Endowing a chatbot with a personality is essential to deliver more realistic conversations. Various persona-based dialogue models have been proposed to generate personalized and diverse responses by utilizing predefined persona information. However, generating personalized responses is still a challenging task since the leverage of predefined persona information is often insufficient. To alleviate this problem, we propose a novel Persona Enhanced Dual Alternating Learning Network (PEDNet) aiming at producing more personalized responses in various open-domain conversation scenarios. PEDNet consists of a Context-Dominate Network (CDNet) and a Persona-Dominate Network (PDNet), which are built upon a common encoder-decoder backbone. CDNet learns to select a proper persona as well as ensure the contextual relevance of the predicted response, while PDNet learns to enhance the utilization of persona information when generating the response by weakening the disturbance of specific content in the conversation context. CDNet and PDNet are trained alternately using a multi-task training approach to equip PEDNet with the both capabilities they have learned. Both automatic and human evaluations on a newly released dialogue dataset Persona-chat demonstrate that our method could deliver more personalized responses than baseline methods.