operatorname{KPP}}$.[]{data-label="fig26"}](fig261.pdf "fig:")   ![Different configurations of the tangency between $\S_D(c)$ and $\S_d(c)$ (continuous lines) or $\tilde{\S}_D(c)$ and $\tilde{\S}_d(c)$ (dashed lines), for $c>c_{\operatorname{KPP}}$.[]{data-label="fig26"}](fig262.pdf "fig:")   ![Different configurations of the tangency between $\S_D(c)$ and $\S_d(c)$ (continuous lines) or $\tilde{\S}_D(c)$ and $\tilde{\S}_d(c)$ (dashed lines), for $c>c_{\operatorname{KPP}}$.[]{data-label="fig26"}](fig263.pdf "fig:")
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Finally, when $D>D_{\operatorname{KPP}}$ we consider at the same time the pairs of curves $\S_D(c)$, $\S_d(c)$ and $\tilde{\S}_D(c)$, $\tilde{\S}_d(c)$. We define $c^*$ to be the smallest value of $c$ for which either the former two curves or the latter two are tangent in a positive $\b$ (see Figure \[fig26\](A) and (B) respectively). The last case 