Population dynamics is often modeled using ODE. For example one common model is logistic growth model:
$$\frac {dx}{dt} = kx\left(1-\frac{x}{C}\right)$$ where $x$ is population size, $k$ is rate constant for growth, $C$ is carrying capacity.
But population is a discrete variable. It is not continuous. It always takes whole numbers. You can have a population of 3000 fishes, but not 3001.2 fishes.
Then how can one use population as a dependent variable in a differential equation?
Integration of the ODE, given above, will give me a function to calculate size of population, $x_t$, at time $t$, when size of the population at $t = 0$, was $x_0$. We can specify a whole number for $x_0$. But, $x_t$ can be a real number with fraction. But a population size is always a whole number.
How does one tackle this anomaly?
There is no anomaly. If the assumption that "the equation so-and-so describes the population size at time $t$" seems to be arbitrary for you, you should start with underlying stochastic system. In particular, assume that $X$ is a discrete random variable that takes only integer values. Specify the rules this variable follows. In this way you get a stochastic process. However, if this process is nonlinear it is usually very difficult to analyze. There is a significant body of work that allows to deduce a system of ordinary differential equations to approximate some characteristics of this process, in particular, the mean of $X$ (not always though). Hence, very often you should actually replace the words "let $N(t)$ be the population size at time $t$" with "let $N(t)$ be the expectation of the population size at time $t$" In this way there is no internal conflict.