SETTING
In this paper the authors study the effects of tourism in a national park in Austria on a endangered species of bird. They model the species size (number of breeding couples) with a continuous function of time $S = S(t)$ satisfying the following logistic ODE (with time dependent carrying capacity) $$ \dot{S}(t) = \gamma S(t)\left(1 - \frac{S(t)}{\beta H(t)}\right), \qquad S(0) = S_0 > 0 $$ where $\gamma, \beta > 0$ are constants and $H = H(t)$, which represent the habitat extent, satisfies the following logistic ODE $$ \dot{H} = \alpha H(t) \left( 1 - \frac{H(t)}{K} \right), \qquad H(0) = H_0 >0, $$ with $K >0$ constant.
QUESTION: How reasonable is to model the dynamic of the size of a population with a continuous function $S$? For a very large population, I wouldn't see any problem, but in the specific case analyzed in the paper, $S_0 = 12$ and they argue in the appendix A.2 that the partridge carrying capacity is of 16 breeding couples. Therefore the actual discrete function that the authors want to model has a very limited range and I wonder, how legitimate it is to use an ODE.
A more realistic model would be to use a discrete state dynamic system to model the population dynamic (actually a stochastic process would probably be even more suitable) and I wonder how legitimate is to approximate such a system with an ODE.
Any suggestion/idea is very welcome! Even heuristic arguments.
PS: I am definitely more familiar with ODE than with discrete state systems.