The entanglement dynamics of two atoms in the one-photon and two-photon Jaynes-Cummings models are studied by employing the concurrence and linear entropy. The effect of the initial entanglement degree of two atoms and the initial state of the cavity field on the time evolution of concurrence is analyzed. The results show that the phenomenon of sudden death of entanglement between two atoms happens when the field is initially in the state |11〉. The length of the death time interval is dependent on the initial entanglement degree of two atoms. Moreover, it is found that the atom and the field are entangled in the overall time evolution process.