How to model species interactions on population size dynamic?
You can write these equations in the form of ODEs but it is not really essential.

This is my analysis (just the model formulation and not literature):

The effects of interactor species may or may not be dependent on the population of the species that is being analysed (recipient). Now you have to model based on how you would realistically expect the interactions to have an effect on a species. A simple assumption would be that the interaction improves or reduces fitness (growth rate), additively. So you can model the effects as:

$$N_{t+1} = (r_0 + \beta_a + \beta_b + \beta_c) N_t$$

where $\beta_x$ denotes the effect of species $x$ which in turn you can model as dependent ($N_tN_{x_t}\alpha_x$) or independent ($N_{x_t}\alpha_x$) of the recipient population. Now, you can also impose a condition that the growth rate cannot increase beyond a certain value and so you can model it as a saturating function. A logistic or a Michaelis-Menten like function would work. For e.g:

$$\begin{align} r &amp;= \frac{r'}{K+r'}\\[1em]
 r' &amp;= r_0 +  \beta_a + \beta_b + \beta_c\end{align}$$

You can likewise adapt model using the logistic equation.