Example of chemical reaction with a given form for the kinetic equations
A chemist friend found an answer for the simplest case:
$$
\begin{cases} 
\frac{\partial u}{\partial t} - d_u \Delta u = \tilde{k}_1 v - \tilde{k}_{-1} u - \tilde{k}_2 uv \\
\frac{\partial v}{\partial t} - d_v \Delta v = \tilde{k}_{-1} u - \tilde{k}_1 v - \tilde{k}_2 uv
\end{cases}
$$


$u$ is the concentration of ethenol;
$v$ is the concentration of ethanal;
the tautomerization of ethenol and ethanal gives the first-order part of the reaction;
the aldol reaction of ethanal with ethenol, giving the 3-hydroxybutanal, gives the second order part.


This is definitely a satisfying answer.

Nevertheless I’m still interested in other answers, especially with autocatalysis of either $u$ or $v$ (that would decrease the first-order consumption of it) or with autoreactions (that would give the term $-k_i u^2$ or $-k_i v^2$). As a matter of fact, the French Wikipedia claims that an aldol reaction of ethanal with itself in presence of a base can also produce 3-hydroxybutanal, so that the same example would also give a term $-k v^2$? But maybe these two aldol reactions correspond in fact to the same elementary mechanism?