Final answer:
In order for a predator-prey relationship to be stable over the long-term, the number of prey must be directly related to the number of predators and a cyclic fluctuation in population sizes must occur.
Step-by-step explanation:
A stable predator-prey relationship can be maintained over the long-term by several factors. One important factor is that the number of prey is directly related to the number of predators so that the two populations remain at the same ratio even though their total population numbers fluctuate. This ensures that neither species goes extinct. Another factor is that increasing prey numbers trigger decreases in predator numbers, which eventually causes a decrease in prey numbers as predators become too sparse, and then the cycle repeats. This cyclic fluctuation in population sizes helps maintain the stability of the relationship.