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.
If you peel this dry skin, the healing process starts all over again with dry skin on your lip. Dehydration is the commonest cause of dry cracked lips. Drink plenty of water and see. The other possibility is iron and Vit B deficiency. Along with multivitamin B (not a multivitamin which also has other vitamins) take an iron preparation and see.