\tilde{\a}_d^{\pm}(c,\b)=\frac{1}{2d}\left(c\pm\sqrt{c^2-{c_{\operatorname{KPP}}}^2-4d^2\b^2}\right).$$ The function $\tilde{\a}_d^-$ satisfies $\p_{\b}\tilde{\a}_d^-(c,\b)>0$, $\p_{c}\tilde{\a}_d^-(c,\b)<0$, for $c>{c_{\operatorname{KPP}}}$ and $\b\in(0,r(c))$, while its value at $\b=0$ satisfies $\tilde{\a}_d^-(c,0)=\a_d^-(c,0)$. Moreover, it can be easily seen that $$\tilde{\S}_d({c_{\operatorname{KPP}}})=\left\{\left(0,\frac{{c_{\operatorname{KPP}}}}{2d}\right)\right\}$$ and $$\lim_{c\ua+\infty}\tilde{a}_d^-(c,\b)=0.$$ As a consequence of these properties, the half-disks delimited by $\tilde{\S}_d(c)$ and contained in the first quadrant of the $(\b,\a)-$plane invade it monotonically as $c$ increases. With these ingredients we are able to give the following result

\[pr41\] There exists $c^*:=c^*_L(D,d,\mu,\nu)$ such that, for every $c>c^*$, Problem admits supersolutions either of the form or , with positive $\a,\b,\g$. Moreover the function $D\mapsto c^*(D)$ is increasing.

By the previous discussion, we have that, in order to find a solution of and therefore a supersolution to , it is sufficient to find an intersection between either $\S_D(c)$ and $\S_d(c)$ or $\tilde{\S}_D(c)$ and $\tilde{\S}_d(c)$ lying in the interior of the first quadrant of the $(\b,\a)-$plane. Due to the monotonicity properties of these curves with respect to $c$ shown above, $c^*$ will be the smallest value of $c$ for which such an intersection exists for all $c>c^*$.

Let us start by examining the case $D$ small (relatively to $d$), in which we consider the curves $\S_D(c)$ and $\S_d(c)$. Recalling that we are assuming , it is 