add some suggestions

Author: vyudu

docs/src/api/core_api.md

@@ -88,7 +88,7 @@ ReactionSystem
 ## [Options for the `@reaction_network` DSL](@id api_dsl_options)
 We have [previously described](@ref dsl_advanced_options) how options permits the user to supply non-reaction information to [`ReactionSystem`](@ref) created through the DSL. Here follows a list
 of all options currently available.
-- [`parameters`]:(@ref dsl_advanced_options_declaring_species_and_parameters) Allows the designation of a set of symbols as system parameter.
+- [`parameters`](@ref dsl_advanced_options_declaring_species_and_parameters): Allows the designation of a set of symbols as system parameters.
 - [`species`](@ref dsl_advanced_options_declaring_species_and_parameters): Allows the designation of a set of symbols as system species.
 - [`variables`](@ref dsl_advanced_options_declaring_species_and_parameters): Allows the designation of a set of symbols as system non-species variables.
 - [`ivs`](@ref dsl_advanced_options_ivs): Allows the designation of a set of symbols as system independent variables.

docs/src/network_analysis/crn_theory.md

@@ -1,6 +1,3 @@
-> Note, currently API functions for network analysis and conservation law analysis
-> do not work with constant species (currently only generated by SBMLToolkit).
-
 # [Chemical Reaction Network Theory](@id network_analysis_structural_aspects)
 
 The systems of ODEs or stochastic chemical kinetics models that arise from chemical
@@ -15,8 +12,8 @@ section
 [Network Analysis and Representations](@ref api_network_analysis).
 
 Broadly, results from chemical reaction network theory relate a purely
-graph-structural property (e.g. deficiency) to dynamical properties of the reaction system
-(e.g. complex balance). The relevant graph here is the one corresponding to the [reaction complex representation](@ref network_analysis_reaction_complexes)
+graph-structural property (e.g. [deficiency](@ref network_analysis_structural_aspects_deficiency)) to dynamical properties of the reaction system
+(e.g. [complex balance](@ref network_analysis_complex_and_detailed_balance)). The relevant graph here is the one corresponding to the [reaction complex representation](@ref network_analysis_reaction_complexes)
 of the network, where the nodes represent the reaction complexes and the edges represent reactions.
 Let us illustrate some of the types of network properties that 
 Catalyst can determine.
@@ -75,13 +72,13 @@ and,
 
 ## [Deficiency of the network](@id network_analysis_structural_aspects_deficiency)
 A famous theorem in Chemical Reaction Network Theory, the Deficiency Zero
-Theorem [^1], allows us to use knowledge of the net stoichiometry matrix and the
+Theorem[^1], allows us to use knowledge of the net stoichiometry matrix and the
 linkage classes of a *mass action* [RRE ODE system](@ref network_analysis_matrix_vector_representation) to draw conclusions about the
 system's possible steady states. In this section we'll see how Catalyst can
 calculate a network's deficiency.
 
 The rank, ``r``, of a reaction network is defined as the dimension of the
-subspace spanned by the net stoichiometry vectors of the reaction-network [^1],
+subspace spanned by the net stoichiometry vectors of the reaction-network[^1],
 i.e. the span of the columns of the net stoichiometry matrix `N`. It corresponds
 to the number of independent species in a chemical reaction network. That is, if
 we calculate the linear conservation laws of a network, and use them to
@@ -275,21 +272,36 @@ Very analogous to the deficiency zero theorem is the deficiency one theorem. The
 2. The sum of the linkage class deficiencies is the total deficiency of the network, and
 3. Each linkage class has at most one terminal linkage class, which is a linkage class that is 1) strongly connected, and 2) has no outgoing reactions.
 
-For the set of reactions $A \to B, B \to A, B \to C$, $\{A, B, C\}$ is a linkage class, and $\{A, B\}$ is a strong linkage class (since A is reachable from B and vice versa). However, $\{A, B\}$ is not a terminal linkage class, because the reaction $B \to C$ goes to a complex outside the linkage class. 
+For the set of reactions $A \to B, B \to A, B \to C$, $\{A, B, C\}$ is a linkage class,
+and $\{A, B\}$ is a strong linkage class (since A is reachable from B and vice versa).
+However, $\{A, B\}$ is not a terminal linkage class, because the reaction $B \to C$ goes
+to a complex outside the linkage class. 
 
-If these conditions are met, then the network will have at most one steady state in each stoichiometric compatibility class for any choice of rate constants and parameters.
-Unlike the deficiency zero theorem, networks obeying the deficiency one theorem are not guaranteed to have stable solutions.
+If these conditions are met, then the network will have at most one steady state in each 
+stoichiometric compatibility class for any choice of rate constants and parameters. Unlike 
+the deficiency zero theorem, networks obeying the deficiency one theorem are not guaranteed 
+to have stable solutions.
 
 ```@docs; canonical=false
 satisfiesdeficiencyone
 ```
 
