Fate is what must happen, while destiny is how people react to fate. For example, Prim's fate is to die a premature death and Katniss is destined to postpone that death and then avenge it. Prim escapes fate when Katniss volunteers as tribute to take her place in the 74th Hunger Games. The chances of Prim winning those Hunger Games would have been slim to none. In the off chance that her and Peeta could have worked together to survive like Katniss and him did, then would not have likely survived the Quarter Quell either. Though Prim was afforded the opportunity to live longer, fate displayed its inevitable power when she died along with many other children due to the bombing in Mockingjay.
According to psychiatrist Carl Jung, archetypes are models of people, behaviors or personalities organized by the collective unconscious. The cat is an archetype for femininity. Katniss struggles with her femininity and she does not get along with Prim's cat, Buttercup. He disgust with the cat and desire for it to be out of her life is a metaphor for her wish to dismiss all the complications associated with being female, including vulnerability. Buttercup parallels Katniss in many ways- watching over and protecting Prim, disliking the Victor Village, and mourning the loss of Prim.
Another archetype prevalent in The Hunger Games is the Phoenix. This "bird on fire" goes through a constant cycle of death and rebirth. They die when it is their time and are reborn from the ashes. Katniss is also characterizes as "on fire." She is trapped in a constant cycle of battles and games. Katniss and all her hope die when her home is destroyed, but a new life for her and Peeta is born from the ashes of District 12.