According to Jain tradition, the successor of Rishabhanatha (also known as Adinath), who is regarded as the first Tirthankara of the present age (Avasarpini), was his son Bharata, who became the first Chakravartin (universal monarch) and king of India (Bharatavarsha is named after him). In the spiritual lineage of Tirthankaras, however, the next (second) Tirthankara after Rishabhanatha is Ajitanatha.

So, in summary:

• Worldly successor (kingdom): Bharata Chakravartin (Rishabhanatha’s son)
• Spiritual successor (Tirthankara): Ajitanatha (2nd Tirthankara)