 ## [Complex and Detailed Balance](@id network_analysis_complex_and_detailed_balance)
-A reaction network's steady state is **complex-balanced** if the total production of each *complex* is zero at the steady state. A reaction network's steady state is **detailed balanced** if every reaction is balanced by its reverse reaction at the steady-state (this corresponds to the usual notion of chemical equilibrium; note that this requires every reaction be reversible). 
-
-Note that detailed balance at a given steady state implies complex balance for that steady state, i.e. detailed balance is a stronger property than complex balance.
-
-Remarkably, having just one positive steady state that is complex (detailed) balance implies that complex (detailed) balance obtains at *every* positive steady state, so we say that a network is complex (detailed) balanced if any one of its steady states are complex (detailed) balanced. Additionally, there will be exactly one steady state in every positive stoichiometric compatibility class, and this steady state is asymptotically stable. (For proofs of these results, please consult Martin Feinberg's *Foundations of Chemical Reaction Network Theory*[^1]). So knowing that a network is complex balanced is really quite powerful.
+A reaction network's steady state is **complex-balanced** if the total production of each 
+*complex* is zero at the steady state. A reaction network's steady state is **detailed balanced** 
+if every reaction is balanced by its reverse reaction at the steady-state (this corresponds to 
+the usual notion of chemical equilibrium; note that this requires every reaction be reversible). 
+
+Note that detailed balance at a given steady state implies complex balance for that steady state,
+i.e. detailed balance is a stronger property than complex balance.
+
+Remarkably, having just one positive steady state that is complex (detailed) balance implies that 
+complex (detailed) balance obtains at *every* positive steady state, so we say that a network 
+is complex (detailed) balanced if any one of its steady states are complex (detailed) balanced. 
+Additionally, there will be exactly one steady state in every positive stoichiometric 
+compatibility class, and this steady state is asymptotically stable. (For proofs of these results,
+please consult Martin Feinberg's *Foundations of Chemical Reaction Network Theory*[^1]). So 
+knowing that a network is complex balanced is really quite powerful.
 
 Let's check whether the deficiency 0 reaction network that we defined above is complex balanced by providing a set of rates:
 ```@example s1
@@ -302,7 +314,12 @@ We can do a similar check for detailed balance.
 isdetailedbalanced(rn, rates)
 ```
 
-The reason that the deficiency zero theorem puts such strong restrictions on the steady state properties of the reaction network is because it implies that the reaction network will be complex balanced for any set of rate constants and parameters. The fact that this holds from a purely structural property of the graph, regardless of kinetics, is what makes it so useful. But in some cases it might be desirable to check complex balance on its own, as for higher deficiency networks.
+The reason that the deficiency zero theorem puts such strong restrictions on the steady state 
+properties of the reaction network is because it implies that the reaction network will be 
+complex balanced for any set of rate constants and parameters. The fact that this holds 
+from a purely structural property of the graph, regardless of kinetics, is what makes it so 
+useful. But in some cases it might be desirable to check complex balance on its own, as for 
+higher deficiency networks.
 
 ```@docs; canonical=false
 iscomplexbalanced

docs/src/network_analysis/odes.md

@@ -39,7 +39,7 @@ g = Graph(repressilator)
 ![Repressilator solution](../assets/repressilator.svg)
 
 We also showed in the [Introduction to Catalyst](@ref introduction_to_catalyst) tutorial that
-the reaction rate equation ODE model for the repressilator is
+the reaction rate equation (RRE) ODE model for the repressilator is
 ```math
 \begin{aligned}
 \frac{dm_1(t)}{dt} =& \frac{\alpha K^{n}}{K^{n} + \left( {P_3}\left( t \right) \right)^{n}} - \delta {m_1}\left( t \right) + \gamma \\
@@ -109,8 +109,8 @@ facilitate analysis of a variety of reaction network properties. Consider a simp
 reaction system like
 ```@example s1
 rn = @reaction_network begin
- k, 2*A + 3*B --> A + 2*C + D
- b, C + D --> 2*A + 3*B
+ k, 2A + 3B --> A + 2C + D
+ b, C + D --> 2A + 3B
 end
 ```
 We can think of the first reaction as converting the *reaction complex*,
@@ -257,7 +257,9 @@ isequal(A_k, B * K)
 ```
 
 Note that we have used `isequal` instead of `==` here because `laplacianmat`
-returns a `Matrix{Num}`, since some of its entries are symbolic rate constants.
+returns a `Matrix{Num}`, since some of its entries are symbolic rate constants
+(symbolic variables and `Num`s cannot be compared using `==`, since `a == b`
+is interpreted as a symbolic expression).
 
 In sum, we have that
 ```math
@@ -286,7 +288,7 @@ fluxmat(rn, parammap)
 laplacianmat(rn, parammap)
 ```
 
-# Symbolic ODE functions
+## Symbolic ODE functions
 In some cases it might be useful to generate the function defining the system of ODEs as a symbolic Julia function that can be used for further analysis. This can be done using Symbolics' [`build_function`](https://docs.sciml.ai/Symbolics/stable/getting_started/#Building-Functions), which takes a symbolic expression and a set of desired arguments, and converts it into a Julia function taking those arguments.
 
 Let's build the full symbolic function corresponding to our ODE system. `build_function` will return two expressions, one for a function that outputs a new vector for the result, and one for a function that modifies the input in-place. Either expression can then be evaluated to return a Julia function.
@@ -327,7 +329,7 @@ f(c, ks)
 
 Note also that `f` can take any vector with the right dimension (i.e. the number of species), not just a vector of `Number`, so it can be used to build, e.g. a vector of polynomials in Nemo for commutative algebraic methods.
 
-# Properties of matrix null spaces
+## Properties of matrix null spaces
 The null spaces of the matrices discussed in this section often have special meaning. Below we will discuss some of these properties.
 
 Recall that we may write the net stoichiometry matrix ``N = YB``.
@@ -336,7 +338,7 @@ Recall that we may write the net stoichiometry matrix ``N = YB``.
 
 [Complex balance](@ref network_analysis_complex_and_detailed_balance) can be compactly formulated as the following: a set of steady state reaction fluxes is complex-balanced if it is in the nullspace of the incidence matrix ``B``.
 
-# API Section for matrices and vectors
+## API Section for matrices and vectors
 We have that: 
 - ``N`` is the `netstoichmat`
 - ``Z`` is the `complexstoichmat`