The "standard" model of a density-regulated population is the logistic growth model, also known as the Verhulst equation:
$\frac{dp}{dt} = rp \left(1-\frac pK\right) = rp - \frac rK p^2,$
where $r$ is the exponential growth rate of the population at low densities, and $K$ is the effective carrying capacity (i.e. the maximum stable population size) of the population. This differential equation has two fixed points, $p = 0$ and $p = K$, and its general solutions are logistic functions of the form:
$p(t) = \frac{K p_0 e^{rt}}{K + p_0 (e^{rt} - 1)}.$
However, the Verhulst equation is not a mechanistic population model, as it does not contain explicit birth and death terms, and there are several different ways to construct mechanistic models which are equivalent to it.
For example, we might assume that the per capita birth rate is fixed but the death rate contains a density-dependent term (e.g. due to conflicts, starvation or diseases), giving a model like:
$\frac{dp}{dt} = bp - dp - cp^2,$
where $b$ and $d$ are the baseline per capita birth and death rates and $cp$ is the density-dependent death rate. Alternatively, we might treat the per capita death rate as fixed but the birth rate as proportional to the availability of some resource (e.g. space to grow for plants) of which a constant amount is taken up by each individual, giving us a model like:
$\frac{dp}{dt} = bp\left(1-\frac pN\right) - dp,$
where, again, $b$ and $d$ are the baseline birth and death rates at small population sizes, and $N$ is the absolute maximum population size (i.e. the size at which the birth rate drops to zero).
Both of these models can be written in the form of the Verhulst equation: in both cases, $r = b-d$, but in the first case $K = r/c = (b-d)/c$, while in the second $K = rN/b = (1-d/b)N$. Thus, the two models respond differently to changes in the parameter values: in the first model, for instance, increasing the birth rate can increase the equilibrium population size $K$ indefinitely, where as in the second model the population size can never surpass $N$ no matter how large $b$ is.