This　paper　analyzes　an　allelopathic　phytoplankton　competition　model,　which　was　proposed　by　Bandyopadhyay　[Dynamical　analysis　of　a　allelopathic　phytoplankton　model,　J　Biol　Syst　14(02):205-217,　2006].　Our　study　refines　the　previous　results　and　finds　at　most　three　positive　equilibria　for　the　system.　The　existence　conditions　of　all　positive　equilibria　and　the　corresponding　stability　cases　are　given　in　the　paper.　Interesting　dynamical　phenomena　such　as　bistability,　saddle-node　bifurcation,　and　cusp　bifurcation　are　found.　It　is　shown　that　the　rate　of　toxin　releases　heavily　influences　the　positive　equilibria　of　the　system　under　certain　conditions.　Numerical　simulations　verify　the　feasibility　of　the　theoretical　results.　　©　2023　World　Scientific　Publishing　Company.