Stationary phase: Broadly speaking the growth and death of the cells are equal in this phase. The growth can cease due to limitations in the nutrition, toxic compounds etc.
In the stationary phase the growth stops due to changes in the medium. Either an essential nutrient has run out or a toxic product has formed. Very often these products cause pH-changes. For aerobic cultures the aeration is very critical. Low aeration can cause limitation in dissolved oxygen when the culture reaches dense values.