Species on earth usually follow a similar growth pattern which scientists have been able to observe and quantify. In general, a species will experience a higher growth rate when at lower population densities until it reaches a plateau at its carrying capacity. The carrying capacity is the number of individuals an environment can sustain indefinitely. The most basic way to describe this model is through a logistic growth curve. It begins exponential and levels out. However, this is not the full story in real life. What tends to happen is a population will overshoot the carrying capacity when times are good and population growth rates are positive. When this happens the environment imply cannot sustain this population and the species feels the impact via two factors. Decreased birth rates from less food and possible increased emigration to other suitable ranges. The growth rate then decreases and the population may undershoot the carrying capacity at which point the cycle may repeat. Populations that over-and-under shoot by very little can be described as dampened oscillations.
An unfortunate problem some species face is the allee effect of population growth. The trend most observed in the wild is when population density is low for a certain area, the growth rate is high because the environment can sustain more individuals than currently present. However, consider a small population that is very dispersed and therefore partially isolated from each other. When it comes time to breed they may not be able to find a mate in time and therefore not produce any young. This is the allee effect. Low population densities mixed with isolation produces a decreased growth rate. This effect can drive many species to extinction fairly quick as it's hard to recover when a population size becomes so small so fast.