In a habitat with limited resources, a population exhibits...

The Verhulst-Pearl logistic growth is described by the equation $\dfrac{dN}{dt} = rN\left(1 - \dfrac{N}{K}\right)$, where $N$ is the population density, $r$ is the intrinsic rate of natural increase, and $K$ is the carrying capacity.

The growth curve reaches an asymptote when the population size stops increasing, meaning the growth rate $\dfrac{dN}{dt}$ becomes zero.

Setting $\dfrac{dN}{dt} = 0$ gives $1 - \dfrac{N}{K} = 0$, which implies $N = K$.

Thus, the asymptote is reached when the population density ($N$) equals the carrying capacity ($K$).

Answer: the population density ($N$) equals the carrying capacity ($K$)