A　discrete　allelopathic　phytoplankton　model　with　infinite　delays　and　feedback　controls　is　studied　in　this　paper.　By　applying　the　discrete　comparison　theorem,　a　set　of　sufficient　conditions　which　guarantees　the　permanence　of　the　system　is　obtained.　Also,　by　constructing　some　suitable　discrete　Lyapunov　functionals,　some　sufficient　conditions　for　the　extinction　of　the　system　are　obtained.　Our　results　extend　and　supplement　some　known　results　and　show　that　the　feedback　controls　and　toxic　substances　play　a　crucial　role　on　the　permanence　and　extinction　of　the　system.　©　2020　Liang　Zhao　et　al.