There is a Summer / Autumn / Winter / Spring pattern in the Temperate zone, also affecting the Desert and the Grassland climatic zones and, a Wet / Dry pattern in the tropical north which includes the Equatorial, Tropical and sub-tropical zones.
Most likely, your lips are suffering from excessive dryness, or chapping. Actually, picking at them and peeling off the skin can worsen the situation, as this can make the dryness worse. It would be better to try applying a good lip balm or lip moisturizing lotion. This should help you break the cycle of dryness and peeling, but only if you can refrain from picking at the skin in the meantime.
That's right, the greasy crap you are slathering on your lips in an effort to keep them from flaking off is actually the very thing that's drying them out. What's worse, you tend to lick your lips a lot when you use lip balm, which also contributes to the dryness.