A　Leslie–Gower　predator–prey　model　with　square　root　response　function　is　considered.　We　prove　that　the　origin　is　unstable　using　blow-up　method,　and　show　that　if　the　positive　equilibrium　is　locally　stable,　then　it　is　globally　asymptotically　stable　based　on　the　divergency　criterion.　Also,　if　the　positive　equilibrium　is　unstable,　the　system　admits　a　unique　stable　limit　cycle.　A　conclusion　is　finally　given　to　show　the　influence　of　the　square　root　response　function　on　the　dynamic　behaviors　of　the　system.　©　2022　Elsevier　Ltd