  The Hilbert number $H(n)$ is defined as the maximum number of limit cycles of
a planar autonomous system of ordinary differential equations (ODEs) with
right-hand sides containing polynomials of degree at most $n \in {\mathbb N}$.
The dynamics of chemical reaction systems with two chemical species can be
(under mass-action kinetics) described by such planar autonomous ODEs, where
$n$ is equal to the maximum order of the chemical reactions in the system.
Generalizations of the Hilbert number $H(n)$ to three different classes of
chemical reaction networks are investigated: (i) chemical systems with
reactions up to the $n$-th order; (ii) systems with up to $n$-molecular
chemical reactions; and (iii) weakly reversible chemical reaction networks. In
each case (i), (ii) and (iii), the question on the number of limit cycles is
considered. Lower bounds on the generalized Hilbert numbers are provided for
both algebraic and non-algebraic limit cycles. Furthermore, given a general
algebraic curve $h(x,y)=0$ of degree $n_h \in {\mathbb N}$ and containing one
or more ovals in the positive quadrant, a chemical system is constructed which
has the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the
dynamics of the constructed chemical system contain polynomials of degree at
most $n=2\,n_h+1.$ Considering $n_h \ge 4,$ the algebraic curve $h(x,y)=0$ can
contain multiple closed components with the maximum number of ovals given by
Harnack's curve theorem as $1+(n_h-1)(n_h-2)/2$, which is equal to 4 for
$n_h=4.$ Algebraic curve $h(x,y)=0$ with $n_h=4$ and the maximum number of four
ovals is used to construct a chemical system which has four stable algebraic
limit cycles.
