In Markovian environment, the nature of entanglement dynamics and entanglement sudden death of coupling qubits in the dissipative environment are investigated in detail using concurrence function. The results show that the dynamic evolution of entanglement is closely related with temperature of the system, the initial entanglement and the initial quantum state. Under normal circumstances, entanglement sudden death always occurs due to environmental effect. A specific area is found through numerical simulation where the decay of entanglement is very slow. Under the zero-temperature approximation, the specific conditions of entanglement sudden death and the equation for calculating the time of entanglement sudden death are both given.