For the given Petri Net, the number of tokens for P_1,P_2 and P_3 is represented by m_1,m_2 and m_3 respectively. Based on the definition of boundedness of a Petri Net by Popova-Zeugmann (2013), the given Petri Net is bounded because m_1 ≤ 2,m_2 ≤ 2 and m_3 ≤ 2. In other words, there is a natural number (2 in this case) such that the number of tokens in P_1,P_2 and P_3 never exceeds the number.
I am also providing an example of an unbounded Petri Net and an example of a bounded Petri Net. Regardless of the initial marking, the Petri Net in Figure 1 is not bounded (or it is unbounded) because m_0 → ∞ when T_0 keeps firing.
For the given initial marking, the Petri Net in Figure 2 is bounded because m_0 ≤ 1 and m_1 ≤ 2. In general, the Petri Net in Figure 2 is bounded: let the initial value of m_0 = q_0 and the initial value of m_1 = q_1; m_0 ≤ q_0 and m_1 ≤ 2q_0+q_1
Popova-Zeugmann, L. (2013). Chapter 2 The Classic Petri Net [Electronic version]. Time and Petri Nets. Berlin; Heidelerg: Springer-Verlag.