Live Virtual Machine (VM) migration among geographically distributed edge clouds is an important strategy for providing low latency and reliable services for mobile end users. VM migration among edge clouds is more challenging than that in cloud computing, because the network bandwidth among edge clouds is more constrained than the cloud data center networks. In this paper, we study the bandwidth allocation among multiple concurrent live VM migrations in edge clouds. This problem is novel in that existing works aim to reduce the migration time for a single live VM migration among the edge clouds, and also ignores the QoS requirement for the service running on the VM in migration. However, our problem considers multiple VM migration tasks, and aims to maximize the average QoS while meeting the migration time constraint for each VM migration task. We formulate the problem as a Non-Linear Programming (NLP) problem which is also shown to be NP-Hard. We develop a new method to solve this problem. In our approach, we first transfer the problem into a Linear Programming (LP) problem by reducing the solution space. Taking the output from the LP solver as an initial solution, we then develop a heuristic to adjust it in order to find a better one to the original NLP problem. Finally, we design a set of evolutionary algorithms to select the optimal initial solution from the LP solver. Extensive simulations show that our proposed method can achieve good QoS and also has a fast convergence speed.