= Solution
The equation is a <conservation law> $n_t+J_x=0$ with <diffusive flux> $J=-D(n_0/n)n_x$. For positive density, insects move down the density gradient, with effective <diffusion coefficient> $Dn_0/n$: dispersal is stronger in low-density regions. This is <logarithmic diffusion>, since the equation is $n_t=Dn_0(\log n)_{xx}$.
Put $\tau=Dt$ and $\xi=x/\tau^\beta$. A <self-similar solution> $n=n_0\tau^{-\beta}g(\xi)$ conserves total number when $\int_{\mathbb R}g=1$. Its width shrinks to zero as $t\downarrow0$; for an integrable nonnegative $g$, changing variables in the integral against a bounded <continuous function> proves convergence to $n_0$ times the <Dirac delta distribution> at the origin.
Substitution gives
$$
-\beta\tau^{-\beta-1}(g+\xi g')
=\tau^{-2\beta}(g'/g)'.
$$
Matching powers for a nonstationary profile forces \b[$\beta=1$]. The profile equation integrates to $g'/g=-\xi g+C$. For a centred even profile, $C=0$, and integration of $g'=-\xi g^2$ gives $g=2/(\xi^2+a^2)$. Normalization requires $2\pi/a=1$, hence
$$
\boxed{g(\xi)=\frac2{\xi^2+4\pi^2},\qquad
n(x,t)=\frac{2n_0Dt}{x^2+4\pi^2D^2t^2}.}
$$
This is $n_0$ times a <Cauchy distribution> density with scale $2\pi Dt$. It is positive, has total mass $n_0$, and directly satisfies the <nonlinear diffusion equation> for $t>0$.
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